Preconditioned infinite GMRES for parameterized linear systems

06/10/2022
by   Siobhán Correnty, et al.
0

We are interested in obtaining approximate solutions to parameterized linear systems of the form A(μ) x(μ) = b for many values of the parameter μ. Here A(μ) is large, sparse, and nonsingular, with a nonlinear analytic dependence on μ. Our approach is based on a companion linearization for parameterized linear systems. The companion matrix is similar to the operator in the infinite Arnoldi method, and we use this to adapt the flexible GMRES setting. In this way, our method returns a function x̃(μ) which is cheap to evaluate for different μ, and the preconditioner is applied only approximately. This novel approach leads to increased freedom to carry out the action of the operation inexactly, which provides performance improvement over the method infinite GMRES, without a loss of accuracy in general. We show that the error of our method is estimated based on the magnitude of the parameter μ, the inexactness of the preconditioning, and the spectrum of the linear companion matrix. Numerical examples from a finite element discretization of a Helmholtz equation with a parameterized material coefficient illustrate the competitiveness of our approach. The simulations are reproducible and publicly available online.

READ FULL TEXT
research
12/08/2022

Preconditioned Chebyshev BiCG for parameterized linear systems

The biconjugate gradient method (BiCG) is one of the most popular short-...
research
02/08/2021

Infinite GMRES for parameterized linear systems

We consider linear parameter-dependent systems A(μ) x(μ) = b for many di...
research
09/24/2021

Compound Krylov subspace methods for parametric linear systems

In this work, we propose a reduced basis method for efficient solution o...
research
09/02/2021

Matrix-oriented FEM formulation for stationary and time-dependent PDEs on x-normal domains

When numerical solution of elliptic and parabolic partial differential e...
research
05/12/2021

Combining Set Propagation with Finite Element Methods for Time Integration in Transient Solid Mechanics Problems

The Finite Element Method (FEM) is the gold standard for spatial discret...
research
09/05/2023

Accurate Solution of the Nonlinear Schrödinger Equation via Conservative Multiple-Relaxation ImEx Methods

The nonlinear Schrödinger (NLS) equation possesses an infinite hierarchy...
research
10/26/2020

A discretize-then-map approach for the treatment of parameterized geometries in model order reduction

We propose a new general approach for the treatment of parameterized geo...

Please sign up or login with your details

Forgot password? Click here to reset