DISCRETIZATION-ERROR-ACCURATE MIXED-PRECISION MULTIGRID SOLVERS

Summary: The study presents an algorithm for solving linear systems efficiently by utilizing different precisions of arithmetic and combining Cholesky factorization with GMRES-based iterative refinement. It introduces matrix shifting and scaling to improve computational performance and avoid numerical issues.

Summary: This paper establishes the first theoretical framework for analyzing the rounding-error effects on multigrid methods using mixed-precision iterative-refinement solvers, providing normwise forward error analysis and introducing the notion of progressive precision for multigrid solvers. The theoretical results show that rounding an exact result to finite precision causes an error in the energy norm proportional to the square root of the matrix condition number K, indicating that the limiting accuracy for both V-cycles and full multigrid is proportional to k(1/2) in energy. Additionally, the loss of convergence rate due to rounding grows in proportion to k(1/2), but is argued to be insignificant in practice.