arXiv subjects

Ángeles Martínez

Publications and source records attributed to Ángeles Martínez.

3 recordsLinked to original sources

arXiv2026

Mixed precision solvers for the all-at-once Runge--Kutta discretization of the heat equation

We study the effect of mixed precision on the numerical integration of the heat equation discretized with a Runge--Kutta method in time. A full space-time discretization is applied, which results in a very large and sparse linear system to be solved for the numerical approximations and the Runge--Kutta stages of all time steps. The linear system is solved by applying a suitable preconditioned iterative method that can be run in parallel. In order to speed up the solution process, the preconditioner is applied using a mixed precision framework. Sequential results show the robustness of the preconditioner, even when applied in mixed precision. Finally, we present numerical evidence of the improved performance of the mixed precision strategy when applied in a parallel environment, achieving up to a 50% reduction in CPU time.

math.NA↗

arXiv2023

Parallel-in-Time Solver for the All-at-Once Runge--Kutta Discretization

In this article, we derive fast and robust parallel-in-time preconditioned iterative methods for the all-at-once linear systems arising upon discretization of time-dependent PDEs. The discretization we employ is based on a Runge--Kutta method in time, for which the development of parallel solvers is an emerging research area in the literature of numerical methods for time-dependent PDEs. By making use of classical theory of block matrices, one is able to derive a preconditioner for the systems considered. The block structure of the preconditioner allows for parallelism in the time variable, as long as one is able to provide an optimal solver for the system of the stages of the method. We thus propose a preconditioner for the latter system based on a singular value decomposition (SVD) of the (real) Runge--Kutta matrix $A_{\mathrm{RK}} = U ΣV^\top$. Supposing $A_{\mathrm{RK}}$ is invertible, we prove that the spectrum of the system for the stages preconditioned by our SVD-based preconditioner is contained within the right-half of the unit circle, under suitable assumptions on the matrix $U^\top V$ (the assumptions are well posed due to the polar decomposition of $A_{\mathrm{RK}}$). We show the numerical efficiency of our SVD-based preconditioner by solving the system of the stages arising from the discretization of the heat equation and the Stokes equations, with sequential time-stepping. Finally, we provide numerical results of the all-at-once approach for both problems, showing the speed-up achieved on a parallel architecture.

math.NA↗

arXiv2020

A New Preconditioning Approach for an Interior Point-Proximal Method of Multipliers for Linear and Convex Quadratic Programming

In this paper, we address the efficient numerical solution of linear and quadratic programming problems, often of large scale. With this aim, we devise an infeasible interior point method, blended with the proximal method of multipliers, which in turn results in a primal-dual regularized interior point method. Application of this method gives rise to a sequence of increasingly ill-conditioned linear systems which cannot always be solved by factorization methods, due to memory and CPU time restrictions. We propose a novel preconditioning strategy which is based on a suitable sparsification of the normal equations matrix in the linear case, and also constitutes the foundation of a block-diagonal preconditioner to accelerate MINRES for linear systems arising from the solution of general quadratic programming problems. Numerical results for a range of test problems demonstrate the robustness of the proposed preconditioning strategy, together with its ability to solve linear systems of very large dimension.

math.NA↗