Continuous analogues of Krylov methods for differential operators

2018/03/29 by Marc Aurèle Gilles, Alex Townsend, Gilles, Marc Aurèle +1
Computer Science · Mathematics · Physics and Astronomy · #47E05 #65F10 #65N35 #Electromagnetic Scattering and Analysis #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.1803.11049

openalex publication_date 2018/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Analogues of the conjugate gradient method, MINRES, and GMRES are derived for solving boundary value problems (BVPs) involving second-order differential operators. Two challenges arise: imposing the boundary conditions on the solution while building up a Krylov subspace, and guaranteeing convergence of the Krylov-based method on unbounded operators. Our approach employs projection operators to guarantee that the boundary conditions are satisfied, and we develop an operator preconditioner that ensures that an approximate solution is computed after a finite number of iterations. The developed Krylov methods are practical iterative BVP solvers that are particularly efficient when a fast operator-function product is available.

Citations

• A Fast and Well-Conditioned Spectral Method

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