From 0621d6ec8ec66212a36c003f14d5516fb37d5a36 Mon Sep 17 00:00:00 2001 From: Alexandru Fikl Date: Tue, 2 Sep 2025 11:31:44 +0300 Subject: [PATCH 01/15] feat: add initial JOSS paper --- doc/paper/.gitignore | 4 + doc/paper/Makefile | 77 ++++ doc/paper/images/Makefile | 41 ++ doc/paper/images/pkdo-2d.png | Bin 0 -> 580339 bytes .../images/quadrature_rule_vr_order_11_2d.png | Bin 0 -> 132147 bytes .../images/quadrature_rule_wv_order_11_2d.png | Bin 0 -> 52554 bytes doc/paper/images/quadrature_rules.png | Bin 0 -> 182401 bytes doc/paper/images/simplices.png | Bin 0 -> 112377 bytes doc/paper/images/simplices.tex | 66 +++ doc/paper/paper.bib | 330 +++++++++++++++ doc/paper/paper.md | 384 ++++++++++++++++++ 11 files changed, 902 insertions(+) create mode 100644 doc/paper/.gitignore create mode 100644 doc/paper/Makefile create mode 100644 doc/paper/images/Makefile create mode 100644 doc/paper/images/pkdo-2d.png create mode 100644 doc/paper/images/quadrature_rule_vr_order_11_2d.png create mode 100644 doc/paper/images/quadrature_rule_wv_order_11_2d.png create mode 100644 doc/paper/images/quadrature_rules.png create mode 100644 doc/paper/images/simplices.png create mode 100644 doc/paper/images/simplices.tex create mode 100644 doc/paper/paper.bib create mode 100644 doc/paper/paper.md diff --git a/doc/paper/.gitignore b/doc/paper/.gitignore new file mode 100644 index 0000000..29cc143 --- /dev/null +++ b/doc/paper/.gitignore @@ -0,0 +1,4 @@ +.inara +joss +*.pdf +*.tex diff --git a/doc/paper/Makefile b/doc/paper/Makefile new file mode 100644 index 0000000..c5b3ecf --- /dev/null +++ b/doc/paper/Makefile @@ -0,0 +1,77 @@ +OUTDIR?=latex.out +TEXMK?=latexmk -verbose -pdflua -output-directory=$(OUTDIR) + +INARA_PATH := .inara +TARGET_FOLDER := $$(pwd)/publishing-artifacts + +JOURNAL := joss +INARA_DATA_PATH := $(INARA_PATH)/data +OPENJOURNALS_PATH := $(INARA_PATH)/resources +ARTICLE_INFO_FILE = $(OPENJOURNALS_PATH)/default-article-info.yaml + +all: paper +.PHONY: all + +help: ## Show this help + @echo -e "\nSpecify a command. The choices are:\n" + @grep -E '^[0-9a-zA-Z_-]+:.*?## .*$$' $(MAKEFILE_LIST) | awk 'BEGIN {FS = ":.*?## "}; {printf " \033[0;36m%-12s\033[m %s\n", $$1, $$2}' + @echo "" +.PHONY: help + +%.pdf: %.tex + PYTHONWARNINGS=ignore $(TEXMK) $< + @cp $(OUTDIR)/$@ . + +paper: paper.md paper.bib ## Compile paper to PDF using JOSS tools + mkdir -p $(TARGET_FOLDER) + make -C $(INARA_PATH) -B resources/footer.csl + INARA_ARTIFACTS_PATH=$(TARGET_FOLDER)/ pandoc \ + --data-dir=$(INARA_DATA_PATH) \ + --defaults=$(INARA_DATA_PATH)/defaults/shared.yaml \ + --defaults=$(INARA_DATA_PATH)/defaults/pdf.yaml \ + --defaults=$(OPENJOURNALS_PATH)/$(JOURNAL)/defaults.yaml \ + --resource-path=$$(pwd):$(OPENJOURNALS_PATH) \ + --metadata=article-info-file=$(ARTICLE_INFO_FILE) \ + --variable=$(JOURNAL) \ + --output=$@.tex \ + $< + make paper.pdf +.PHONY: paper + +inara: ## Download JOSS tools (inara) for compilation + @if [ ! -d "$(INARA_PATH)" ]; then \ + git clone git@github.com:openjournals/inara.git $(INARA_PATH); \ + else \ + cd $(INARA_PATH) && git pull origin main; \ + fi + mkdir -p joss + cp $(INARA_PATH)/resources/joss/logo.png joss/logo.png +.PHONY: inara + +tidy: paper.bib ## Clean up bibliography (with bibtex-tidy) + @bibtex-tidy \ + --modify \ + --sort \ + --sort-fields \ + --drop-all-caps \ + --merge last \ + --numeric \ + --strip-enclosing-braces \ + --trailing-commas \ + --blank-lines \ + --remove-empty-fields \ + --remove-dupe-fields \ + --wrap \ + $< +.PHONY: tidy + +clean: ## Remove temporary files generated by JOSS tools + rm -rf $(TARGET_FOLDER) + rm -rf $(OUTDIR) +.PHONY: clean + +purge: clean + rm -rf paper.pdf + rm -rf .inara +.PHONY: purge + diff --git a/doc/paper/images/Makefile b/doc/paper/images/Makefile new file mode 100644 index 0000000..93a31eb --- /dev/null +++ b/doc/paper/images/Makefile @@ -0,0 +1,41 @@ +OUTDIR?=latex.out +TEXMK?=latexmk -verbose -pdf -output-directory=$(OUTDIR) +PYTHON?=python -X dev + +all: figures + +help: ## Show this help + @echo -e "\nSpecify a command. The choices are:\n" + @grep -E '^[0-9a-zA-Z_-]+:.*?## .*$$' $(MAKEFILE_LIST) | awk 'BEGIN {FS = ":.*?## "}; {printf " \033[0;36m%-12s\033[m %s\n", $$1, $$2}' + @echo "" +.PHONY: help + +%.pdf: %.tex + PYTHONWARNINGS=ignore $(TEXMK) $< + @cp $(OUTDIR)/$@ . + +%.png: %.pdf + magick \ + -density 600 \ + $< \ + -quality 100 \ + -background white \ + -alpha remove \ + $@ + +quadrature_rules.png: + magick \ + quadrature_rule_vr_order_11_2d.png \ + \( -size 900x`identify -format "%h" quadrature_rule_vr_order_11_2d.png` xc:white \) \ + quadrature_rule_wv_order_11_2d.png \ + +append \ + -background white \ + quadrature_rules.png + +figures: simplices quadratures ## Compile all figures + for img in simplices.png pkdo-2d.png quadrature_rules.png; do \ + magick "$${img}" -trim +repage "$${img}"; \ + done + +simplices: simplices.png +quadratures: quadrature_rules.png diff --git a/doc/paper/images/pkdo-2d.png b/doc/paper/images/pkdo-2d.png new file mode 100644 index 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z$cHfFY4+T)oZtfdkEVIsxaUscU_=#x!B_@47Fyo8qgT*g_@(=9sT~RQf4Pv^T%I}dIl7wKJ%C* zTm1T=M>FKK4R*WjchB4ww)pt-lfB4rmC64Vi|J5sHp@JS;^wUF7HDXeavgOvJMx^f zLs`jtBy2<&A|e&cM3U@%6C9By zUN&S@V!ToBD3@W!c}`;bXRAabHz@E9Y;A8wa>|_7oLhz%(H=myf_{TM04rP8Wj{m>X;B%J_oPcwA^TAN< zI|QFRK+=RkunImTvErwQ|9UtOf1jiK&D}eTy&JDUr^%3kC(Z`Cxdggvy863=zYxj? zl~olEsw*fTv^=P;sRI5}J8. + +# Statement of need + +High-order accurate calculus operations on unstructured discretizations are +crucial to many solvers of differential and integral equations, but their uses +can also extend to computer graphics and Computer-Aided Design (CAD). This +functionality is often embedded in an ad-hoc manner in larger codes, leading to +limited scope and a lack of reusability. `modepy` addresses this need by +providing a robust, reusable, generalizable, and composable implementation. +FIAT [@FIAT] is an early software tool aiming to address a similar need. It +offers a comprehensive set of finite elements and basis functions, but largely +focuses on tabulation. FInAT [@FInAT], a somewhat recent refinement, adds a +focus on algebraic expressions for basis functions, exposing details needed for +efficient tensor product evaluation. Both are, however, tightly interwoven into +the FENiCS/Firedrake ecosystems (e.g. by depending on UFL, the "Unified Form +Language" used for the expression of variational forms). `StartUpDG.jl` +[@StartUpDG] provides functionality with some overlap with `modepy`, but with a +focus on the needs of discontinuous Galerkin FEM. `QuadPy` [@QuadPy] also has +some overlap with `modepy`, in that it can provide quadrature rules, and, while +it has comprehensive coverage, it is regrettably no longer open-source, and it +lacks `modepy`'s composability. `minterpy` [@Wicaksono2025], meanwhile, deals +exclusively with polynomial interpolation, with a focus on sparse grids. + +A significant complicating factor in providing a flexible implementation is +that numerical solvers for differential equations (and likely many other +candidate users) must satisfy rigorous cost constraints, as simulation fidelity +is always traded off against computational cost. As a result, such software +commonly adopts cutting-edge high-performance computing (HPC) techniques, such +as GPU computation and distributed-memory approaches. To facilitate separation +of those concerns from the numerical methods, `modepy` adopts a two-pronged +approach. In many cases, it suffices for operations to be represented as data +in matrix or tabular form, so that no actual execution of `modepy` code is +needed in a cost-constrained setting. + +A prominent exception to this is the evaluation of basis functions. To +facilitate this, `modepy` allows its function evaluation to be "traced" (in the +sense of lazy/deferred evaluation), so that an expression graph can be +obtained. This graph is represented with the help of the `pymbolic` [@Pymbolic] +software library. This, in turn, can interoperate with Python ASTs +[@Python_AST], SymPy [@SymPy], SymEngine [@SymEngine], as well as a number of +other pieces of software for symbolic computation. Another case where a purely +data-driven approach falls short is the setting of tensor-product elements. In +this case, for example, a mass matrix $\boldsymbol M$ with entries +$$ +M_{ij}=\int_{[-1,1]^d} \phi_i(\boldsymbol r)\phi_j(\boldsymbol r) \mathrm{d}\boldsymbol r +$$ +permits a Kronecker product factorization +$$ +\boldsymbol M = \boldsymbol M^{1D}\otimes \cdots \otimes \boldsymbol M^{1D}, +\qquad \text{with} \qquad +M^{1D}_{ij} = \int_{[-1,1]} \phi_i(r)\phi_j(r) \mathrm{d}r, +$$ +assuming a suitable index order of nodes $\boldsymbol{r}_i$ and basis functions +$\{\phi_i\}$. In $d$ dimensions, the use of this factorization reduces the +asymptotic complexity of a matrix-vector product $\boldsymbol M \boldsymbol u$ +from $O(N_p^{2d})$ to $O(N_p^{d+1})$, where $N_p$ is the number of degrees of +freedom in one dimension, and $d$ is the number of dimensions. To take +advantage of this factorization, an abstraction such as `modepy`'s must expose +additional information, including numbering of degrees of freedom. In +`modepy`'s case, this is accomplished via reshaping operation that converts +arrays of degrees of freedom back and forth between a flat (one-dimensional) +and a structured ($d$-dimensional) representation. This reshaping is applicable +to any array type (including GPU or other high-performance arrays) as long as +`numpy`-compatible reshaping is supported. + +# Overview + +The high-level concepts available in `modepy` are shapes (i.e the geometry of a +domain), modes (i.e. the basis functions), and nodes (i.e. the degrees of +freedom). These are implemented in a user-extensible fashion using the +`singledispatch` mechanism, with inspiration taken from common idiomatic usage +in Julia [@Bezanson2017]. Given a function space `space` on a `shape`, a set of +nodal degrees of freedom can be obtained via +`edge_clustered_nodes_for_space(space, shape)`, with an implementation selected +based on the type of `space`. While the example shows that multiple dispatch +could be beneficial, we have decided to forgo it to avoid potential soundness +issues [@Julia2018]. + +## Shapes + +The geometry of a reference element is described in `modepy` by the `Shape` +class. Built-in support exists for `Simplex` and `Hypercube` geometries, +encompassing the commonly used interval, triangle, tetrahedron, quadrilateral, +and hexahedral shapes (see \autoref{FigureSimplices}). `TensorProductShape` can +be used to compose additional shapes, such as prisms, that are useful in +specific applications and sometimes generated by meshing software (such as +`gmsh` [@Geuzaine2009]). An example that constructs a prism and performs some +standard operations on it is given in the next section. + +![Domains corresponding to the one-, two-, and three-dimensional simplices.](images/simplices.png){#FigureSimplices width="80%"} + +To support a wide array of applications, shapes can be queried and manipulated +in several ways. The `faces_for_shape` function can be used to return a list of +faces, which are shapes themselves, that allow access to domain normals and +mappings back to the reference element. + +## Modes and Spaces + +To perform calculus operations, each reference element can be equipped with +(finite-dimensional) function spaces described by the `FunctionSpace` class. +These represent a finite-dimensional space functions $\phi_i: D \to +\mathbb{R}$, where $D$ is the reference element domain, but make no specific +choice of basis. The function spaces mirror the shape definitions and are +intended to work in tandem. In fact, the `space_for_shape` function provides a +default choice of space for each shape. Predefined choices include the `PN` +space, containing polynomials of total degree at most `order`, and the `QN` +space, containing polynomials of maximum degree at most `order`. As with +shapes, these spaces can also be combined using `TensorProductSpace` to +describe functional spaces for other domains or in higher dimensions. + +For calculations, each space can be equipped with a default basis using the +`basis_for_space` function. `modepy` provides two special types of basis +functions: orthonormal (accessed through `orthonormal_basis_for_space`) and +monomial (accessed through `monomial_basis_for_space`). The basis is provided +in the form of a `Basis` class, which allows access to the basis functions, +their derivatives (per axis) and opaque `mode_ids`. As noted before, the basis +functions and their derivatives can be evaluated directly or can be traced +using the `pymbolic` library to aid in code generation. + +![PKDO basis functions for the triangle.](images/pkdo-2d.png){#FigurePKDO width="80%"} + +By default, the `Simplex` shape is equipped with a `PN` functional space. This +function space uses the Proriol-Koornwinder-Dubiner-Owens (PKDO) basis (see +\autoref{FigurePKDO}) as defined in [@Dubiner1991] for dimensions up to 3. The +`Hypercube` is equipped with the `QN` functional space that uses a tensor +product of standard Legendre polynomials. Additionally, general Jacobi +polynomials are available and can be used as a basis for variants of `QN` with +different weights, depending on the application. + +## Nodes + +A final component in an FEM discretization (or 'Ciarlet triple' [@Brenner2007, +Section 3.1]) is a set of 'degrees of freedom' ('DOFs'), that is, a vector of +numbers (technically, functionals) that uniquely define a certain element in +the span of a basis. The literature considers various types of degrees of +freedom; the two most common are modal DOFs (i.e. basis coefficients) and nodal +DOFs (i.e. function or derivative values at certain points). `modepy` provides +easy access to both, while paying particular attention to their usability in +the high-order setting. Due to Runge's phenomenon [@Trefethen2020], simple +equispaced nodal DOFs rapidly become unusable with increasing polynomial order +due to poor conditioning. As such, `modepy` offers access to many types of +nodes with better properties for each reference element. There also exists an +ability to construct one's own set of nodes given a set of functions in 2D via +a procedure based on eigenvalues of multiplication operators [@Vioreanu2014]. + +On each shape, three default families of nodes are provided: equispaced +(accessed through `equispaced_nodes_for_space`), edge-clustered (accessed +through `edge_clustered_nodes_for_space`) and randomly distributed (accessed +through `random_nodes_for_space`). + +For more generality, `modepy` can directly interoperate with the +`recursivenodes` library described in [@Isaac2020], which offers +well-conditioned nodes on the simplex that obey a 'recursive' property, where +nodes on a face of a higher-dimensional shape are exactly the lower-dimensional +nodes. In addition, it offers many many tunable parameters (e.g. nodes on the +boundary vs. not). On simplices, the "warp-and-blend" nodes [@Warburton2007] +are also available for dimensions three and lower. On the hypercube, standard +tensor product nodes are constructed from one-dimensional +Legendre-Gauss-Lobatto nodes. As for the basis functions, the full family of +Jacobi-Gauss and Jacobi-Gauss-Lobatto quadrature points is available for +specific applications. + +Taken together, the functionality described thus far provides comprehensive +support for interpolation, that is, the evaluation of a polynomial defined by +its nodal DOFs, at any point, and further the ability to obtain any derivative +of the interpolating polynomial. + +## Quadrature + +While nodal degrees of freedom must be *unisolvent* or *interpolatory* (i.e. +uniquely determine a function in a discrete function space), this requirement +does not exist in the setting of quadrature, where the objective is to +accurately approximate the value of an integral of a function on a shape. Using +function values at points $\boldsymbol{x}_i$ ("nodes") and associated weights +$w_i$, quadrature rules are generically expressed as +$$ +\int_{D} f(\boldsymbol x) \mathrm{d}\boldsymbol{x} \approx + \sum_{k = 0}^N f(\boldsymbol{x}_k) w_k. +$$ + +Besides the standard building blocks consisting of domains, basis functions and +node sets, `modepy` also offers a wide array of quadrature rules that can be +used on each domain. The quadrature rules are provided as implementations of +the `Quadrature` superclass. + +For the interval, a broad selection of quadrature rules are present: +Clenshaw--Curtis, Fejér, and Jacobi-Gauss. The latter class includes the +commonly used Chebyshev and Legendre types, among others. Lobatto (that is, +endpoint-included) variants of these are also available. The line segment +quadratures can be used as part of the `TensorProductQuadrature` class to +define different higher-dimensional rules. Additionally, several +state-of-the-art higher-dimensional rules are available: + +* Grundmann--Möller [@Grundmann1978] rules for the $n$-simplex. +* Vioreanu--Rokhlin [@Vioreanu2014] rules for the two- and + three-dimensional simplex (see \autoref{FigureQuadrature}). The quadrature + nodes from these rules are also suitable for interpolation. +* Xiao--Gimbutas [@Xiao2010] rules for the two- and three-dimensional + simplex. +* Jáskowiec--Sukumar [@Jaskowiec2021] rules for the tetrahedron. +* Witherden--Vincent [@Witherden2015] rules for the two- and + three-dimensional hypercube (see \autoref{FigureQuadrature}). These rules + have significantly fewer nodes than an equivalent tensor product rule, while + maintaining positive weights. + +![(left) Vioreanu--Rokhlin quadrature points of order 11 and (right) Witherden--Vincent quadrature points of order 11.](images/quadrature_rules.png){#FigureQuadrature width="50%"} + +Most of the rules for simplices and hypercubes are available to high order +(i.e. $\ge 20$). However, `modepy` also has functionality that allows +constructing desired quadrature nodes on a given domain. This can be found in +`modepy.quadrature.construction`. + +## Matrices + +While `modepy` is sufficiently flexible to accommodate different applications, +it also provides functionality specific to FEM needs. In particular, it allows +constructing a variety of matrix operators that define necessary resampling +operators and bilinear forms used in FEM codes. + +The Vandermonde matrix used in interpolation is provided by the `vandermonde` +function. This can be used to obtain point-to-point interpolants using +`resampling_matrix`. In a similar fashion, the `differentiation_matrices` +function can be used to construct derivative operators along each reference +axis. + +A specific bilinear form matrix can be obtained using +`nodal_quadrature_bilinear_form_matrix`, which requires providing the usual +test functions, trial functions and corresponding nodes for the input and +output. This allows constructing very general families of bilinear forms with +or without oversampling. One such example is the standard mass matrix, which +can be obtained more easily using the `mass_matrix` function. If the provided +functions are derivatives of the basis functions, standard weak forms of +differentiation operators can also be obtained. + +# A Practical Example + +In the following example, we demonstrate how to use the library to define a +non-default tensor product shape --- the prism --- and construct a first +derivative in the $x$ direction on this reference element. To produce the +derivative matrix operator, we go through all the objects described in the +previous section: nodes, modes, and quadrature rules. + +```python +import numpy as np +import modepy as mp + +# Define the shape on which we will operate +line = mp.Simplex(1) +triangle = mp.Simplex(2) +prism = mp.TensorProductShape((triangle, line)) + +assert prism.dim == 3 + +# Define a function space for the prism of order 12 +n = 12 +space = mp.TensorProductSpace((mp.PN(triangle.dim, n), mp.PN(line.dim, n))) + +assert space.order == n +assert space.spatial_dim == 3 +assert space.space_dim == (n + 1) * (n + 1) * (n + 2) // 2 + +# Define a basis function for the prism +basis = mp.orthonormal_basis_for_space(space, prism) + +# Define a point set for the prism +nodes = mp.edge_clustered_nodes_for_space(space, prism) + +# Define a quadrature rule for the prism +quadrature = mp.TensorProductQuadrature([ + mp.VioreanuRokhlinSimplexQuadrature(n, triangle.dim), + mp.LegendreGaussQuadrature(n), +]) + +# Define a bilinear form: weak derivative in the i=x direction +i = 0 +weak_d = mp.nodal_quadrature_bilinear_form_matrix( + quadrature, + test_functions=basis.derivatives(i), + trial_functions=basis.functions, + nodal_interp_functions_test=basis.functions, + nodal_interp_functions_trial=basis.functions, + input_nodes=nodes, + output_nodes=nodes, +) +inv_mass = mp.inverse_mass_matrix(basis, nodes) + +# Compute derivative and check exact solution +x = nodes[i] +f = 1.0 - np.sin(x) ** 3 +f_ref = -3.0 * np.cos(x) * np.sin(x) ** 2 +f_approx = inv_mass @ weak_d.T @ f + +error = np.linalg.norm(f_approx - f_ref) / np.linalg.norm(f_ref) +assert error < 1.0e-4 +``` + +# Acknowledgements + +A. Fikl was supported by the Office of Naval Research (ONR) as part of the +Multidisciplinary University Research Initiatives (MURI) Program, under Grant +Number *N00014-16-1-2617*. A. Klöckner was supported by the US National Science +Foundation under award number DMS-2410943, and by the US Department of Energy +under award number DE-NA0003963. From 8f723dc54486e7b71ffbd0bf3d6f6971354ebfc1 Mon Sep 17 00:00:00 2001 From: Alexandru Fikl Date: Tue, 2 Sep 2025 11:48:35 +0300 Subject: [PATCH 02/15] ci: add build for JOSS paper --- .github/workflows/joss.yml | 34 ++++++++++++++++++++++++++++++++++ 1 file changed, 34 insertions(+) create mode 100644 .github/workflows/joss.yml diff --git a/.github/workflows/joss.yml b/.github/workflows/joss.yml new file mode 100644 index 0000000..a4416cc --- /dev/null +++ b/.github/workflows/joss.yml @@ -0,0 +1,34 @@ +name: JOSS Paper +on: + push: + paths: + - doc/paper/** + - .github/workflows/joss.yml + schedule: + - cron: '17 3 * * 0' + +concurrency: + group: ${{ github.workflow }}-${{ github.event.pull_request.number || github.ref }} + cancel-in-progress: true + +jobs: + paper: + runs-on: ubuntu-latest + name: Paper Draft + steps: + - name: Checkout + uses: actions/checkout@v5 + + - name: Build draft PDF + uses: openjournals/openjournals-draft-action@master + with: + journal: joss + paper-path: doc/paper/paper.md + + - name: Upload + uses: actions/upload-artifact@v4 + with: + name: paper + path: doc/paper/paper.pdf + +# vim: sw=4 From 0d86c154bcfcb2353a926acbb0c04d2b12e2472f Mon Sep 17 00:00:00 2001 From: Alexandru Fikl Date: Tue, 2 Sep 2025 11:53:37 +0300 Subject: [PATCH 03/15] chore: ignore paper.bib for typos --- pyproject.toml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/pyproject.toml b/pyproject.toml index 367484f..8004d19 100644 --- a/pyproject.toml +++ b/pyproject.toml @@ -131,7 +131,7 @@ nd = "nd" zerod = "zerod" [tool.typos.files] -extend-exclude = [] +extend-exclude = ["doc/paper/paper.bib"] [tool.basedpyright] pythonVersion = "3.10" From 52109c1314b3e6b91e8dc22663efd30656474fbd Mon Sep 17 00:00:00 2001 From: Andreas Kloeckner Date: Wed, 3 Sep 2025 11:48:51 -0500 Subject: [PATCH 04/15] Fix finite-dimensional repetition in 'modes and spaces' --- doc/paper/paper.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/doc/paper/paper.md b/doc/paper/paper.md index 3b7988b..b60ee45 100644 --- a/doc/paper/paper.md +++ b/doc/paper/paper.md @@ -178,8 +178,8 @@ mappings back to the reference element. ## Modes and Spaces To perform calculus operations, each reference element can be equipped with -(finite-dimensional) function spaces described by the `FunctionSpace` class. -These represent a finite-dimensional space functions $\phi_i: D \to +function spaces described by the `FunctionSpace` class. +These represent a finite-dimensional space of functions $\phi_i: D \to \mathbb{R}$, where $D$ is the reference element domain, but make no specific choice of basis. The function spaces mirror the shape definitions and are intended to work in tandem. In fact, the `space_for_shape` function provides a From 68e00c6b2ad9973129a02b4d0dde08e361b869fe Mon Sep 17 00:00:00 2001 From: Alexandru Fikl Date: Thu, 4 Sep 2025 09:23:00 +0300 Subject: [PATCH 05/15] Fix recursivenodes reference and add doi --- doc/paper/paper.bib | 8 ++++++-- 1 file changed, 6 insertions(+), 2 deletions(-) diff --git a/doc/paper/paper.bib b/doc/paper/paper.bib index 19c6cae..6129400 100644 --- a/doc/paper/paper.bib +++ b/doc/paper/paper.bib @@ -115,8 +115,12 @@ @article{Isaac2020 }, author = {Tobin Isaac}, year = 2020, - eprint = {2002.09421v1}, - eprinttype = {arXiv}, + journal = {{SIAM} Journal on Scientific Computing}, + publisher = {Society for Industrial \& Applied Mathematics {(SIAM)}}, + volume = 42, + pages = {A4046--A4062}, + doi = {10.1137/20m1321802}, + issue = 6, } @article{Jaskowiec2021, From cbc0f6a4dfcec0bb6ff00989a3d4da56dfc7dcf7 Mon Sep 17 00:00:00 2001 From: Alexandru Fikl Date: Thu, 4 Sep 2025 09:27:31 +0300 Subject: [PATCH 06/15] Add References section for bibliography --- doc/paper/paper.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/doc/paper/paper.md b/doc/paper/paper.md index b60ee45..8a700b5 100644 --- a/doc/paper/paper.md +++ b/doc/paper/paper.md @@ -382,3 +382,5 @@ Multidisciplinary University Research Initiatives (MURI) Program, under Grant Number *N00014-16-1-2617*. A. Klöckner was supported by the US National Science Foundation under award number DMS-2410943, and by the US Department of Energy under award number DE-NA0003963. + +# References From 94fbac157cfc97df228817236cfdd3fc87bf15bc Mon Sep 17 00:00:00 2001 From: Alexandru Fikl Date: Fri, 12 Sep 2025 11:07:00 +0300 Subject: [PATCH 07/15] Remove example and trim down overview --- doc/paper/paper.md | 232 ++++++++++----------------------------------- 1 file changed, 48 insertions(+), 184 deletions(-) diff --git a/doc/paper/paper.md b/doc/paper/paper.md index 8a700b5..5a552d2 100644 --- a/doc/paper/paper.md +++ b/doc/paper/paper.md @@ -146,16 +146,11 @@ to any array type (including GPU or other high-performance arrays) as long as # Overview -The high-level concepts available in `modepy` are shapes (i.e the geometry of a +The high-level concepts available in `modepy` are shapes (i.e. a reference domain), modes (i.e. the basis functions), and nodes (i.e. the degrees of freedom). These are implemented in a user-extensible fashion using the `singledispatch` mechanism, with inspiration taken from common idiomatic usage -in Julia [@Bezanson2017]. Given a function space `space` on a `shape`, a set of -nodal degrees of freedom can be obtained via -`edge_clustered_nodes_for_space(space, shape)`, with an implementation selected -based on the type of `space`. While the example shows that multiple dispatch -could be beneficial, we have decided to forgo it to avoid potential soundness -issues [@Julia2018]. +in Julia [@Bezanson2017]. ## Shapes @@ -163,126 +158,68 @@ The geometry of a reference element is described in `modepy` by the `Shape` class. Built-in support exists for `Simplex` and `Hypercube` geometries, encompassing the commonly used interval, triangle, tetrahedron, quadrilateral, and hexahedral shapes (see \autoref{FigureSimplices}). `TensorProductShape` can -be used to compose additional shapes, such as prisms, that are useful in -specific applications and sometimes generated by meshing software (such as -`gmsh` [@Geuzaine2009]). An example that constructs a prism and performs some -standard operations on it is given in the next section. +be used to compose additional shapes, that are useful in specific applications +and sometimes generated by meshing software (such as `gmsh` [@Geuzaine2009]). ![Domains corresponding to the one-, two-, and three-dimensional simplices.](images/simplices.png){#FigureSimplices width="80%"} -To support a wide array of applications, shapes can be queried and manipulated -in several ways. The `faces_for_shape` function can be used to return a list of -faces, which are shapes themselves, that allow access to domain normals and -mappings back to the reference element. - ## Modes and Spaces -To perform calculus operations, each reference element can be equipped with -function spaces described by the `FunctionSpace` class. -These represent a finite-dimensional space of functions $\phi_i: D \to -\mathbb{R}$, where $D$ is the reference element domain, but make no specific -choice of basis. The function spaces mirror the shape definitions and are -intended to work in tandem. In fact, the `space_for_shape` function provides a -default choice of space for each shape. Predefined choices include the `PN` -space, containing polynomials of total degree at most `order`, and the `QN` -space, containing polynomials of maximum degree at most `order`. As with -shapes, these spaces can also be combined using `TensorProductSpace` to -describe functional spaces for other domains or in higher dimensions. - -For calculations, each space can be equipped with a default basis using the -`basis_for_space` function. `modepy` provides two special types of basis -functions: orthonormal (accessed through `orthonormal_basis_for_space`) and -monomial (accessed through `monomial_basis_for_space`). The basis is provided -in the form of a `Basis` class, which allows access to the basis functions, -their derivatives (per axis) and opaque `mode_ids`. As noted before, the basis -functions and their derivatives can be evaluated directly or can be traced -using the `pymbolic` library to aid in code generation. +To perform calculus operations, each reference element can be equipped with a +function space described by the `FunctionSpace` class. These represent a +finite-dimensional space of functions $\phi_i: D \to \mathbb{R}$, where $D$ is +the reference element domain, and no specific choice of basis. Predefined +choices include the `PN` space, containing polynomials of total degree at most +`order`, and the `QN` space, containing polynomials of maximum degree at most +`order`. As with shapes, these spaces can be combined using +`TensorProductSpace`. + +The basis is provided in the form of a `Basis` class, which allows access to +the basis functions, their derivatives (per axis) and opaque `mode_ids`. As +noted before, the basis functions and their derivatives can be evaluated +directly or can be traced using the `pymbolic` library to aid in code +generation. Various standard basis functions are provided, such as the +monomials, general Jacobi polynomials, and the +Proriol-Koornwinder-Dubiner-Owens (PKDO) basis from [@Dubiner1991] (see +\autoref{FigurePKDO}). ![PKDO basis functions for the triangle.](images/pkdo-2d.png){#FigurePKDO width="80%"} -By default, the `Simplex` shape is equipped with a `PN` functional space. This -function space uses the Proriol-Koornwinder-Dubiner-Owens (PKDO) basis (see -\autoref{FigurePKDO}) as defined in [@Dubiner1991] for