Optimize nt_batch automatically to improve performance

[gauravharsha]

GW kernel uses cublas::gemm_strided_batched which performs the best when batch size is large.
This PR proposes automatic optimization of nt_batch to achieve high optimal performance, with the following logic:

• Set default value: nt_batch = 0
• If nt_batch == 0, optimize value for better performance.
• Otherwise, use specified value.

Optimization logic:

• Keep at least 2 streams, then maximize nt_batch.
• If optimized nt_batch is large but n_tau - nt_batch < n_tau / 4 (i.e., second batch is small), we opt instead for nt_batch = n_tau / 2 -- this is debatable but for both small and large applications, this shouldn't make much of a difference.

[Copilot]

Pull Request Overview

This PR adds automatic optimization of the nt_batch parameter for the CUDA GW solver to maximize GPU performance. Previously, users had to manually specify this value, which required knowledge of GPU memory constraints and optimal batch sizing.

• Changed default nt_batch from 1 to 0 (triggers automatic optimization)
• Added optimize_ntbatch() function to calculate optimal batch size based on available GPU memory
• Extended test coverage to verify both automatic optimization and manual specification modes

Reviewed Changes

Copilot reviewed 4 out of 4 changed files in this pull request and generated 9 comments.

File	Description
src/green/gpu/gpu_factory.h	Updated nt_batch parameter default from 1 to 0 and clarified documentation
src/green/gpu/gw_gpu_kernel.h	Added declaration for optimize_ntbatch() function with documentation
src/gw_gpu_kernel.cpp	Implemented optimize_ntbatch() logic and integrated it into memory checking; updated warning/error messages
test/cu_solver_test.cpp	Extended test coverage to test both automatic optimization (nt_batch="0") and manual setting (nt_batch="1")

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[Copilot]

Pull Request Overview

Copilot reviewed 4 out of 4 changed files in this pull request and generated 4 comments.

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[egull]

Looks good to me. Print nt_batch so we have it in the log (ignore if it's done elsewhere), and then let's try it out!

[gauravharsha]

Merging now. More tests and benchmarks under way, but the output of unit tests for Hydrogen on ancient Quadro P1000 already show improvements with optimization:

Low memory mode   |   Precision   |   GFLOPS w/ optimization  |   GFLOPS w/ nt_batch=1
-------------------------------------------------------------------------------------------
No                |   Double      |   2.81423                 |   0.505108
Yes               |   Double      |   2.88435                 |   0.514456
No                |   Single      |   23.6168                 |   1.44349
Yes               |   Single      |   27.5014                 |   1.44404