Probabilistic domain decomposition algorithms for the numerical solution of large-scale elliptic boundary value problems
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2024-03
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2024-03-14
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Abstract
1.5. Organization and overview.
This thesis is organized as follows. In chapter 2, I give a brief overview of the mathematical approaches and algorithms that have been used or considered during the formulation, analysis and implementation of the different approaches to PDDSparse. The chapter runs through the methods to tackle the probabilistic representation of BVPs numerically (sections 2.1). Then, it goes on to the idea of implementing substructuring techniques in PDDSparse (section 2.2) and gets deeper into the GMRES method and preconditioning techniques to solve the PDDSparse linear system (section 2.3). A brief overview of pseudospectral methods (section 2.4) is given before finally discussing some aspects of implementing the algorithms in supercomputing facilities (section 2.5). Then, the four articles of the compendium of publications are included:
• In the first paper (chapter 3), a thorough formulation of PDDSparse is given for the case of a domain discretized into a mesh of square non-overlapping subdomains. The analysis of PDDSparse error motivates a three-stage algorithm to estimate such error and optimize the computational resources invested in its execution. This first approach to PDDSparse is illustrated with the results of numerical experiments conducted in Galileo100 which show the scalability and overall performance of the algorithm.
• The second paper (chapter 4) revolves around the G matrix characterization. It contains a proof of the invertibility of G and a bound on its condition number that relates the algebraic properties of PDDSparse linear system with quantities derived from its stochastic formulation. The analysis carried out in this paper motivates the formulation of a preconditioner to improve the convergence of Krylov subspace methods. All this material is supported by a set of numerical experiments conducted in LUMI-C.
• In the third paper (chapter 5), PDDSparse is used as the linear solver for two iterative schemes aimed at solving elliptic BVPs with positive reaction coefficients and semilinear BVPs, which are problems whose probabilistic representations can not be obtained via the Feynman-Kac formula. Those iterative algorithms are presented in this work alongside some proof of concept results attained in the supercomputer Marconi100.
• In the last article of the compendium (chapter 6), PDDSparse is reformulated to solve a BVP by discretizing the domain with a mesh of overlapping circles instead of non-overlapping squares. This new approach avoids the Monte Carlo computations for many linear system coefficients. In this work, the parallel performance of the RAS preconditioner on solving the PDDSparse linear system with GMRES is also tested. Numerical results carried out in the supercomputer Fugaku are reported, supporting the validity of this approach to PDDSparse.
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Mención Internacional en el título de doctor
Keywords
Deterministic domain decomposition, Probabilistic domain decomposition, High-performance computing, Strong scalability, Feynman-Kac formula