Compound Krylov subspace methods for parametric linear systems

2021/09/24 by Antti Autio, Autio, Antti, Antti Hannukainen +1
Computer Science · Decision Sciences · Engineering · #65F10 #65N15 #65N30 #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design

paper · pdf · doi:10.48550/arxiv.2109.12206

openalex publication_date 2021/09/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this work, we propose a reduced basis method for efficient solution of parametric linear systems. The coefficient matrix is assumed to be a linear matrix-valued function that is symmetric and positive definite for admissible values of the parameter \mathbfσ∈ ℝ^s. We propose a solution strategy where one first computes a basis for the appropriate compound Krylov subspace and then uses this basis to compute a subspace solution for multiple \mathbfσ. Three kinds of compound Krylov subspaces are discussed. Error estimate is given for the subspace solution from each of these spaces. Theoretical results are demonstrated by numerical examples related to solving parameter dependent elliptic PDEs using the finite element method (FEM).

Related

• Reduced Krylov Basis Methods for Parametric Partial Differential Equations
• Krylov-Simplex method that minimizes the residual in ℓ₁-norm or ℓ_∞-norm
• Deflation and augmentation techniques in Krylov subspace methods for the solution of linear systems
• Krylov-subspace recycling via the POD-augmented conjugate-gradient method
• Equivalence of Krylov Subspace Methods for Skew-Symmetric Linear Systems