Use modified Gram-Schmidt orthogonalization in GMRES

[michelebucelli]

Fixes #599.

Current situation

The GMRES method, as implemented in gmres.cpp, used unmodified Gram-Schmidt orthogonalization. Multiple experiments indicate that the numerical instability related to that propagates to other parts of the solver. See #599 for additional details.

Release Notes

1. GMRES uses modified Gram-Schmidt orthogonalization;
2. the orthogonalization step has been extracted to two dedicated functions (one for scalar and one for vector problems), for improved readability and to remove duplication between the two vector versions of GMRES (gmres and gmres_v);
3. Addressed a @todo about replacing value = value + ... with value += ... in FourierInterpolation.

Documentation

The new functions that wrap the orthogonalization step have Doxygen documentation.

None of these changes is user-facing.

Testing

All automatic tests are still passing.

Code of Conduct & Contributing Guidelines

• I agree to follow the Code of Conduct and Contributing Guidelines.

[claude]

Claude Code Review

This pull request is from a fork — automated review is disabled. A repository maintainer can comment @claude review to run a one-time review.

[Copilot]

Pull request overview

This pull request updates the GMRES linear solver to use modified Gram–Schmidt (MGS) orthogonalization (extracted into dedicated helper functions) to improve numerical stability and reduce divergence between MPI runs, and also performs a small cleanup in Fourier interpolation evaluation.

Changes:

• Replaced classical Gram–Schmidt with modified Gram–Schmidt in GMRES and refactored the orthogonalization step into shared helper functions.
• Added a “lucky breakdown” handling path based on comparing the subdiagonal Hessenberg entry to machine epsilon-scaled norms.
• Simplified arithmetic in FourierInterpolation by using compound assignment operators (-= / +=).

Reviewed changes

Copilot reviewed 2 out of 2 changed files in this pull request and generated 2 comments.

File	Description
Code/Source/linear_solver/gmres.cpp	Introduces MGS orthogonalization helpers and replaces the previous orthogonalization logic in GMRES variants.
Code/Source/solver/FourierInterpolation.cpp	Uses compound assignment operators for trend subtraction and Fourier series accumulation.

Suppressed comments (2)

Code/Source/linear_solver/gmres.cpp:426

• The new lucky-breakdown handling in orthogonalize_* can set h(i+1,i)=0.0, which makes the subsequent Givens-rotation normalization tmp = sqrt(h(i,i)^2 + h(i+1,i)^2) potentially become 0.0. That leads to division by zero when computing c(i) and s(i) and can produce NaNs.

      orthogonalize_s(lhs, nNo, mynNo, i, u, h);

Code/Source/linear_solver/gmres.cpp:598

• The new lucky-breakdown handling in orthogonalize_* can set h(i+1,i)=0.0, which makes the subsequent Givens-rotation normalization tmp = sqrt(h(i,i)^2 + h(i+1,i)^2) potentially become 0.0. That leads to division by zero when computing c(i) and s(i) and can produce NaNs.

      orthogonalize_v(lhs, dof, nNo, mynNo, i, u, h);

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[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 72.71%. Comparing base (a3413e0) to head (d3c66cc).

Additional details and impacted files

@@            Coverage Diff             @@
##             main     #600      +/-   ##
==========================================
- Coverage   72.73%   72.71%   -0.02%     
==========================================
  Files         255      255              
  Lines       39290    39273      -17     
  Branches     6726     6725       -1     
==========================================
- Hits        28576    28557      -19     
- Misses      10472    10473       +1     
- Partials      242      243       +1     

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[aabrown100-git]

This looks good to me, pending resolution of copilot's comments

[ktbolt]

I would like to see some tests on real problems to collect data on performance changes, improved convergence, etc.

[michelebucelli]

Moving this to draft, pending the discussion at #599.