dimensions up to 3. The -`Hypercube` is equipped with the `QN` functional space that uses a tensor -product of standard Legendre polynomials. Additionally, general Jacobi -polynomials are available and can be used as a basis for variants of `QN` with -different weights, depending on the application. - ## Nodes A final component in an FEM discretization (or 'Ciarlet triple' [@Brenner2007, -Section 3.1]) is a set of 'degrees of freedom' ('DOFs'), that is, a vector of -numbers (technically, functionals) that uniquely define a certain element in -the span of a basis. The literature considers various types of degrees of -freedom; the two most common are modal DOFs (i.e. basis coefficients) and nodal -DOFs (i.e. function or derivative values at certain points). `modepy` provides -easy access to both, while paying particular attention to their usability in -the high-order setting. Due to Runge's phenomenon [@Trefethen2020], simple -equispaced nodal DOFs rapidly become unusable with increasing polynomial order -due to poor conditioning. As such, `modepy` offers access to many types of -nodes with better properties for each reference element. There also exists an -ability to construct one's own set of nodes given a set of functions in 2D via -a procedure based on eigenvalues of multiplication operators [@Vioreanu2014]. - -On each shape, three default families of nodes are provided: equispaced -(accessed through `equispaced_nodes_for_space`), edge-clustered (accessed -through `edge_clustered_nodes_for_space`) and randomly distributed (accessed -through `random_nodes_for_space`). - -For more generality, `modepy` can directly interoperate with the -`recursivenodes` library described in [@Isaac2020], which offers -well-conditioned nodes on the simplex that obey a 'recursive' property, where -nodes on a face of a higher-dimensional shape are exactly the lower-dimensional -nodes. In addition, it offers many many tunable parameters (e.g. nodes on the -boundary vs. not). On simplices, the "warp-and-blend" nodes [@Warburton2007] -are also available for dimensions three and lower. On the hypercube, standard -tensor product nodes are constructed from one-dimensional -Legendre-Gauss-Lobatto nodes. As for the basis functions, the full family of -Jacobi-Gauss and Jacobi-Gauss-Lobatto quadrature points is available for -specific applications. - -Taken together, the functionality described thus far provides comprehensive -support for interpolation, that is, the evaluation of a polynomial defined by -its nodal DOFs, at any point, and further the ability to obtain any derivative -of the interpolating polynomial. +Section 3.1]) is a set of 'degrees of freedom' ('DOFs') that uniquely define a +certain function in the span of a basis. `modepy` provides easy access to modal +DOFs (i.e. basis coefficients) and nodal DOFs (i.e. function or derivative +values at a point). As equispaced nodal DOFs are not usable in the high-order +setting [@Trefethen2020], `modepy` offers access to many types of nodes with +better properties for each reference element. It can also construct custom +nodes sets for a given set of basis functions in 2D via a procedure based on +eigenvalues of multiplication operators [@Vioreanu2014]. + +On simplices, the "warp-and-blend" nodes [@Warburton2007] are available, and on +the hypercube, standard tensor product nodes are constructed from +one-dimensional Legendre-Gauss(-Lobatto) nodes. `modepy` can also directly +interoperate with the `recursivenodes` library described in [@Isaac2020], which +offers additional well-conditioned nodes on the simplex. ## Quadrature -While nodal degrees of freedom must be *unisolvent* or *interpolatory* (i.e. -uniquely determine a function in a discrete function space), this requirement -does not exist in the setting of quadrature, where the objective is to -accurately approximate the value of an integral of a function on a shape. Using -function values at points $\boldsymbol{x}_i$ ("nodes") and associated weights -$w_i$, quadrature rules are generically expressed as -$$ -\int_{D} f(\boldsymbol x) \mathrm{d}\boldsymbol{x} \approx - \sum_{k = 0}^N f(\boldsymbol{x}_k) w_k. -$$ - -Besides the standard building blocks consisting of domains, basis functions and -node sets, `modepy` also offers a wide array of quadrature rules that can be -used on each domain. The quadrature rules are provided as implementations of -the `Quadrature` superclass. - -For the interval, a broad selection of quadrature rules are present: -Clenshaw--Curtis, Fejér, and Jacobi-Gauss. The latter class includes the -commonly used Chebyshev and Legendre types, among others. Lobatto (that is, -endpoint-included) variants of these are also available. The line segment -quadratures can be used as part of the `TensorProductQuadrature` class to -define different higher-dimensional rules. Additionally, several -state-of-the-art higher-dimensional rules are available: +Besides the standard building blocks above, `modepy` also offers a wide array +of quadrature rules that can be used on each reference element. The quadrature +rules are provided as implementations of the `Quadrature` superclass. For the +interval, a broad selection of quadrature rules are present: Clenshaw--Curtis, +Fejér, and Jacobi-Gauss(-Lobatto). Additionally, several state-of-the-art +higher-dimensional rules are available: * Grundmann--Möller [@Grundmann1978] rules for the $n$-simplex. * Vioreanu--Rokhlin [@Vioreanu2014] rules for the two- and - three-dimensional simplex (see \autoref{FigureQuadrature}). The quadrature - nodes from these rules are also suitable for interpolation. + three-dimensional simplex (see \autoref{FigureQuadrature}). * Xiao--Gimbutas [@Xiao2010] rules for the two- and three-dimensional simplex. * Jáskowiec--Sukumar [@Jaskowiec2021] rules for the tetrahedron. * Witherden--Vincent [@Witherden2015] rules for the two- and - three-dimensional hypercube (see \autoref{FigureQuadrature}). These rules - have significantly fewer nodes than an equivalent tensor product rule, while - maintaining positive weights. + three-dimensional hypercube (see \autoref{FigureQuadrature}). ![(left) Vioreanu--Rokhlin quadrature points of order 11 and (right) Witherden--Vincent quadrature points of order 11.](images/quadrature_rules.png){#FigureQuadrature width="50%"} @@ -296,84 +233,11 @@ constructing desired quadrature nodes on a given domain. This can be found in While `modepy` is sufficiently flexible to accommodate different applications, it also provides functionality specific to FEM needs. In particular, it allows constructing a variety of matrix operators that define necessary resampling -operators and bilinear forms used in FEM codes. - -The Vandermonde matrix used in interpolation is provided by the `vandermonde` -function. This can be used to obtain point-to-point interpolants using -`resampling_matrix`. In a similar fashion, the `differentiation_matrices` -function can be used to construct derivative operators along each reference -axis. - -A specific bilinear form matrix can be obtained using -`nodal_quadrature_bilinear_form_matrix`, which requires providing the usual -test functions, trial functions and corresponding nodes for the input and -output. This allows constructing very general families of bilinear forms with -or without oversampling. One such example is the standard mass matrix, which -can be obtained more easily using the `mass_matrix` function. If the provided -functions are derivatives of the basis functions, standard weak forms of -differentiation operators can also be obtained. - -# A Practical Example - -In the following example, we demonstrate how to use the library to define a -non-default tensor product shape --- the prism --- and construct a first -derivative in the $x$ direction on this reference element. To produce the -derivative matrix operator, we go through all the objects described in the -previous section: nodes, modes, and quadrature rules. - -```python -import numpy as np -import modepy as mp - -# Define the shape on which we will operate -line = mp.Simplex(1) -triangle = mp.Simplex(2) -prism = mp.TensorProductShape((triangle, line)) - -assert prism.dim == 3 - -# Define a function space for the prism of order 12 -n = 12 -space = mp.TensorProductSpace((mp.PN(triangle.dim, n), mp.PN(line.dim, n))) - -assert space.order == n -assert space.spatial_dim == 3 -assert space.space_dim == (n + 1) * (n + 1) * (n + 2) // 2 - -# Define a basis function for the prism -basis = mp.orthonormal_basis_for_space(space, prism) - -# Define a point set for the prism -nodes = mp.edge_clustered_nodes_for_space(space, prism) - -# Define a quadrature rule for the prism -quadrature = mp.TensorProductQuadrature([ - mp.VioreanuRokhlinSimplexQuadrature(n, triangle.dim), - mp.LegendreGaussQuadrature(n), -]) - -# Define a bilinear form: weak derivative in the i=x direction -i = 0 -weak_d = mp.nodal_quadrature_bilinear_form_matrix( - quadrature, - test_functions=basis.derivatives(i), - trial_functions=basis.functions, - nodal_interp_functions_test=basis.functions, - nodal_interp_functions_trial=basis.functions, - input_nodes=nodes, - output_nodes=nodes, -) -inv_mass = mp.inverse_mass_matrix(basis, nodes) - -# Compute derivative and check exact solution -x = nodes[i] -f = 1.0 - np.sin(x) ** 3 -f_ref = -3.0 * np.cos(x) * np.sin(x) ** 2 -f_approx = inv_mass @ weak_d.T @ f - -error = np.linalg.norm(f_approx - f_ref) / np.linalg.norm(f_ref) -assert error < 1.0e-4 -``` +operators and bilinear forms used in FEM codes. A specific bilinear form matrix +can be obtained using `nodal_quadrature_bilinear_form_matrix`, which requires +providing the usual test functions, trial functions and corresponding nodes for +the input and output. This allows constructing very general families of +bilinear forms with or without oversampling. # Acknowledgements From d3866d08e85947d40b9f9ec66dbbcd0717519268 Mon Sep 17 00:00:00 2001 From: Alexandru Fikl Date: Wed, 17 Sep 2025 11:50:39 +0300 Subject: [PATCH 08/15] Trim down Summary and Statement of Need --- doc/paper/paper.md | 147 ++++++++++++++++----------------------------- 1 file changed, 53 insertions(+), 94 deletions(-) diff --git a/doc/paper/paper.md b/doc/paper/paper.md index 5a552d2..66ec0d0 100644 --- a/doc/paper/paper.md +++ b/doc/paper/paper.md @@ -41,108 +41,67 @@ bibliography: paper.bib # Summary -`modepy` provides a means for constructing geometric shapes for reference -elements, equipping them with appropriate approximation spaces, and numerically -performing calculus operations (derivatives, integrals) on those spaces. +`modepy` is a Python library for defining reference elements, equipping them +with appropriate approximation spaces, and numerically performing calculus +operations (derivatives, integrals) on those spaces. It is written in pure +modern Python 3, offering comprehensive type annotations and full documentation +with minimal runtime dependencies (NumPy being a primary dependency). `modepy` focuses on high-order accuracy -- given an element size $h$, this refers to the asymptotic decay of the approximation error as $O(h^n)$, for $n -\ge 3$, assuming sufficient smoothness of the solution being approximated. If -we consider a problem in $d$ dimensions, the number of unknowns scales as -$O(h^{-d})$. In the absence of high-order approximation, say at $n=1$, even -modest accuracy increases (e.g. an additional significant digit) can incur -significant cost (a $1000\times$ increase in the problem size in 3D). -Therefore, if accuracy is desired at manageable cost, high-order methods are -crucial. - -A popular approach for accurate approximation of functions on geometrically -complex domains is the use of *unstructured discretizations*, which represent -the geometry as a typically disjoint union (a "mesh") of primitive geometric -shapes, most often simplices and quadrilaterals. Given the means to perform -calculus operations on these *reference elements* and mapping functions from -them to the *global* elements, calculus operations become available on the -entire domain. Notably, one can approximate the reference-to-global coordinate -mapping function itself with the same high-order machinery. Altogether, this -paves the way for the solution of integral and (partial) differential +\ge 3$, assuming sufficient smoothness of the solution being approximated -- at +manageable computational cost. These capabilities enable High-Performance +Computing (HPC) approaches to solving integral and (partial) differential equations. Those, in turn, can be used to model many phenomena in the physical -world, including fluid flow, electromagnetism, and solid mechanics. The finite -element method (FEM) is a popular example of this, including its continuous and -discontinuous flavors. Further examples include collocation, spectral, and -Nyström methods. - -`modepy` has been used to construct FEM solvers [@Grudge; @PyNucleus] and -integral equation solvers [@Pytential]. It is written in pure modern Python 3, -offering comprehensive type annotations and full documentation with minimal -runtime dependencies, of which numpy is perhaps the most significant. Versions -going back to as early as 2013 offer broad compatibility with older versions of -Python, including Python 2.7. - -`modepy` is licensed under the MIT license and available on Github at -. +world, including fluid flow, electromagnetism, and solid mechanics. A popular +example of this is the finite element method (FEM), including its continuous +and discontinuous flavors, but general collocation, spectral, and Nyström +methods are also supported. As such, `modepy` has been used to construct FEM +solvers [@Grudge; @PyNucleus] and integral equation solvers [@Pytential] that +run on both CPUs and GPUs. + +`modepy` is licensed under the MIT license and available on GitHub at +. Versions dating back to 2013 maintain +broad compatibility with older versions of Python, including Python 2.7, +ensuring broad accessibility. # Statement of need High-order accurate calculus operations on unstructured discretizations are -crucial to many solvers of differential and integral equations, but their uses -can also extend to computer graphics and Computer-Aided Design (CAD). This -functionality is often embedded in an ad-hoc manner in larger codes, leading to -limited scope and a lack of reusability. `modepy` addresses this need by -providing a robust, reusable, generalizable, and composable implementation. -FIAT [@FIAT] is an early software tool aiming to address a similar need. It -offers a comprehensive set of finite elements and basis functions, but largely -focuses on tabulation. FInAT [@FInAT], a somewhat recent refinement, adds a -focus on algebraic expressions for basis functions, exposing details needed for -efficient tensor product evaluation. Both are, however, tightly interwoven into -the FENiCS/Firedrake ecosystems (e.g. by depending on UFL, the "Unified Form -Language" used for the expression of variational forms). `StartUpDG.jl` -[@StartUpDG] provides functionality with some overlap with `modepy`, but with a -focus on the needs of discontinuous Galerkin FEM. `QuadPy` [@QuadPy] also has -some overlap with `modepy`, in that it can provide quadrature rules, and, while -it has comprehensive coverage, it is regrettably no longer open-source, and it -lacks `modepy`'s composability. `minterpy` [@Wicaksono2025], meanwhile, deals -exclusively with polynomial interpolation, with a focus on sparse grids. - -A significant complicating factor in providing a flexible implementation