arXiv subjects

Petr Vacek

Publications and source records attributed to Petr Vacek.

3 recordsLinked to original sources

arXiv2025

Mixed precision multigrid with smoothing based on incomplete Cholesky factorization

Multigrid methods are popular iterative methods for solving large-scale sparse systems of linear equations. We present a mixed precision formulation of the multigrid V-cycle method with general assumptions on the finite precision errors coming from the application of coarsest-level solver and smoothing. Inspired by existing analysis, we derive a bound on the relative finite precision error of the V-cycle which gives insight into how the finite precision errors from the individual components of the method may affect the overall finite precision error. We use the result to study V-cycle methods with smoothing based on incomplete Cholesky factorization. The results imply that in certain settings, the precisions used for applying the incomplete Cholesky smoothing can be significantly lower than the precision used for computing the residual, restriction, prolongation, and correction on a given level. We perform numerical experiments using simulated floating point arithmetic with the MATLAB Advanpix toolbox as well as experiments on GPUs using the Ginkgo library. The experiments illustrate the theoretical findings and show that in the considered settings, the incomplete Cholesky smoothing can be applied in relatively low precisions, resulting in significant improvements in the execution time (up to 28% less), energy savings (up to 26% less) and required memory (up to 16% less) in comparison with the uniform double precision variant.

math.NA↗

arXiv2024

A posteriori error estimates based on multilevel decompositions with large problems on the coarsest level

Multilevel methods represent a powerful approach in numerical solution of partial differential equations. The multilevel structure can also be used to construct estimates for total and algebraic errors of computed approximations. This paper deals with residual-based error estimates that are based on properties of quasi-interpolation operators, stable-splittings, or frames. We focus on the settings where the system matrix on the coarsest level is still large and the associated terms in the estimates can only be approximated. We show that the way in which the error term associated with the coarsest level is approximated is substantial. It can significantly affect both the efficiency (accuracy) of the overall error estimates and their robustness with respect to the size of the coarsest problem. The newly proposed approximation of the coarsest-level term is based on using the conjugate gradient method with an appropriate stopping criterion. We prove that the resulting estimates are efficient and robust with respect to the size of the coarsest-level problem. Numerical experiments illustrate the theoretical findings.

math.NA↗

arXiv2023

The effect of approximate coarsest-level solves on the convergence of multigrid V-cycle methods

The multigrid V-cycle method is a popular method for solving systems of linear equations. It computes an approximate solution by using smoothing on fine levels and solving a system of linear equations on the coarsest level. Solving on the coarsest level depends on the size and difficulty of the problem. If the size permits, it is typical to use a direct method based on LU or Cholesky decomposition. In settings with large coarsest-level problems, approximate solvers such as iterative Krylov subspace methods, or direct methods based on low-rank approximation, are often used. The accuracy of the coarsest-level solver is typically determined based on the experience of the users with the concrete problems and methods. In this paper we present an approach to analyzing the effects of approximate coarsest-level solves on the convergence of the V-cycle method for symmetric positive definite problems. Using these results, we derive coarsest-level stopping criterion through which we may control the difference between the approximation computed by a V-cycle method with approximate coarsest-level solver and the approximation which would be computed if the coarsest-level problems were solved exactly. The coarsest-level stopping criterion may thus be set up such that the V-cycle method converges to a chosen finest-level accuracy in (nearly) the same number of V-cycle iterations as the V-cycle method with exact coarsest-level solver. We also utilize the theoretical results to discuss how the convergence of the V-cycle method may be affected by the choice of a tolerance in a coarsest-level stopping criterion based on the relative residual norm.

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