Newton-Raphson preconditioner for Krylov type solvers on GPU devices.

Area of Science:

• Numerical Analysis
• High-Performance Computing
• Scientific Computing

Background:

• Conventional preconditioners for Krylov solvers perform well on CPUs but struggle on GPGPUs due to implementation complexity.
• Powerful preconditioners often require significant memory and computational resources, hindering their use on modern supercomputers.

Purpose of the Study:

• To develop and evaluate a new Newton-Raphson method-based preconditioner for Krylov solvers optimized for GPGPU.
• To address the limitations of traditional preconditioners on GPGPU architectures.

Main Methods:

• Developed a preconditioner based on the limited memory BFGS (L-BFGS) Hessian approximation technique.
• Utilized the Generalized Conjugate Residual (GCR) method, a flexible Krylov solver, due to its compatibility with variable preconditioning matrices.
• Implemented L-BFGS using optimized BLAS libraries for efficient GPGPU execution.

Main Results:

• The L-BFGS preconditioner demonstrated robustness, successfully converging where conventional methods like diagonal scaling and SSOR failed.
• Achieved performance improvements of over 10x compared to conventional preconditioners on CPUs in optimal scenarios.
• Showcased excellent performance on GPGPUs due to the preconditioner's reliance on simple operations.
• Mathematical analysis confirmed improved matrix conditioning through calculated condition numbers of preconditioned matrices.

Conclusions:

• The developed L-BFGS preconditioner offers a robust and efficient solution for Krylov solvers on GPGPUs.
• This approach overcomes the limitations of traditional preconditioners, providing significant speedups and enhanced numerical stability.
• The method is particularly effective with flexible Krylov solvers like GCR.