is -that numerical solvers for differential equations (and likely many other -candidate users) must satisfy rigorous cost constraints, as simulation fidelity -is always traded off against computational cost. As a result, such software -commonly adopts cutting-edge high-performance computing (HPC) techniques, such -as GPU computation and distributed-memory approaches. To facilitate separation -of those concerns from the numerical methods, `modepy` adopts a two-pronged -approach. In many cases, it suffices for operations to be represented as data -in matrix or tabular form, so that no actual execution of `modepy` code is -needed in a cost-constrained setting. - -A prominent exception to this is the evaluation of basis functions. To -facilitate this, `modepy` allows its function evaluation to be "traced" (in the -sense of lazy/deferred evaluation), so that an expression graph can be -obtained. This graph is represented with the help of the `pymbolic` [@Pymbolic] -software library. This, in turn, can interoperate with Python ASTs -[@Python_AST], SymPy [@SymPy], SymEngine [@SymEngine], as well as a number of -other pieces of software for symbolic computation. Another case where a purely -data-driven approach falls short is the setting of tensor-product elements. In -this case, for example, a mass matrix $\boldsymbol M$ with entries -$$ -M_{ij}=\int_{[-1,1]^d} \phi_i(\boldsymbol r)\phi_j(\boldsymbol r) \mathrm{d}\boldsymbol r -$$ -permits a Kronecker product factorization -$$ -\boldsymbol M = \boldsymbol M^{1D}\otimes \cdots \otimes \boldsymbol M^{1D}, -\qquad \text{with} \qquad -M^{1D}_{ij} = \int_{[-1,1]} \phi_i(r)\phi_j(r) \mathrm{d}r, -$$ -assuming a suitable index order of nodes $\boldsymbol{r}_i$ and basis functions -$\{\phi_i\}$. In $d$ dimensions, the use of this factorization reduces the -asymptotic complexity of a matrix-vector product $\boldsymbol M \boldsymbol u$ -from $O(N_p^{2d})$ to $O(N_p^{d+1})$, where $N_p$ is the number of degrees of -freedom in one dimension, and $d$ is the number of dimensions. To take -advantage of this factorization, an abstraction such as `modepy`'s must expose -additional information, including numbering of degrees of freedom. In -`modepy`'s case, this is accomplished via reshaping operation that converts -arrays of degrees of freedom back and forth between a flat (one-dimensional) -and a structured ($d$-dimensional) representation. This reshaping is applicable -to any array type (including GPU or other high-performance arrays) as long as -`numpy`-compatible reshaping is supported. +crucial to differential and integral equations solvers, as well as computer +graphics and Computer-Aided Design (CAD) applications. This functionality is +often embedded in an ad-hoc manner in larger codes, restricting scope and +reusability. `modepy` addresses this need by providing a reusable, +generalizable, and composable implementation. + +There are several other candidate libraries in the literature with similar +goals, but important limitations. FInAT [@FInAT] (and the earlier [@FIAT]) is a +offers reference elements and basis functions, but is tightly coupled to the +FENiCS/Firedrake ecosystem. Similarly, `StartUpDG.jl` [@StartUpDG] has a focus +on the needs of discontinuous Galerkin FEM in the Trixi framework. `QuadPy` +[@QuadPy] provides access to quadrature rules, but it is no longer open source +and lacks `modepy`'s composability. `minterpy` [@Wicaksono2025], meanwhile, +deals exclusively with polynomial interpolation, with a focus on sparse grids. + +To facilitate separation of concerns from the chosen numerical methods, +`modepy` adopts a two-pronged approach that allows flexibility in trading +simulation fidelity against computation cost. First, it usually suffices for +operations to be represented as data in matrix or tabular form, so that no +actual execution of `modepy` code is needed in a cost-constrained setting. For +example, nodes and bilinear forms on reference elements can generally be +pre-computed and tabulated. + +However, this data-driven approach falls short in the setting of tensor product +elements. In this case, for example, a mass matrix $\boldsymbol M$ permits a +Kronecker product factorization that significantly reduces the asymptotic +complexity of a matrix-vector product in higher dimensions. `modepy` exposes +additional information that allows reshaping degrees of freedom arrays to take +advantage of such factorizations. Another prominent exception is the evaluation +of basis functions. To facilitate efficient evaluation, `modepy` allows its +functions to be "traced", in the sense of lazy or deferred evaluation. The +resulting expression graph is represented by the `pymbolic` [@Pymbolic] +software library, that can interoperate with Python ASTs [@Python_AST], SymPy +[@SymPy], SymEngine [@SymEngine], etc., for straightforward code generation. # Overview From 8dbd1ebb32c1d032e7b514a107cddb0445bce37f Mon Sep 17 00:00:00 2001 From: Andreas Kloeckner Date: Fri, 19 Sep 2025 10:06:35 -0500 Subject: [PATCH 09/15] Configure typos --- doc/paper/.gitignore | 1 + doc/paper/_typos.toml | 7 +++++++ 2 files changed, 8 insertions(+) create mode 100644 doc/paper/_typos.toml diff --git a/doc/paper/.gitignore b/doc/paper/.gitignore index 29cc143..ec6a328 100644 --- a/doc/paper/.gitignore +++ b/doc/paper/.gitignore @@ -2,3 +2,4 @@ joss *.pdf *.tex +latex.out diff --git a/doc/paper/_typos.toml b/doc/paper/_typos.toml new file mode 100644 index 0000000..13254b7 --- /dev/null +++ b/doc/paper/_typos.toml @@ -0,0 +1,7 @@ +[files] +extend-exclude = ["*.bib"] + + +[default.extend-words] +# the space +PN = "PN" From dd57407746ddcd1a614379aeb0a4cae3810199c1 Mon Sep 17 00:00:00 2001 From: Andreas Kloeckner Date: Fri, 19 Sep 2025 10:07:07 -0500 Subject: [PATCH 10/15] Another shortening pass --- doc/paper/paper.bib | 519 ++++++++++++++++++++++---------------------- doc/paper/paper.md | 166 ++++++-------- 2 files changed, 332 insertions(+), 353 deletions(-) diff --git a/doc/paper/paper.bib b/doc/paper/paper.bib index 6129400..17b826b 100644 --- a/doc/paper/paper.bib +++ b/doc/paper/paper.bib @@ -1,334 +1,345 @@ @article{Bezanson2017, - title = {Julia: {A} Fresh Approach to Numerical Computing}, - author = {Jeff Bezanson and Alan Edelman and Stefan Karpinski and Viral B. Shah}, - year = 2017, - journal = {{SIAM} Review}, - publisher = {Society for Industrial \& Applied Mathematics {(SIAM)}}, - volume = 59, - pages = {65--98}, - doi = {10.1137/141000671}, - issue = 1, + title = {Julia: {A} Fresh Approach to Numerical Computing}, + author = {Jeff Bezanson and Alan Edelman and Stefan Karpinski and Viral B. + Shah}, + year = 2017, + journal = {{SIAM} Review}, + publisher = {Society for Industrial \& Applied Mathematics {(SIAM)}}, + volume = 59, + pages = {65--98}, + doi = {10.1137/141000671}, + issue = 1, } @book{Brenner2007, - title = {The Mathematical Theory of Finite Element Methods}, - author = {Susanne Brenner and Ridgway Scott}, - year = 2007, - publisher = {Springer Science \& Business Media}, - isbn = 9780387759333, + title = {The Mathematical Theory of Finite Element Methods}, + author = {Susanne Brenner and Ridgway Scott}, + year = 2007, + publisher = {Springer Science \& Business Media}, + isbn = 9780387759333, } @article{Dubiner1991, - title = {Spectral Methods on Triangles and Other Domains}, - author = {M. Dubiner}, - year = 1991, - journal = {Journal of Scientific Computing}, - publisher = {Springer Science and Business Media {LLC}}, - volume = 6, - pages = {345--390}, - doi = {10.1007/bf01060030}, - issue = 4, + title = {Spectral Methods on Triangles and Other Domains}, + author = {M. Dubiner}, + year = 1991, + journal = {Journal of Scientific Computing}, + publisher = {Springer Science and Business Media {LLC}}, + volume = 6, + pages = {345--390}, + doi = {10.1007/bf01060030}, + issue = 4, } @software{FIAT, - title = {{Fiat}: 2025.4.0}, - author = { - Mikl\'{o}s Homolya and Pablo Brubeck and David A. Ham and Robert Kirby and - Lawrence Mitchell and Jan Blechta and Marie E Rognes and Garth N. Wells and - Anders Logg and cyruscycheng21 and Thomas H. Gibson and Johannes Ring and - Ivan Yashchuk and Justincrum and k-b-oelgaard and ksagiyam and Chris - Richardson and Nico Schl\"{o}mer and Reuben W. Nixon-Hill and Lizao Li and - Aslak Bergersen and Connor Ward and Nicholas Barton and Colin J Cotter and - Patrick E. Farrell and Jack S. Hale and FAznaran and Matthew Knepley and - Florian Rathgeber and Matthew Scroggs - }, - year = 2025, - month = apr, - publisher = {Zenodo}, - doi = {10.5281/zenodo.15302339}, - url = {https://doi.org/10.5281/zenodo.15302339}, - version = {2025.4.0}, + title = {{Fiat}: 2025.4.0}, + author = { Mikl\'{o}s Homolya and Pablo Brubeck and David A. Ham and Robert + Kirby and Lawrence Mitchell and Jan Blechta and Marie E Rognes and + Garth N. Wells and Anders Logg and cyruscycheng21 and Thomas H. + Gibson and Johannes Ring and Ivan Yashchuk and Justincrum and + k-b-oelgaard and ksagiyam and Chris Richardson and Nico Schl\"{o} + mer and Reuben W. Nixon-Hill and Lizao Li and Aslak Bergersen and + Connor Ward and Nicholas Barton and Colin J Cotter and Patrick E. + Farrell and Jack S. Hale and FAznaran and Matthew Knepley and + Florian Rathgeber and Matthew Scroggs }, + year = 2025, + month = apr, + publisher = {Zenodo}, + doi = {10.5281/zenodo.15302339}, + url = {https://doi.org/10.5281/zenodo.15302339}, + version = {2025.4.0}, } @software{FInAT, - title = {{FInAT}: {A} smarter library of finite elements}, - author = { - David A. Ham and Mikl\'{o}s Homolya and Robert Kirby and Lawrence Mitchell - and Pablo Brubeck and cyruscycheng21 and FAznaran and ksagiyam and Matthew - Scroggs and Patrick E. Farrell and Justincrum and Connor Ward and celdred - and Thomas Bendall and Jack Betteridge and FabianL1908 - }, - year = 2025, - month = mar, - publisher = {Zenodo}, - doi = {10.5281/zenodo.15114385}, - url = {https://doi.org/10.5281/zenodo.15114385}, - version = {Firedrake\_20250331.0}, + title = {{FInAT}: {A} smarter library of finite elements}, + author = { David A. Ham and Mikl\'{o}s Homolya and Robert Kirby and Lawrence + Mitchell and Pablo Brubeck and cyruscycheng21 and FAznaran and + ksagiyam and Matthew Scroggs and Patrick E. Farrell and Justincrum + and Connor Ward and celdred and Thomas Bendall and Jack Betteridge + and FabianL1908 }, + year = 2025, + month = mar, + publisher = {Zenodo}, + doi = {10.5281/zenodo.15114385}, + url = {https://doi.org/10.5281/zenodo.15114385}, + version = {Firedrake\_20250331.0}, } @article{Geuzaine2009, - title = { - {Gmsh}: {A} 3-D finite element mesh generator with built-in pre-and - post-processing facilities - }, - author = {Geuzaine, Christophe and Remacle, Jean-Fran{\c{c}}ois}, - year = 2009, - journal = {International journal for numerical methods in engineering}, - publisher = {Wiley Online Library}, - volume = 79, - number = 11, - pages = {1309--1331}, + title = { {Gmsh}: {A} 3-D finite element mesh generator with built-in + pre-and post-processing facilities }, + author = {Geuzaine, Christophe and Remacle, Jean-Fran{\c{c}}ois}, + year = 2009, + journal = {International journal for numerical methods in engineering}, + publisher = {Wiley Online Library}, + volume = 79, + number = 11, + pages = {1309--1331}, } @software{Grudge, - title = {{Grudge}: {A}n environment for Discontinuous Galerkin discretizations}, - author = {Andreas Kl\"{o}ckner and others}, - year = 2025, - url = {https://github.com/inducer/grudge}, - version = {hash:193100a}, + title = {{Grudge}: {A}n environment for Discontinuous Galerkin + discretizations}, + author = {Andreas Kl\"{o}ckner and others}, + year = 2025, + url = {https://github.com/inducer/grudge}, + version = {hash:193100a}, } @article{Grundmann1978, - title = {Invariant Integration Formulas for the N-Simplex by Combinatorial Methods}, - author = {A. Grundmann and H. M. M\"{o}ller}, - year = 1978, - journal = {{SIAM} Journal on Numerical Analysis}, - publisher = {Society for Industrial \& Applied Mathematics {(SIAM)}}, - volume = 15, - pages = {282--290}, - doi = {10.1137/0715019}, - issue = 2, + title = {Invariant Integration Formulas for the N-Simplex by Combinatorial + Methods}, + author = {A. Grundmann and H. M. M\"{o}ller}, + year = 1978, + journal = {{SIAM} Journal on Numerical Analysis}, + publisher = {Society for Industrial \& Applied Mathematics {(SIAM)}}, + volume = 15, + pages = {282--290}, + doi = {10.1137/0715019}, + issue = 2, } @book{Hesthaven2007, - title = {Nodal Discontinuous {Galerkin} Methods}, - author = {J. S. Hesthaven and T. Warburton}, - year = 2007, - publisher = {Springer Science \& Business Media}, - isbn = 9780387720678, + title = {Nodal Discontinuous {Galerkin} Methods}, + author = {J. S. Hesthaven and T. Warburton}, + year = 2007, + publisher = {Springer Science \& Business Media}, + isbn = 9780387720678, } @article{Isaac2020, - title = { - Recursive, Parameter-Free, Explicitly Defined Interpolation Nodes for - Simplices - }, - author = {Tobin Isaac}, - year = 2020, - journal = {{SIAM} Journal on Scientific Computing}, - publisher = {Society for Industrial \& Applied Mathematics {(SIAM)}}, - volume = 42, - pages = {A4046--A4062}, - doi = {10.1137/20m1321802}, - issue = 6, + title = { Recursive, Parameter-Free, Explicitly Defined Interpolation Nodes + for Simplices }, + author = {Tobin Isaac}, + year = 2020, + journal = {{SIAM} Journal on Scientific Computing}, + publisher = {Society for Industrial \& Applied Mathematics {(SIAM)}}, + volume = 42, + pages = {A4046--A4062}, + doi = {10.1137/20m1321802}, + issue = 6, } @article{Jaskowiec2021, - title = {High-order Symmetric Cubature Rules for Tetrahedra and Pyramids}, - author = {J. Ja\'{s}kowiec and N. Sukumar}, - year = 2021, - journal = {International Journal for Numerical Methods in Engineering}, - publisher = {Wiley}, - volume = 122, - pages = {148--171}, - doi = {10.1002/nme.6528}, - issue = 1, + title = {High-order Symmetric Cubature Rules for Tetrahedra and Pyramids}, + author = {J. Ja\'{s}kowiec and N. Sukumar}, + year = 2021, + journal = {International Journal for Numerical Methods in Engineering}, + publisher = {Wiley}, + volume = 122, + pages = {148--171}, + doi = {10.1002/nme.6528}, + issue = 1, } @article{Julia2018, - title = {Julia Subtyping: {A} Rational Reconstruction}, - author = { - Francesco Zappa Nardelli and Julia Belyakova and Artem Pelenitsyn and - Benjamin Chung and Jeff Bezanson and Jan Vitek - }, - year = {2018}, - journal = {Proceedings of the {ACM} on Programming Languages}, - publisher = {Association for Computing {Machinery} {(ACM)}}, - volume = {2}, - pages = {1--27}, - doi = {10.1145/3276483}, - url = {https://dl.acm.org/doi/10.1145/3276483}, - issue = {{OOPSLA}}, + title = {Julia Subtyping: {A} Rational Reconstruction}, + author = { Francesco Zappa Nardelli and Julia Belyakova and Artem Pelenitsyn + and Benjamin Chung and Jeff Bezanson and Jan Vitek }, + year = {2018}, + journal = {Proceedings of the {ACM} on Programming Languages}, + publisher = {Association for Computing {Machinery} {(ACM)}}, + volume = {2}, + pages = {1--27}, + doi = {10.1145/3276483}, + url = {https://dl.acm.org/doi/10.1145/3276483}, + issue = {{OOPSLA}}, } @software{Pymbolic, - title = {{Pymbolic}: {A} package to do symbolic operations for code generation}, - author = { - Kloeckner, Andreas and Wala, Matt and Fernando, Isuru and Kulkarni, Kaushik - and Fikl, Alex and Weiner, Zach and Kempf, Dominic and Ham, David A. and - Mitchell, Lawrence and Wilcox, Lucas C and Diener, Matthias and