Replace nested sparse solver calls in ripple solver with direct iterative methods

[krystophny]

Summary

Refactor ripple solver to use sparse iterative methods directly for the full problem using a matrix-free approach, eliminating nested solver calls.

Background

Currently the ripple solver uses Arnoldi-Richardson iterations where each iteration requires solving a sparse linear system. We can reformulate to solve the entire problem with a single iterative solver using matrix-free methods.

Dependencies

• Depends on Add iterative solver methods to sparse solver module #48 (sparse iterative solvers) for base implementations
• Depends on Refactor ripple solver for testability and modularity #45 (ripple solver refactoring) for clean structure
• Depends on Add testing and code coverage infrastructure #50 (testing infrastructure)

Tasks

• Analyze current Arnoldi formulation to understand the full operator
• Implement matrix-free operator for the complete ripple problem
• Adapt BiCGSTAB for matrix-free operation
• Adapt GMRES for matrix-free operation
• Adapt IDR(s) for matrix-free operation
• Develop physics-aware preconditioners
• Compare convergence and performance with nested approach
• Ensure physics results remain identical

Implementation Strategy

1. Build on iterative solvers from Add iterative solver methods to sparse solver module #48
2. Instead of assembling sparse matrices, implement matrix-vector products directly
3. The operator applies the full physics without intermediate linear solves
4. Use Krylov methods with this matrix-free operator
5. May need specialized preconditioners for convergence

Matrix-Free Approach

• No explicit matrix assembly
• Operator computes action of full ripple problem
• Leverages physics structure for efficiency
• Reduces memory requirements significantly

Expected Benefits

• Eliminate nested solver inefficiency
• Lower memory usage (no matrix storage)
• Better scalability
• More direct convergence control
• Potential for physics-based optimizations

Challenges

• Matrix-free preconditioners are more complex
• Need careful operator implementation
• Convergence may require problem-specific tuning

Testing Requirements

• Verify identical physics results
• Compare performance with current method
• Test convergence for various problem sizes
• Memory usage comparison
• Validate with golden records

Related Work

• Extends iterative solvers from Add iterative solver methods to sparse solver module #48 to matrix-free operation
• Alternative to Add iterative solver options (BiCGSTAB, GMRES, IDR(s)) to ripple solver #47 (which keeps Arnoldi structure)