Direct optimization of BPX preconditioners

Ivan Oseledets, Vladimir Fanaskov

    Research output: Contribution to journalArticlepeer-review


    We consider an automatic construction of locally optimal preconditioners for positive definite linear systems. To achieve this goal, we introduce a differentiable loss function that does not explicitly include the estimation of minimal eigenvalue. Nevertheless, the resulting optimization problem is equivalent to a direct minimization of the condition number. To demonstrate our approach, we construct a parametric family of modified BPX preconditioners. Namely, we define a set of empirical basis functions for coarse finite element spaces and tune them to achieve better condition number. For considered model equations (that includes Poisson, Helmholtz, Convection–diffusion, Biharmonic, and others), we achieve from two to twenty times smaller condition numbers for symmetric positive definite linear systems.

    Original languageEnglish
    Article number113811
    JournalJournal of Computational and Applied Mathematics
    Publication statusPublished - 1 Mar 2022


    • Boundary value problems
    • BPX
    • Multigrid
    • Numerical linear algebra
    • PDE
    • Preconditioning


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