Kapyshin, - Pavlo and Raksi, Reno and Gibson, Thomas H. - }, - year = 2024, - month = dec, - publisher = {Zenodo}, - doi = {10.5281/zenodo.14526145}, - url = {https://doi.org/10.5281/zenodo.14526145}, - version = {v2024.2.2}, + title = {{Pymbolic}: {A} package to do symbolic operations for code + generation}, + author = { Kloeckner, Andreas and Wala, Matt and Fernando, Isuru and + Kulkarni, Kaushik and Fikl, Alex and Weiner, Zach and Kempf, + Dominic and Ham, David A. and Mitchell, Lawrence and Wilcox, Lucas + C and Diener, Matthias and Kapyshin, Pavlo and Raksi, Reno and + Gibson, Thomas H. }, + year = 2024, + month = dec, + publisher = {Zenodo}, + doi = {10.5281/zenodo.14526145}, + url = {https://doi.org/10.5281/zenodo.14526145}, + version = {v2024.2.2}, } @software{PyNucleus, - title = { - {PyNucleus}: {A} finite element code that specifically targets nonlocal - operators - }, - author = {Glusa, Christian}, - year = 2021, - url = {https://github.com/sandialabs/PyNucleus}, - institution = {Sandia National Lab (SNL-NM), Albuquerque, NM (United States)}, + title = { {PyNucleus}: {A} finite element code that specifically targets + nonlocal operators }, + author = {Glusa, Christian}, + year = 2021, + url = {https://github.com/sandialabs/PyNucleus}, + institution = {Sandia National Lab (SNL-NM), Albuquerque, NM (United States) + }, } @software{Pytential, - title = {{Pytential}: {E}valuate layer and volume potentials accurately}, - author = {Andreas Kloeckner and others}, - year = 2025, - month = may, - url = {https://github.com/inducer/pytential}, - version = {hash:7395b97f}, + title = {{Pytential}: {E}valuate layer and volume potentials accurately}, + author = {Andreas Kloeckner and others}, + year = 2025, + month = may, + url = {https://github.com/inducer/pytential}, + version = {hash:7395b97f}, } @software{Python_AST, - title = {{ast} --- Abstract Syntax Trees}, - author = {{Python Software Foundation}}, - year = 2024, - url = {https://docs.python.org/3/library/ast.html}, + title = {{ast} --- Abstract Syntax Trees}, + author = {{Python Software Foundation}}, + year = 2024, + url = {https://docs.python.org/3/library/ast.html}, } @software{QuadPy, - title = {{QuadPy}: {N}umerical integration (quadrature, cubature) in {Python}}, - author = { - Nico Schl\"{o}mer and Nick Papior and Darius Arnold and Jan Blechta and - Rasmus Zetter - }, - year = 2021, - publisher = {Zenodo}, - doi = {10.5281/zenodo.5541216}, - version = {v0.16.10}, + title = {{QuadPy}: {N}umerical integration (quadrature, cubature) in {Python + }}, + author = { Nico Schl\"{o}mer and Nick Papior and Darius Arnold and Jan + Blechta and Rasmus Zetter }, + year = 2021, + publisher = {Zenodo}, + doi = {10.5281/zenodo.5541216}, + version = {v0.16.10}, } @software{StartUpDG, - title = {{StartUpDG.jl}: {R}eference elements and physical meshes for {DG}}, - author = { - Jesse Chan and David Knapp and Mason McCallum and Hendrik Ranocha and - Vincent X. Wang and Johannes Markert - }, - year = 2024, - url = {https://github.com/jlchan/StartUpDG.jl}, - version = {v1.1.5}, + title = {{StartUpDG.jl}: {R}eference elements and physical meshes for {DG}}, + author = { Jesse Chan and David Knapp and Mason McCallum and Hendrik Ranocha + and Vincent X. Wang and Johannes Markert }, + year = 2024, + url = {https://github.com/jlchan/StartUpDG.jl}, + version = {v1.1.5}, } @software{SymEngine, - title = {{SymEngine}: {A} fast symbolic manipulation library written in C++}, - author = { - O. \v{C}ert\'{\i}k and I. Fernando and S. Garg and T. Rathnayake and others - }, - year = 2013, - url = {https://github.com/symengine/symengine}, + title = {{SymEngine}: {A} fast symbolic manipulation library written in C++}, + author = { O. \v{C}ert\'{\i}k and I. Fernando and S. Garg and T. Rathnayake + and others }, + year = 2013, + url = {https://github.com/symengine/symengine}, } @article{SymPy, - title = {{SymPy}: {S}ymbolic computing in Python}, - author = { - Meurer, Aaron and Smith, Christopher P and Paprocki, Mateusz and - \v{C}ert\'{\i}k, Ond\v{r}ej and Kirpichev, Sergey B and Rocklin, Matthew - and Kumar, AMiT and Ivanov, Sergiu and Moore, Jason K and Singh, Sartaj and - others - }, - year = 2017, - journal = {PeerJ Computer Science}, - publisher = {PeerJ Inc.}, - volume = 3, - pages = {e103}, + title = {{SymPy}: {S}ymbolic computing in Python}, + author = { Meurer, Aaron and Smith, Christopher P and Paprocki, Mateusz and + \v{C}ert\'{\i}k, Ond\v{r}ej and Kirpichev, Sergey B and Rocklin, + Matthew and Kumar, AMiT and Ivanov, Sergiu and Moore, Jason K and + Singh, Sartaj and others }, + year = 2017, + journal = {PeerJ Computer Science}, + publisher = {PeerJ Inc.}, + volume = 3, + pages = {e103}, } @book{Trefethen2020, - title = {Approximation Theory and Approximation Practice}, - author = {Lloyd N. Trefethen}, - year = 2020, - publisher = {Society for Industrial and Applied Mathematics}, - isbn = 9781611975932, + title = {Approximation Theory and Approximation Practice}, + author = {Lloyd N. Trefethen}, + year = 2020, + publisher = {Society for Industrial and Applied Mathematics}, + isbn = 9781611975932, } @article{Vioreanu2014, - title = {Spectra of Multiplication Operators as a Numerical Tool}, - author = {B. Vioreanu and V. Rokhlin}, - year = 2014, - journal = {{SIAM} Journal on Scientific Computing}, - publisher = {Society for Industrial \& Applied Mathematics {(SIAM)}}, - volume = 36, - pages = {A267--a288}, - doi = {10.1137/110860082}, - issue = 1, + title = {Spectra of Multiplication Operators as a Numerical Tool}, + author = {B. Vioreanu and V. Rokhlin}, + year = 2014, + journal = {{SIAM} Journal on Scientific Computing}, + publisher = {Society for Industrial \& Applied Mathematics {(SIAM)}}, + volume = 36, + pages = {A267--a288}, + doi = {10.1137/110860082}, + issue = 1, } @article{Waldvogel2006, - title = { - Fast Construction of the {Fej\'{e}r} and {Clenshaw–Curtis} Quadrature Rules - }, - author = {J. Waldvogel}, - year = 2006, - journal = {BIT Numerical Mathematics}, - publisher = {Springer Science and Business Media {LLC}}, - volume = 46, - pages = {195--202}, - doi = {10.1007/s10543-006-0045-4}, - issue = 1, + title = { Fast Construction of the {Fej\'{e}r} and {Clenshaw–Curtis} + Quadrature Rules }, + author = {J. Waldvogel}, + year = 2006, + journal = {BIT Numerical Mathematics}, + publisher = {Springer Science and Business Media {LLC}}, + volume = 46, + pages = {195--202}, + doi = {10.1007/s10543-006-0045-4}, + issue = 1, } @article{Warburton2007, - title = {An Explicit Construction of Interpolation Nodes on the Simplex}, - author = {T. Warburton}, - year = 2007, - journal = {Journal of Engineering Mathematics}, - publisher = {Springer Nature}, - volume = 56, - pages = {247--262}, - doi = {10.1007/s10665-006-9086-6}, - issue = 3, + title = {An Explicit Construction of Interpolation Nodes on the Simplex}, + author = {T. Warburton}, + year = 2007, + journal = {Journal of Engineering Mathematics}, + publisher = {Springer Nature}, + volume = 56, + pages = {247--262}, + doi = {10.1007/s10665-006-9086-6}, + issue = 3, } @article{Wicaksono2025, - title = {{Minterpy}: {M}ultivariate polynomial interpolation in {Python}}, - author = { - Damar Wicaksono and Uwe Hernandez Acosta and Sachin Krishnan Thekke Veettil - and Jannik Kissinger and Michael Hecht - }, - year = 2025, - journal = {Journal of Open Source Software}, - publisher = {The Open Journal}, - volume = 10, - number = 109, - pages = 7702, - doi = {10.21105/joss.07702}, - url = {https://doi.org/10.21105/joss.07702}, + title = {{Minterpy}: {M}ultivariate polynomial interpolation in {Python}}, + author = { Damar Wicaksono and Uwe Hernandez Acosta and Sachin Krishnan + Thekke Veettil and Jannik Kissinger and Michael Hecht }, + year = 2025, + journal = {Journal of Open Source Software}, + publisher = {The Open Journal}, + volume = 10, + number = 109, + pages = 7702, + doi = {10.21105/joss.07702}, + url = {https://doi.org/10.21105/joss.07702}, } @article{Witherden2015, - title = { - On the Identification of Symmetric Quadrature Rules for Finite Element - Methods - }, - author = {F. D. Witherden and P. E. Vincent}, - year = 2015, - journal = {Computers \& Mathematics With Applications}, - publisher = {Elsevier {BV}}, - volume = 69, - pages = {1232--1241}, - doi = {10.1016/j.camwa.2015.03.017}, - issue = 10, + title = { On the Identification of Symmetric Quadrature Rules for Finite + Element Methods }, + author = {F. D. Witherden and P. E. Vincent}, + year = 2015, + journal = {Computers \& Mathematics With Applications}, + publisher = {Elsevier {BV}}, + volume = 69, + pages = {1232--1241}, + doi = {10.1016/j.camwa.2015.03.017}, + issue = 10, } @article{Xiao2010, - title = { - A Numerical Algorithm for the Construction of Efficient Quadrature Rules in - Two and Higher Dimensions - }, - author = {H. Xiao and Z. Gimbutas}, - year = 2010, - journal = {Computers \& Mathematics With Applications}, - publisher = {Elsevier {BV}}, - volume = 59, - pages = {663--676}, - doi = {10.1016/j.camwa.2009.10.027}, - issue = 2, + title = { A Numerical Algorithm for the Construction of Efficient Quadrature + Rules in Two and Higher Dimensions }, + author = {H. Xiao and Z. Gimbutas}, + year = 2010, + journal = {Computers \& Mathematics With Applications}, + publisher = {Elsevier {BV}}, + volume = 59, + pages = {663--676}, + doi = {10.1016/j.camwa.2009.10.027}, + issue = 2, +} + +@article{Orszag1980, + title = {Spectral methods for problems in complex geometries}, + volume = {37}, + issn = {0021-9991}, + url = {https://www.sciencedirect.com/science/article/pii/0021999180900054}, + doi = {10.1016/0021-9991(80)90005-4}, + abstract = {The properties of spectral methods are surveyed and their + extension to solve problems in complex geometries is developed. A + new iteration procedure is introduced to solve efficiently the + full matrix equations resulting from spectral approximations to + nonconstant coefficient boundary-value problems in complex + geometries. It is shown that the work required to solve these + spectral equations exceeds that of solving the lowest-order + finite-difference approximation to the same problem by only O(N + log N).}, + number = {1}, + urldate = {2025-09-19}, + journal = {Journal of Computational Physics}, + author = {Orszag, Steven A}, + month = aug, + year = {1980}, + pages = {70--92}, + file = {ScienceDirect Full Text + PDF:/home/andreas/Zotero/storage/YKEGQJ4D/Orszag - 1980 - Spectral + methods for problems in complex + geometries.pdf:application/pdf;ScienceDirect + Snapshot:/home/andreas/Zotero/storage/8RUFDWUR/0021999180900054.html:text/html + }, } diff --git a/doc/paper/paper.md b/doc/paper/paper.md index 66ec0d0..b74bacb 100644 --- a/doc/paper/paper.md +++ b/doc/paper/paper.md @@ -43,39 +43,37 @@ bibliography: paper.bib `modepy` is a Python library for defining reference elements, equipping them with appropriate approximation spaces, and numerically performing calculus -operations (derivatives, integrals) on those spaces. It is written in pure -modern Python 3, offering comprehensive type annotations and full documentation -with minimal runtime dependencies (NumPy being a primary dependency). - -`modepy` focuses on high-order accuracy -- given an element size $h$, this -refers to the asymptotic decay of the approximation error as $O(h^n)$, for $n -\ge 3$, assuming sufficient smoothness of the solution being approximated -- at -manageable computational cost. These capabilities enable High-Performance -Computing (HPC) approaches to solving integral and (partial) differential -equations. Those, in turn, can be used to model many phenomena in the physical -world, including fluid flow, electromagnetism, and solid mechanics. A popular -example of this is the finite element method (FEM), including its continuous -and discontinuous flavors, but general collocation, spectral, and Nyström -methods are also supported. As such, `modepy` has been used to construct FEM -solvers [@Grudge; @PyNucleus] and integral equation solvers [@Pytential] that -run on both CPUs and GPUs. - -`modepy` is licensed under the MIT license and available on GitHub at -. Versions dating back to 2013 maintain -broad compatibility with older versions of Python, including Python 2.7, -ensuring broad accessibility. +operations (derivatives, integrals) on those spaces. It is written in pure, +type-annotated Python 3, offering comprehensive documentation +and minimal runtime dependencies (mainly NumPy). + +`modepy` focuses on high-order accuracy --- given an element size $h$, this refers to +the asymptotic decay of the approximation error as $O(h^n)$, for $n \ge 3$, assuming +sufficient smoothness of the solution being approximated. For a problem in $d$ +dimensions, the number of unknowns scales as $O(h^{-d})$. Therefore, if accuracy is +desired at manageable cost, high-order methods are crucial. + +A popular approach (e.g. in the finite element method, or FEM) for accurate +approximation of functions on geometrically complex domains is the use of *unstructured +discretizations*, which represent the geometry as a typically disjoint union (a "mesh") +of primitive geometric shapes, most often simplices and quadrilaterals. Given the means +to perform calculus operations on these *reference elements* and mapping functions +from them to the *global* elements, calculus operations become available on the entire +domain. These primitives are chiefly useful in the numerical solution of integral and +(partial) differential equations. Additional applications include computer graphics, +CAD, and robotics. Those, in turn, can be used to model many physical phenomena, +including fluid flow, electromagnetism, and solid mechanics. `modepy` has been used to +construct FEM solvers [@Grudge; @PyNucleus] and integral equation solvers [@Pytential] +that run on both CPUs and GPUs. # Statement of need -High-order accurate calculus operations on unstructured discretizations are -crucial to differential and integral equations solvers, as well as computer -graphics and Computer-Aided Design (CAD) applications. This functionality is -often embedded in an ad-hoc manner in larger codes, restricting scope and -reusability. `modepy` addresses this need by providing a reusable, -generalizable, and composable implementation. +`modepy`'s functionality, as outlined above, is often embedded in an ad-hoc manner +in larger codes, restricting scope and reusability. `modepy` addresses this need by +providing a reusable, generalizable, and composable implementation. -There are several other candidate libraries in the literature with similar -goals, but important limitations. FInAT [@FInAT] (and the earlier [@FIAT]) is a +There are several other libraries in the literature with similar +goals, but important limitations and differences. FInAT [@FInAT] (and the earlier [@FIAT]) is a offers reference elements and basis functions, but is tightly coupled to the FENiCS/Firedrake ecosystem. Similarly, `StartUpDG.jl` [@StartUpDG] has a focus on the needs of discontinuous Galerkin FEM in the Trixi framework. `QuadPy` @@ -83,33 +81,33 @@ on the needs of discontinuous Galerkin FEM in the Trixi framework. `QuadPy` and lacks `modepy`'s composability. `minterpy` [@Wicaksono2025], meanwhile, deals exclusively with polynomial interpolation, with a focus on sparse grids. -To facilitate separation of concerns from the chosen numerical methods, -`modepy` adopts a two-pronged approach that allows flexibility in trading -simulation fidelity against computation cost. First, it usually suffices for -operations to be represented as data in matrix or tabular form, so that no -actual execution of `modepy` code is needed in a cost-constrained setting. For -example, nodes and bilinear forms on reference elements can generally be -pre-computed and tabulated. - -However, this data-driven approach falls short in the setting of tensor product -elements. In this case, for example, a mass matrix $\boldsymbol M$ permits a -Kronecker product factorization that significantly reduces the asymptotic -complexity of a matrix-vector product in higher dimensions. `modepy` exposes -additional information that allows reshaping degrees of freedom arrays to take -advantage of such factorizations. Another prominent exception is the evaluation -of basis functions. To facilitate efficient evaluation, `modepy` allows its -functions to be "traced", in the sense of lazy or deferred evaluation. The -resulting expression graph is represented by the `pymbolic` [@Pymbolic] -software library, that can interoperate with Python ASTs [@Python_AST], SymPy -[@SymPy], SymEngine [@SymEngine], etc., for straightforward code generation. +The solvers served by `modepy` typically have tight cost constraints, often adopting HPC +techniques (GPU, MPI, etc.). To facilitate separation of implementation/high-performance +concerns from the core numerical method, `modepy` adopts a two-pronged approach. First, +if it suffices to represent operations as data in matrix or tabular form, execution +of `modepy` code is not needed in a cost-constrained setting. For example, nodes and +bilinear forms on reference elements can generally be pre-computed and tabulated. +Second, if this tabulation approach falls short, `modepy` provides data structures to +reveal additional internal structure. + +Tensor product elements provide an example of this. In this instance, many operator +matrices permit a Kronecker product factorization that significantly reduces the +asymptotic complexity of a matrix-vector product in higher dimensions [@Orszag1980]. +`modepy` exposes functionality that allows reshaping degrees of freedom arrays to take +advantage of such factorizations. Another prominent example is the evaluation of basis +functions at points known only at runtime. To facilitate efficient evaluation, `modepy` +allows its functions to be "traced", in the sense of lazy or deferred evaluation. +The resulting expression graph is represented by the `pymbolic` [@Pymbolic] software +library, that can interoperate with Python ASTs [@Python_AST], SymPy [@SymPy], SymEngine +[@SymEngine], etc., for straightforward generation of high-performance code. # Overview -The high-level concepts available in `modepy` are shapes (i.e. a reference -domain), modes (i.e. the basis functions), and nodes (i.e. the degrees of +The high-level concepts available in `modepy` are shapes (i.e. reference +domains), modes (i.e. the basis functions), and nodes (i.e. the degrees of freedom). These are implemented in a user-extensible fashion using the `singledispatch` mechanism, with inspiration taken from common idiomatic usage -in Julia [@Bezanson2017]. +in Julia [@Bezanson2017]. ## Shapes @@ -117,8 +115,8 @@ The geometry of a reference element is described in `modepy` by the `Shape` class. Built-in support exists for `Simplex` and `Hypercube` geometries, encompassing the commonly used interval, triangle, tetrahedron, quadrilateral, and hexahedral shapes (see \autoref{FigureSimplices}). `TensorProductShape` can -be used to compose additional shapes, that are useful in specific applications -and sometimes generated by meshing software (such as `gmsh` [@Geuzaine2009]). +be used to compose additional shapes (e.g. prims, as generated by, +e.g. `gmsh` [@Geuzaine2009]). ![Domains corresponding to the one-, two-, and three-dimensional simplices.](images/simplices.png){#FigureSimplices width="80%"} @@ -131,14 +129,9 @@ the reference element domain, and no specific choice of basis. Predefined choices include the `PN` space, containing polynomials of total degree at most `order`, and the `QN` space, containing polynomials of maximum degree at most `order`. As with shapes, these spaces can be combined using -`TensorProductSpace`. - -The basis is provided in the form of a `Basis` class, which allows access to -the basis functions, their derivatives (per axis) and opaque `mode_ids`. As -noted before, the basis functions and their derivatives can be evaluated -directly or can be traced using the `pymbolic` library to aid in code -generation. Various standard basis functions are provided, such as the -monomials, general Jacobi polynomials, and the +`TensorProductSpace`. A `Basis` objects is available separately, +giving access to basis functions and their derivatives, for, e.g., +the monomials, general Jacobi polynomials, and the Proriol-Koornwinder-Dubiner-Owens (PKDO) basis from [@Dubiner1991] (see \autoref{FigurePKDO}). @@ -146,17 +139,11 @@ Proriol-Koornwinder-Dubiner-Owens (PKDO) basis from [@Dubiner1991] (see ## Nodes -A final component in an FEM discretization (or 'Ciarlet triple' [@Brenner2007, -Section 3.1]) is a set of 'degrees of freedom' ('DOFs') that uniquely define a -certain function in the span of a basis. `modepy` provides easy access to modal +A final component in an FEM discretization [@Brenner2007, +Section 3.1] is a set of 'degrees of freedom' ('DOFs') that uniquely identify a +certain function in the span of a basis. `modepy` supports modal DOFs (i.e. basis coefficients) and nodal DOFs (i.e. function or derivative -values at a point). As equispaced nodal DOFs are not usable in the high-order -setting [@Trefethen2020], `modepy` offers access to many types of nodes with -better properties for each reference element. It can also construct custom -nodes sets for a given set of basis functions in 2D via a procedure based on -eigenvalues of multiplication operators [@Vioreanu2014]. - -On simplices, the "warp-and-blend" nodes [@Warburton2007] are available, and on +values at a point). On simplices, the "warp-and-blend" nodes [@Warburton2007] are available, and on the hypercube, standard tensor product nodes are constructed from one-dimensional Legendre-Gauss(-Lobatto) nodes. `modepy` can also directly interoperate with the `recursivenodes` library described in [@Isaac2020], which @@ -164,39 +151,20 @@ offers additional well-conditioned nodes on the simplex. ## Quadrature -Besides the standard building blocks above, `modepy` also offers a wide array -of quadrature rules that can be used on each reference element. The quadrature -rules are provided as implementations of the `Quadrature` superclass. For the -interval, a broad selection of quadrature rules are present: Clenshaw--Curtis, -Fejér, and Jacobi-Gauss(-Lobatto). Additionally, several state-of-the-art -higher-dimensional rules are available: - -* Grundmann--Möller [@Grundmann1978] rules for the $n$-simplex. -* Vioreanu--Rokhlin [@Vioreanu2014] rules for the two- and - three-dimensional simplex (see \autoref{FigureQuadrature}). -* Xiao--Gimbutas [@Xiao2010] rules for the two- and three-dimensional - simplex. -* Jáskowiec--Sukumar [@Jaskowiec2021] rules for the tetrahedron. -* Witherden--Vincent [@Witherden2015] rules for the two- and - three-dimensional hypercube (see \autoref{FigureQuadrature}). +`modepy` also offers a wide array of quadrature rules that can be used on each +reference element. For the interval, Clenshaw--Curtis, Fejér, and Jacobi-Gauss(-Lobatto) +are on offer. Many more state-of-the-art rules are available, typically up to high +order [@Grundmann1978;@Vioreanu2014;@Xiao2010;@Jaskowiec2021;@Witherden2015] (see +\autoref{FigureQuadrature}). There is also functionality [@Vioreanu2014] to allow +constructing novel quadratures on a given domain. ![(left) Vioreanu--Rokhlin quadrature points of order 11 and (right) Witherden--Vincent quadrature points of order 11.](images/quadrature_rules.png){#FigureQuadrature width="50%"} -Most of the rules for simplices and hypercubes are available to high order -(i.e. $\ge 20$). However, `modepy` also has functionality that allows -constructing desired quadrature nodes on a given domain. This can be found in -`modepy.quadrature.construction`. - ## Matrices -While `modepy` is sufficiently flexible to accommodate different applications, -it also provides functionality specific to FEM needs. In particular, it allows -constructing a variety of matrix operators that define necessary resampling -operators and bilinear forms used in FEM codes. A specific bilinear form matrix -can be obtained using `nodal_quadrature_bilinear_form_matrix`, which requires -providing the usual test functions, trial functions and corresponding nodes for -the input and output. This allows constructing very general families of -bilinear forms with or without oversampling. +`modepy`'s functionality is rounded out by various tabulation/matrix generation +functions, including the ability to tabulate operator matrices for fairly general +bilinear forms. # Acknowledgements From bb9bc54f354f1faa814bc9669ef3661e989a1605 Mon Sep 17 00:00:00 2001 From: Alexandru Fikl Date: Fri, 19 Sep 2025 19:21:47 +0300 Subject: [PATCH 11/15] chore: rename typos.toml --- doc/paper/{_typos.toml => typos.toml} | 0 1 file changed, 0 insertions(+), 0 deletions(-) rename doc/paper/{_typos.toml => typos.toml} (100%) diff --git a/doc/paper/_typos.toml b/doc/paper/typos.toml similarity index 100% rename from doc/paper/_typos.toml rename to doc/paper/typos.toml From c916db29ab0cb442206107b9b52b8ce86342125d Mon Sep 17 00:00:00 2001 From: Alexandru Fikl Date: Fri, 19 Sep 2025 19:55:26 +0300 Subject: [PATCH 12/15] feat(paper): small rephrasing and reformatting --- doc/paper/paper.md | 154 ++++++++++++++++++++++----------------------- 1 file changed, 74 insertions(+), 80 deletions(-) diff --git a/doc/paper/paper.md b/doc/paper/paper.md index b74bacb..0ada3d9 100644 --- a/doc/paper/paper.md +++ b/doc/paper/paper.md @@ -41,54 +41,53 @@ bibliography: paper.bib # Summary -`modepy` is a Python library for defining reference elements, equipping them -with appropriate approximation spaces, and numerically performing calculus -operations (derivatives, integrals) on those spaces. It is written in pure, -type-annotated Python 3, offering comprehensive documentation -and minimal runtime dependencies (mainly NumPy). - -`modepy` focuses on high-order accuracy --- given an element size $h$, this refers to -the asymptotic decay of the approximation error as $O(h^n)$, for $n \ge 3$, assuming -sufficient smoothness of the solution being approximated. For a problem in $d$ -dimensions, the number of unknowns scales as $O(h^{-d})$. Therefore, if accuracy is -desired at manageable cost, high-order methods are crucial. - -A popular approach (e.g. in the finite element method, or FEM) for accurate -approximation of functions on geometrically complex domains is the use of *unstructured -discretizations*, which represent the geometry as a typically disjoint union (a "mesh") -of primitive geometric shapes, most often simplices and quadrilaterals. Given the means +`modepy` is a Python library for defining reference elements, equipping them with +appropriate approximation spaces, and numerically performing calculus operations +(derivatives, integrals) on those spaces. It is written in pure, type-annotated Python +3, offering comprehensive documentation and minimal runtime dependencies (mainly NumPy). + +`modepy` focuses on high-order accuracy --- given an element size $h$, this refers +to the asymptotic decay of the approximation error as $O(h^n)$, for $n \ge 3$, +assuming sufficient smoothness of the solution being approximated. For a problem +in $d$ dimensions, the number of unknowns scales as $O(h^{-d})$. Therefore, if +accuracy is desired at manageable cost, high-order methods are crucial. + +A popular approach for accurate approximation of functions on geometrically complex +domains is the use of *unstructured discretizations*, e.g. in the Finite Element +Method (FEM). The geometry is typically represented as a disjoint union (a "mesh") of +primitive geometric shapes, most often simplices and quadrilaterals. Given the means to perform calculus operations on these *reference elements* and mapping functions from them to the *global* elements, calculus operations become available on the entire -domain. These primitives are chiefly useful in the numerical solution of integral and +domain. These primitives are chiefly useful in the numerical solution of integral and (partial) differential equations. Additional applications include computer graphics, -CAD, and robotics. Those, in turn, can be used to model many physical phenomena, -including fluid flow, electromagnetism, and solid mechanics. `modepy` has been used to -construct FEM solvers [@Grudge; @PyNucleus] and integral equation solvers [@Pytential] -that run on both CPUs and GPUs. +Computer Aided Design (CAD), and robotics. Those, in turn, can be used to model many +physical phenomena, including fluid flow, electromagnetism, and solid mechanics. +`modepy` has been used to construct FEM solvers [@Grudge; @PyNucleus] and integral +equation solvers [@Pytential] that run on both CPUs and GPUs. # Statement of need -`modepy`'s functionality, as outlined above, is often embedded in an ad-hoc manner -in larger codes, restricting scope and reusability. `modepy` addresses this need by -providing a reusable, generalizable, and composable implementation. +The functionality outlined above is often embedded in an ad-hoc manner in +larger codes, restricting scope and reusability. `modepy` addresses this +need by providing a reusable, generalizable, and composable implementation. -There are several other libraries in the literature with similar -goals, but important limitations and differences. FInAT [@FInAT] (and the earlier [@FIAT]) is a -offers reference elements and basis functions, but is tightly coupled to the -FENiCS/Firedrake ecosystem. Similarly, `StartUpDG.jl` [@StartUpDG] has a focus -on the needs of discontinuous Galerkin FEM in the Trixi framework. `QuadPy` +There are several other libraries in the literature with similar goals, but +important differences and limitations. FInAT [@FInAT] (and the earlier FIAT +[@FIAT]) offers reference elements and basis functions, but is tightly coupled +to the FEniCS/Firedrake ecosystem. Similarly, `StartUpDG.jl` [@StartUpDG] has a +focus on the needs of discontinuous Galerkin FEM in the Trixi framework. `QuadPy` [@QuadPy] provides access to quadrature rules, but it is no longer open source -and lacks `modepy`'s composability. `minterpy` [@Wicaksono2025], meanwhile, -deals exclusively with polynomial interpolation, with a focus on sparse grids. - -The solvers served by `modepy` typically have tight cost constraints, often adopting HPC -techniques (GPU, MPI, etc.). To facilitate separation of implementation/high-performance -concerns from the core numerical method, `modepy` adopts a two-pronged approach. First, -if it suffices to represent operations as data in matrix or tabular form, execution -of `modepy` code is not needed in a cost-constrained setting. For example, nodes and -bilinear forms on reference elements can generally be pre-computed and tabulated. -Second, if this tabulation approach falls short, `modepy` provides data structures to -reveal additional internal structure. +and lacks `modepy`'s composability. `minterpy` [@Wicaksono2025], meanwhile, deals +exclusively with polynomial interpolation, with a focus on sparse grids. + +The solvers served by `modepy` typically have tight cost constraints, often adopting +HPC techniques (GPU, MPI, etc.). To facilitate separation of implementation and +high-performance concerns from the core numerical method, `modepy` adopts a two-pronged +approach. First, if it suffices to represent operations as data in matrix or tabular +form, execution of `modepy` code is not needed in a cost-constrained setting. For +example, nodes and bilinear forms on reference elements can generally be pre-computed +and tabulated. Second, if this tabulation approach falls short, `modepy` provides +data structures to reveal additional internal structure. Tensor product elements provide an example of this. In this instance, many operator matrices permit a Kronecker product factorization that significantly reduces the @@ -103,68 +102,63 @@ library, that can interoperate with Python ASTs [@Python_AST], SymPy [@SymPy], S # Overview -The high-level concepts available in `modepy` are shapes (i.e. reference -domains), modes (i.e. the basis functions), and nodes (i.e. the degrees of -freedom). These are implemented in a user-extensible fashion using the -`singledispatch` mechanism, with inspiration taken from common idiomatic usage -in Julia [@Bezanson2017]. +The high-level concepts available in `modepy` are shapes (i.e. reference domains), +modes (i.e. the basis functions), and nodes (i.e. the degrees of freedom). These are +implemented in a user-extensible fashion using the `singledispatch` mechanism, with +inspiration taken from common idiomatic usage in Julia [@Bezanson2017]. ## Shapes -The geometry of a reference element is described in `modepy` by the `Shape` -class. Built-in support exists for `Simplex` and `Hypercube` geometries, -encompassing the commonly used interval, triangle, tetrahedron, quadrilateral, -and hexahedral shapes (see \autoref{FigureSimplices}). `TensorProductShape` can -be used to compose additional shapes (e.g. prims, as generated by, -e.g. `gmsh` [@Geuzaine2009]). +The geometry of a reference element is described in `modepy` by the `Shape` class. +Built-in support exists for `Simplex` and `Hypercube` geometries, encompassing +the commonly used interval, triangle, tetrahedron, quadrilateral, and hexahedral +shapes (see \autoref{FigureSimplices}). `TensorProductShape` can be used to compose +additional shapes (e.g. prims, as generated by, e.g. `gmsh` [@Geuzaine2009]). ![Domains corresponding to the one-, two-, and three-dimensional simplices.](images/simplices.png){#FigureSimplices width="80%"} ## Modes and Spaces -To perform calculus operations, each reference element can be equipped with a -function space described by the `FunctionSpace` class. These represent a -finite-dimensional space of functions $\phi_i: D \to \mathbb{R}$, where $D$ is -the reference element domain, and no specific choice of basis. Predefined -choices include the `PN` space, containing polynomials of total degree at most -`order`, and the `QN` space, containing polynomials of maximum degree at most -`order`. As with shapes, these spaces can be combined using -`TensorProductSpace`. A `Basis` objects is available separately, -giving access to basis functions and their derivatives, for, e.g., -the monomials, general Jacobi polynomials, and the -Proriol-Koornwinder-Dubiner-Owens (PKDO) basis from [@Dubiner1991] (see -\autoref{FigurePKDO}). +To perform calculus operations, each reference element can be equipped with a function +space described by the `FunctionSpace` class. These represent a finite-dimensional +space of functions $\phi_i: D \to \mathbb{R}$, where $D$ is the reference element +domain, and no specific choice of basis. Predefined choices include the `PN` space, +containing polynomials of total degree at most $N$, and the `QN` space, containing +polynomials of maximum degree at most $N$. As with shapes, these spaces can be +combined using `TensorProductSpace`. A `Basis` objects is available separately, +giving access to basis functions and their derivatives, for, e.g., the monomials, +general Jacobi polynomials, and the Proriol-Koornwinder-Dubiner-Owens (PKDO) basis +from [@Dubiner1991] (see \autoref{FigurePKDO}). ![PKDO basis functions for the triangle.](images/pkdo-2d.png){#FigurePKDO width="80%"} ## Nodes -A final component in an FEM discretization [@Brenner2007, -Section 3.1] is a set of 'degrees of freedom' ('DOFs') that uniquely identify a -certain function in the span of a basis. `modepy` supports modal -DOFs (i.e. basis coefficients) and nodal DOFs (i.e. function or derivative -values at a point). On simplices, the "warp-and-blend" nodes [@Warburton2007] are available, and on -the hypercube, standard tensor product nodes are constructed from -one-dimensional Legendre-Gauss(-Lobatto) nodes. `modepy` can also directly -interoperate with the `recursivenodes` library described in [@Isaac2020], which -offers additional well-conditioned nodes on the simplex. +A final component in an FEM discretization [@Brenner2007, Section 3.1] is a set of +'degrees of freedom' ('DOFs') that uniquely identify a certain function in the span of +a basis. `modepy` supports modal DOFs (i.e. basis coefficients) and nodal DOFs (i.e. +function or derivative values at a point). On simplices, the "warp-and-blend" nodes +[@Warburton2007] are available, and on the hypercube, standard tensor product nodes +are constructed from one-dimensional Legendre-Gauss(-Lobatto) nodes. `modepy` can also +directly interoperate with the `recursivenodes` library described in [@Isaac2020], +which offers additional well-conditioned nodes on the simplex. ## Quadrature -`modepy` also offers a wide array of quadrature rules that can be used on each -reference element. For the interval, Clenshaw--Curtis, Fejér, and Jacobi-Gauss(-Lobatto) -are on offer. Many more state-of-the-art rules are available, typically up to high -order [@Grundmann1978;@Vioreanu2014;@Xiao2010;@Jaskowiec2021;@Witherden2015] (see -\autoref{FigureQuadrature}). There is also functionality [@Vioreanu2014] to allow +`modepy` also offers a wide array of quadrature rules that can be used on each reference +element. For the interval, Clenshaw--Curtis, Fejér, and Jacobi-Gauss(-Lobatto) are +provided. Many more state-of-the-art rules are available, typically up to high order +$n > 20$ from [@Grundmann1978;@Vioreanu2014;@Xiao2010;@Jaskowiec2021;@Witherden2015] +(see \autoref{FigureQuadrature}). There is also functionality [@Vioreanu2014] to allow constructing novel quadratures on a given domain. ![(left) Vioreanu--Rokhlin quadrature points of order 11 and (right) Witherden--Vincent quadrature points of order 11.](images/quadrature_rules.png){#FigureQuadrature width="50%"} ## Matrices -`modepy`'s functionality is rounded out by various tabulation/matrix generation -functions, including the ability to tabulate operator matrices for fairly general -bilinear forms. +`modepy`'s functionality is rounded out by various tabulation and matrix +generation functions. This includes the ability to tabulate operator matrices +for fairly general bilinear forms used in FEM. # Acknowledgements From 576cc3e98738044dfaf9ba1bea672e8a7c66eb17 Mon Sep 17 00:00:00 2001 From: Alexandru Fikl Date: Fri, 19 Sep 2025 20:02:18 +0300 Subject: [PATCH 13/15] feat(paper): clean up and fix up bibliography --- doc/paper/paper.bib | 532 ++++++++++++++++++++++---------------------- 1 file changed, 267 insertions(+), 265 deletions(-) diff --git a/doc/paper/paper.bib b/doc/paper/paper.bib index 17b826b..c17d460 100644 --- a/doc/paper/paper.bib +++ b/doc/paper/paper.bib @@ -1,345 +1,347 @@ @article{Bezanson2017, - title = {Julia: {A} Fresh Approach to Numerical Computing}, - author = {Jeff Bezanson and Alan Edelman and Stefan Karpinski and Viral B. - Shah}, - year = 2017, - journal = {{SIAM} Review}, - publisher = {Society for Industrial \& Applied Mathematics {(SIAM)}}, - volume = 59, - pages = {65--98}, - doi = {10.1137/141000671}, - issue = 1, + title = {Julia: {A} Fresh Approach to Numerical Computing}, + author = {Jeff Bezanson and Alan Edelman and Stefan Karpinski and Viral B. Shah}, + year = 2017, + journal = {{SIAM} Review}, + publisher = {Society for Industrial \& Applied Mathematics {(SIAM)}}, + volume = 59, + pages = {65--98}, + doi = {10.1137/141000671}, + issue = 1, } @book{Brenner2007, - title = {The Mathematical Theory of Finite Element Methods}, - author = {Susanne Brenner and Ridgway Scott}, - year = 2007, - publisher = {Springer Science \& Business Media}, - isbn = 9780387759333, + title = {The Mathematical Theory of Finite Element Methods}, + author = {Susanne Brenner and Ridgway Scott}, + year = 2007, + publisher = {Springer Science \& Business Media}, + isbn = 9780387759333, } @article{Dubiner1991, - title = {Spectral Methods on Triangles and Other Domains}, - author = {M. Dubiner}, - year = 1991, - journal = {Journal of Scientific Computing}, - publisher = {Springer Science and Business Media {LLC}}, - volume = 6, - pages = {345--390}, - doi = {10.1007/bf01060030}, - issue = 4, + title = {Spectral Methods on Triangles and Other Domains}, + author = {M. Dubiner}, + year = 1991, + journal = {Journal of Scientific Computing}, + publisher = {Springer Science and Business Media {LLC}}, + volume = 6, + pages = {345--390}, + doi = {10.1007/bf01060030}, + issue = 4, } @software{FIAT, - title = {{Fiat}: 2025.4.0}, - author = { Mikl\'{o}s Homolya and Pablo Brubeck and David A. Ham and Robert - Kirby and Lawrence Mitchell and Jan Blechta and Marie E Rognes and - Garth N. Wells and Anders Logg and cyruscycheng21 and Thomas H. - Gibson and Johannes Ring and Ivan Yashchuk and Justincrum and - k-b-oelgaard and ksagiyam and Chris Richardson and Nico Schl\"{o} - mer and Reuben W. Nixon-Hill and Lizao Li and Aslak Bergersen and - Connor Ward and Nicholas Barton and Colin J Cotter and Patrick E. - Farrell and Jack S. Hale and FAznaran and Matthew Knepley and - Florian Rathgeber and Matthew Scroggs }, - year = 2025, - month = apr, - publisher = {Zenodo}, - doi = {10.5281/zenodo.15302339}, - url = {https://doi.org/10.5281/zenodo.15302339}, - version = {2025.4.0}, + title = {{FIAT}: 2025.4.0}, + author = { + Mikl\'{o}s Homolya and Pablo Brubeck and David A. Ham and Robert Kirby and + Lawrence Mitchell and Jan Blechta and Marie E Rognes and Garth N. Wells and + Anders Logg and cyruscycheng21 and Thomas H. Gibson and Johannes Ring and + Ivan Yashchuk and Justincrum and k-b-oelgaard and ksagiyam and Chris + Richardson and Nico Schl\"{o} mer and Reuben W. Nixon-Hill and Lizao Li and + Aslak Bergersen and Connor Ward and Nicholas Barton and Colin J Cotter and + Patrick E. Farrell and Jack S. Hale and FAznaran and Matthew Knepley and + Florian Rathgeber and Matthew Scroggs + }, + year = 2025, + month = apr, + publisher = {Zenodo}, + doi = {10.5281/zenodo.15302339}, + url = {https://doi.org/10.5281/zenodo.15302339}, + version = {2025.4.0}, } @software{FInAT, - title = {{FInAT}: {A} smarter library of finite elements}, - author = { David A. Ham and Mikl\'{o}s Homolya and Robert Kirby and Lawrence - Mitchell and Pablo Brubeck and cyruscycheng21 and FAznaran and - ksagiyam and Matthew Scroggs and Patrick E. Farrell and Justincrum - and Connor Ward and celdred and Thomas Bendall and Jack Betteridge - and FabianL1908 }, - year = 2025, - month = mar, - publisher = {Zenodo}, - doi = {10.5281/zenodo.15114385}, - url = {https://doi.org/10.5281/zenodo.15114385}, - version = {Firedrake\_20250331.0}, + title = {{FInAT}: {A} smarter library of finite elements}, + author = { + David A. Ham and Mikl\'{o}s Homolya and Robert Kirby and Lawrence Mitchell + and Pablo Brubeck and cyruscycheng21 and FAznaran and ksagiyam and Matthew + Scroggs and Patrick E. Farrell and Justincrum and Connor Ward and celdred + and Thomas Bendall and Jack Betteridge and FabianL1908 + }, + year = 2025, + month = mar, + publisher = {Zenodo}, + doi = {10.5281/zenodo.15114385}, + url = {https://doi.org/10.5281/zenodo.15114385}, + version = {Firedrake\_20250331.0}, } @article{Geuzaine2009, - title = { {Gmsh}: {A} 3-D finite element mesh generator with built-in - pre-and post-processing facilities }, - author = {Geuzaine, Christophe and Remacle, Jean-Fran{\c{c}}ois}, - year = 2009, - journal = {International journal for numerical methods in engineering}, - publisher = {Wiley Online Library}, - volume = 79, - number = 11, - pages = {1309--1331}, + title = { + {Gmsh}: {A} 3-D finite element mesh generator with built-in pre-and + post-processing facilities + }, + author = {Geuzaine, Christophe and Remacle, Jean-Fran{\c{c}}ois}, + year = 2009, + journal = {International Journal for Numerical Methods in Engineering}, + publisher = {Wiley Online Library}, + volume = 79, + number = 11, + doi = {10.1002/nme.2579}, + pages = {1309--1331}, } @software{Grudge, - title = {{Grudge}: {A}n environment for Discontinuous Galerkin - discretizations}, - author = {Andreas Kl\"{o}ckner and others}, - year = 2025, - url = {https://github.com/inducer/grudge}, - version = {hash:193100a}, + title = {{Grudge}: {A}n environment for Discontinuous {Galerkin} discretizations}, + author = {Andreas Kl\"{o}ckner and others}, + year = 2025, + url = {https://github.com/inducer/grudge}, + version = {hash:193100a}, } @article{Grundmann1978, - title = {Invariant Integration Formulas for the N-Simplex by Combinatorial - Methods}, - author = {A. Grundmann and H. M. M\"{o}ller}, - year = 1978, - journal = {{SIAM} Journal on Numerical Analysis}, - publisher = {Society for Industrial \& Applied Mathematics {(SIAM)}}, - volume = 15, - pages = {282--290}, - doi = {10.1137/0715019}, - issue = 2, + title = {Invariant Integration Formulas for the N-Simplex by Combinatorial Methods}, + author = {A. Grundmann and H. M. M\"{o}ller}, + year = 1978, + journal = {{SIAM} Journal on Numerical Analysis}, + publisher = {Society for Industrial \& Applied Mathematics {(SIAM)}}, + volume = 15, + pages = {282--290}, + doi = {10.1137/0715019}, + issue = 2, } @book{Hesthaven2007, - title = {Nodal Discontinuous {Galerkin} Methods}, - author = {J. S. Hesthaven and T. Warburton}, - year = 2007, - publisher = {Springer Science \& Business Media}, - isbn = 9780387720678, + title = {Nodal Discontinuous {Galerkin} Methods}, + author = {J. S. Hesthaven and T. Warburton}, + year = 2007, + publisher = {Springer Science \& Business Media}, + isbn = 9780387720678, } @article{Isaac2020, - title = { Recursive, Parameter-Free, Explicitly Defined Interpolation Nodes - for Simplices }, - author = {Tobin Isaac}, - year = 2020, - journal = {{SIAM} Journal on Scientific Computing}, - publisher = {Society for Industrial \& Applied Mathematics {(SIAM)}}, - volume = 42, - pages = {A4046--A4062}, - doi = {10.1137/20m1321802}, - issue = 6, + title = { + Recursive, Parameter-Free, Explicitly Defined Interpolation Nodes for + Simplices + }, + author = {Tobin Isaac}, + year = 2020, + journal = {{SIAM} Journal on Scientific Computing}, + publisher = {Society for Industrial \& Applied Mathematics {(SIAM)}}, + volume = 42, + pages = {A4046--a4062}, + doi = {10.1137/20m1321802}, + issue = 6, } @article{Jaskowiec2021, - title = {High-order Symmetric Cubature Rules for Tetrahedra and Pyramids}, - author = {J. Ja\'{s}kowiec and N. Sukumar}, - year = 2021, - journal = {International Journal for Numerical Methods in Engineering}, - publisher = {Wiley}, - volume = 122, - pages = {148--171}, - doi = {10.1002/nme.6528}, - issue = 1, + title = {High-order Symmetric Cubature Rules for Tetrahedra and Pyramids}, + author = {J. Ja\'{s}kowiec and N. Sukumar}, + year = 2021, + journal = {International Journal for Numerical Methods in Engineering}, + publisher = {Wiley}, + volume = 122, + pages = {148--171}, + doi = {10.1002/nme.6528}, + issue = 1, } @article{Julia2018, - title = {Julia Subtyping: {A} Rational Reconstruction}, - author = { Francesco Zappa Nardelli and Julia Belyakova and Artem Pelenitsyn - and Benjamin Chung and Jeff Bezanson and Jan Vitek }, - year = {2018}, - journal = {Proceedings of the {ACM} on Programming Languages}, - publisher = {Association for Computing {Machinery} {(ACM)}}, - volume = {2}, - pages = {1--27}, - doi = {10.1145/3276483}, - url = {https://dl.acm.org/doi/10.1145/3276483}, - issue = {{OOPSLA}}, + title = {Julia Subtyping: {A} Rational Reconstruction}, + author = { + Francesco Zappa Nardelli and Julia Belyakova and Artem Pelenitsyn and + Benjamin Chung and Jeff Bezanson and Jan Vitek + }, + year = 2018, + journal = {Proceedings of the {ACM} on Programming Languages}, + publisher = {Association for Computing {Machinery} {(ACM)}}, + volume = 2, + pages = {1--27}, + doi = {10.1145/3276483}, + url = {https://dl.acm.org/doi/10.1145/3276483}, + issue = {Oopsla}, +} + +@article{Orszag1980, + title = {Spectral Methods for Problems in Complex Geometries}, + author = {Steven A. Orszag}, + year = {1980}, + journal = {Journal of Computational Physics}, + publisher = {Elsevier {BV}}, + volume = {37}, + pages = {70--92}, + doi = {10.1016/0021-9991(80)90005-4}, + issue = {1}, } @software{Pymbolic, - title = {{Pymbolic}: {A} package to do symbolic operations for code - generation}, - author = { Kloeckner, Andreas and Wala, Matt and Fernando, Isuru and - Kulkarni, Kaushik and Fikl, Alex and Weiner, Zach and Kempf, - Dominic and Ham, David A. and Mitchell, Lawrence and Wilcox, Lucas - C and Diener, Matthias and Kapyshin, Pavlo and Raksi, Reno and - Gibson, Thomas H. }, - year = 2024, - month = dec, - publisher = {Zenodo}, - doi = {10.5281/zenodo.14526145}, - url = {https://doi.org/10.5281/zenodo.14526145}, - version = {v2024.2.2}, + title = {{Pymbolic}: {A} package to do symbolic operations for code generation}, + author = { + Kl\"{o}ckner, Andreas and Wala, Matt and Fernando, Isuru and Kulkarni, Kaushik + and Fikl, Alex and Weiner, Zach and Kempf, Dominic and Ham, David A. and + Mitchell, Lawrence and Wilcox, Lucas C and Diener, Matthias and Kapyshin, + Pavlo and Raksi, Reno and Gibson, Thomas H. + }, + year = 2024, + month = dec, + publisher = {Zenodo}, + doi = {10.5281/zenodo.14526145}, + url = {https://doi.org/10.5281/zenodo.14526145}, + version = {v2024.2.2}, } @software{PyNucleus, - title = { {PyNucleus}: {A} finite element code that specifically targets - nonlocal operators }, - author = {Glusa, Christian}, - year = 2021, - url = {https://github.com/sandialabs/PyNucleus}, - institution = {Sandia National Lab (SNL-NM), Albuquerque, NM (United States) - }, + title = { + {PyNucleus}: {A} finite element code that specifically targets nonlocal + operators + }, + author = {Glusa, Christian}, + year = 2021, + url = {https://github.com/sandialabs/PyNucleus}, + institution = {Sandia National Lab (SNL-NM), Albuquerque, NM (United States)}, } @software{Pytential, - title = {{Pytential}: {E}valuate layer and volume potentials accurately}, - author = {Andreas Kloeckner and others}, - year = 2025, - month = may, - url = {https://github.com/inducer/pytential}, - version = {hash:7395b97f}, + title = {{Pytential}: {E}valuate layer and volume potentials accurately}, + author = {Andreas Kl\"{o}ckner and others}, + year = 2025, + month = may, + url = {https://github.com/inducer/pytential}, + version = {hash:7395b97f}, } @software{Python_AST, - title = {{ast} --- Abstract Syntax Trees}, - author = {{Python Software Foundation}}, - year = 2024, - url = {https://docs.python.org/3/library/ast.html}, + title = {{ast} --- Abstract Syntax Trees}, + author = {{Python Software Foundation}}, + year = 2024, + url = {https://docs.python.org/3/library/ast.html}, } @software{QuadPy, - title = {{QuadPy}: {N}umerical integration (quadrature, cubature) in {Python - }}, - author = { Nico Schl\"{o}mer and Nick Papior and Darius Arnold and Jan - Blechta and Rasmus Zetter }, - year = 2021, - publisher = {Zenodo}, - doi = {10.5281/zenodo.5541216}, - version = {v0.16.10}, + title = {{QuadPy}: {N}umerical integration (quadrature, cubature) in {Python}}, + author = { + Nico Schl\"{o}mer and Nick Papior and Darius Arnold and Jan Blechta and + Rasmus Zetter + }, + year = 2021, + publisher = {Zenodo}, + doi = {10.5281/zenodo.5541216}, + version = {v0.16.10}, } @software{StartUpDG, - title = {{StartUpDG.jl}: {R}eference elements and physical meshes for {DG}}, - author = { Jesse Chan and David Knapp and Mason McCallum and Hendrik Ranocha - and Vincent X. Wang and Johannes Markert }, - year = 2024, - url = {https://github.com/jlchan/StartUpDG.jl}, - version = {v1.1.5}, + title = {{StartUpDG.jl}: {R}eference elements and physical meshes for {DG}}, + author = { + Jesse Chan and David Knapp and Mason McCallum and Hendrik Ranocha and + Vincent X. Wang and Johannes Markert + }, + year = 2024, + url = {https://github.com/jlchan/StartUpDG.jl}, + version = {v1.1.5}, } @software{SymEngine, - title = {{SymEngine}: {A} fast symbolic manipulation library written in C++}, - author = { O. \v{C}ert\'{\i}k and I. Fernando and S. Garg and T. Rathnayake - and others }, - year = 2013, - url = {https://github.com/symengine/symengine}, + title = {{SymEngine}: {A} fast symbolic manipulation library written in {C++}}, + author = { + O. \v{C}ert\'{\i}k and I. Fernando and S. Garg and T. Rathnayake and others + }, + year = 2013, + url = {https://github.com/symengine/symengine}, } @article{SymPy, - title = {{SymPy}: {S}ymbolic computing in Python}, - author = { Meurer, Aaron and Smith, Christopher P and Paprocki, Mateusz and - \v{C}ert\'{\i}k, Ond\v{r}ej and Kirpichev, Sergey B and Rocklin, - Matthew and Kumar, AMiT and Ivanov, Sergiu and Moore, Jason K and - Singh, Sartaj and others }, - year = 2017, - journal = {PeerJ Computer Science}, - publisher = {PeerJ Inc.}, - volume = 3, - pages = {e103}, + title = {{SymPy}: {S}ymbolic computing in {Python}}, + author = { + Meurer, Aaron and Smith, Christopher P and Paprocki, Mateusz and + \v{C}ert\'{\i}k, Ond\v{r}ej and Kirpichev, Sergey B and Rocklin, Matthew + and Kumar, AMiT and Ivanov, Sergiu and Moore, Jason K and Singh, Sartaj and + others + }, + year = 2017, + journal = {PeerJ Computer Science}, + publisher = {PeerJ Inc.}, + volume = 3, + pages = {e103}, } @book{Trefethen2020, - title = {Approximation Theory and Approximation Practice}, - author = {Lloyd N. Trefethen}, - year = 2020, - publisher = {Society for Industrial and Applied Mathematics}, - isbn = 9781611975932, + title = {Approximation Theory and Approximation Practice}, + author = {Lloyd N. Trefethen}, + year = 2020, + publisher = {Society for Industrial and Applied Mathematics}, + isbn = 9781611975932, } @article{Vioreanu2014, - title = {Spectra of Multiplication Operators as a Numerical Tool}, - author = {B. Vioreanu and V. Rokhlin}, - year = 2014, - journal = {{SIAM} Journal on Scientific Computing}, - publisher = {Society for Industrial \& Applied Mathematics {(SIAM)}}, - volume = 36, - pages = {A267--a288}, - doi = {10.1137/110860082}, - issue = 1, + title = {Spectra of Multiplication Operators as a Numerical Tool}, + author = {B. Vioreanu and V. Rokhlin}, + year = 2014, + journal = {{SIAM} Journal on Scientific Computing}, + publisher = {Society for Industrial \& Applied Mathematics {(SIAM)}}, + volume = 36, + pages = {A267--a288}, + doi = {10.1137/110860082}, + issue = 1, } @article{Waldvogel2006, - title = { Fast Construction of the {Fej\'{e}r} and {Clenshaw–Curtis} - Quadrature Rules }, - author = {J. Waldvogel}, - year = 2006, - journal = {BIT Numerical Mathematics}, - publisher = {Springer Science and Business Media {LLC}}, - volume = 46, - pages = {195--202}, - doi = {10.1007/s10543-006-0045-4}, - issue = 1, + title = { + Fast Construction of the {Fej\'{e}r} and {Clenshaw–Curtis} Quadrature Rules + }, + author = {J. Waldvogel}, + year = 2006, + journal = {BIT Numerical Mathematics}, + publisher = {Springer Science and Business Media {LLC}}, + volume = 46, + pages = {195--202}, + doi = {10.1007/s10543-006-0045-4}, + issue = 1, } @article{Warburton2007, - title = {An Explicit Construction of Interpolation Nodes on the Simplex}, - author = {T. Warburton}, - year = 2007, - journal = {Journal of Engineering Mathematics}, - publisher = {Springer Nature}, - volume = 56, - pages = {247--262}, - doi = {10.1007/s10665-006-9086-6}, - issue = 3, + title = {An Explicit Construction of Interpolation Nodes on the Simplex}, + author = {T. Warburton}, + year = 2007, + journal = {Journal of Engineering Mathematics}, + publisher = {Springer Nature}, + volume = 56, + pages = {247--262}, + doi = {10.1007/s10665-006-9086-6}, + issue = 3, } @article{Wicaksono2025, - title = {{Minterpy}: {M}ultivariate polynomial interpolation in {Python}}, - author = { Damar Wicaksono and Uwe Hernandez Acosta and Sachin Krishnan - Thekke Veettil and Jannik Kissinger and Michael Hecht }, - year = 2025, - journal = {Journal of Open Source Software}, - publisher = {The Open Journal}, - volume = 10, - number = 109, - pages = 7702, - doi = {10.21105/joss.07702}, - url = {https://doi.org/10.21105/joss.07702}, + title = {{Minterpy}: {M}ultivariate polynomial interpolation in {Python}}, + author = { + Damar Wicaksono and Uwe Hernandez Acosta and Sachin Krishnan Thekke Veettil + and Jannik Kissinger and Michael Hecht + }, + year = 2025, + journal = {Journal of Open Source Software}, + publisher = {The Open Journal}, + volume = 10, + number = 109, + pages = 7702, + doi = {10.21105/joss.07702}, + url = {https://doi.org/10.21105/joss.07702}, } @article{Witherden2015, - title = { On the Identification of Symmetric Quadrature Rules for Finite - Element Methods }, - author = {F. D. Witherden and P. E. Vincent}, - year = 2015, - journal = {Computers \& Mathematics With Applications}, - publisher = {Elsevier {BV}}, - volume = 69, - pages = {1232--1241}, - doi = {10.1016/j.camwa.2015.03.017}, - issue = 10, + title = { + On the Identification of Symmetric Quadrature Rules for Finite Element + Methods + }, + author = {F. D. Witherden and P. E. Vincent}, + year = 2015, + journal = {Computers \& Mathematics With Applications}, + publisher = {Elsevier {BV}}, + volume = 69, + pages = {1232--1241}, + doi = {10.1016/j.camwa.2015.03.017}, + issue = 10, } @article{Xiao2010, - title = { A Numerical Algorithm for the Construction of Efficient Quadrature - Rules in Two and Higher Dimensions }, - author = {H. Xiao and Z. Gimbutas}, - year = 2010, - journal = {Computers \& Mathematics With Applications}, - publisher = {Elsevier {BV}}, - volume = 59, - pages = {663--676}, - doi = {10.1016/j.camwa.2009.10.027}, - issue = 2, -} - -@article{Orszag1980, - title = {Spectral methods for problems in complex geometries}, - volume = {37}, - issn = {0021-9991}, - url = {https://www.sciencedirect.com/science/article/pii/0021999180900054}, - doi = {10.1016/0021-9991(80)90005-4}, - abstract = {The properties of spectral methods are surveyed and their - extension to solve problems in complex geometries is developed. A - new iteration procedure is introduced to solve efficiently the - full matrix equations resulting from spectral approximations to - nonconstant coefficient boundary-value problems in complex - geometries. It is shown that the work required to solve these - spectral equations exceeds that of solving the lowest-order - finite-difference approximation to the same problem by only O(N - log N).}, - number = {1}, - urldate = {2025-09-19}, - journal = {Journal of Computational Physics}, - author = {Orszag, Steven A}, - month = aug, - year = {1980}, - pages = {70--92}, - file = {ScienceDirect Full Text - PDF:/home/andreas/Zotero/storage/YKEGQJ4D/Orszag - 1980 - Spectral - methods for problems in complex - geometries.pdf:application/pdf;ScienceDirect - Snapshot:/home/andreas/Zotero/storage/8RUFDWUR/0021999180900054.html:text/html - }, + title = { + A Numerical Algorithm for the Construction of Efficient Quadrature Rules in + Two and Higher Dimensions + }, + author = {H. Xiao and Z. Gimbutas}, + year = 2010, + journal = {Computers \& Mathematics With Applications}, + publisher = {Elsevier {BV}}, + volume = 59, + pages = {663--676}, + doi = {10.1016/j.camwa.2009.10.027}, + issue = 2, } From f55c19a9486bb21a7bac804ebfa74241c2faabe4 Mon Sep 17 00:00:00 2001 From: Alexandru Fikl Date: Tue, 20 Jan 2026 22:43:21 +0200 Subject: [PATCH 14/15] feat(paper): update names to match code + Zenodo archive --- doc/paper/paper.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/doc/paper/paper.md b/doc/paper/paper.md index 0ada3d9..628a26c 100644 --- a/doc/paper/paper.md +++ b/doc/paper/paper.md @@ -6,7 +6,7 @@ tags: - approximation theory authors: - given-names: Andreas - surname: Klöckner + surname: Kloeckner corresponding: true orcid: 0000-0003-1228-519X affiliation: 1 @@ -14,15 +14,15 @@ authors: surname: Fikl orcid: 0000-0002-0552-5936 affiliation: 2 - - given-names: Jacob Xiaoyu + - given-names: Xiaoyu surname: Wei orcid: 0000-0001-7063-7865 affiliation: 3 - - given-names: Thomas H. + - given-names: Thomas surname: Gibson orcid: 0000-0002-7978-6848 affiliation: 4 - - given-names: Addison J. + - given-names: Addison surname: Alvey-Blanco affiliation: 1 affiliations: From eaca9dae31dba1cf523a2e856d44d55fd3871ea1 Mon Sep 17 00:00:00 2001 From: Alexandru Fikl Date: Tue, 27 Jan 2026 16:41:48 +0200 Subject: [PATCH 15/15] feat(paper): address additional comments from review --- doc/paper/paper.bib | 1 + doc/paper/paper.md | 6 +++--- 2 files changed, 4 insertions(+), 3 deletions(-) diff --git a/doc/paper/paper.bib b/doc/paper/paper.bib index c17d460..1495f3d 100644 --- a/doc/paper/paper.bib +++ b/doc/paper/paper.bib @@ -250,6 +250,7 @@ @article{SymPy year = 2017, journal = {PeerJ Computer Science}, publisher = {PeerJ Inc.}, + doi = {10.7717/peerj-cs.103}, volume = 3, pages = {e103}, } diff --git a/doc/paper/paper.md b/doc/paper/paper.md index 628a26c..7d4ecd3 100644 --- a/doc/paper/paper.md +++ b/doc/paper/paper.md @@ -125,10 +125,10 @@ space of functions $\phi_i: D \to \mathbb{R}$, where $D$ is the reference elemen domain, and no specific choice of basis. Predefined choices include the `PN` space, containing polynomials of total degree at most $N$, and the `QN` space, containing polynomials of maximum degree at most $N$. As with shapes, these spaces can be -combined using `TensorProductSpace`. A `Basis` objects is available separately, +combined using `TensorProductSpace`. A `Basis` object is available separately, giving access to basis functions and their derivatives, for, e.g., the monomials, general Jacobi polynomials, and the Proriol-Koornwinder-Dubiner-Owens (PKDO) basis -from [@Dubiner1991] (see \autoref{FigurePKDO}). +from @Dubiner1991 (see \autoref{FigurePKDO}). ![PKDO basis functions for the triangle.](images/pkdo-2d.png){#FigurePKDO width="80%"} @@ -140,7 +140,7 @@ a basis. `modepy` supports modal DOFs (i.e. basis coefficients) and nodal DOFs ( function or derivative values at a point). On simplices, the "warp-and-blend" nodes [@Warburton2007] are available, and on the hypercube, standard tensor product nodes are constructed from one-dimensional Legendre-Gauss(-Lobatto) nodes. `modepy` can also -directly interoperate with the `recursivenodes` library described in [@Isaac2020], +directly interoperate with the `recursivenodes` library described in @Isaac2020, which offers additional well-conditioned nodes on the simplex. ## Quadrature