DeepAI AI Chat
Log In Sign Up

A study of defect-based error estimates for the Krylov approximation of φ-functions

01/31/2020
by   Tobias Jawecki, et al.
TU Wien
0

Prior recent work, devoted to the study of polynomial Krylov techniques for the approximation of the action of the matrix exponential e^tAv, is extended to the case of associated φ-functions (which occur within the class of exponential integrators). In particular, a posteriori error bounds and estimates, based on the notion of the defect (residual) of the Krylov approximation are considered. Computable error bounds and estimates are discussed and analyzed. This includes a new error bound which favorably compares to existing error bounds in specific cases. The accuracy of various error bounds is characterized in relation to corresponding Ritz values of A. Ritz values yield properties of the spectrum of A (specific properties are known a priori, e.g. for Hermitian or skew-Hermitian matrices) in relation to the actual starting vector v and can be computed. This gives theoretical results together with criteria to quantify the achieved accuracy on the run. For other existing error estimates the reliability and performance is studied by similar techniques. Effects of finite precision (floating point arithmetic) are also taken into account.

READ FULL TEXT

page 1

page 2

page 3

page 4

06/17/2021

Error bounds for Lanczos-based matrix function approximation

We analyze the Lanczos method for matrix function approximation (Lanczos...
12/22/2022

A review of maximum-norm a posteriori error bounds for time-semidiscretisations of parabolic equations

A posteriori error estimates in the maximum norm are studied for various...
04/29/2021

Fast Multiscale Diffusion on Graphs

Diffusing a graph signal at multiple scales requires computing the actio...
06/10/2020

A comparison of limited-memory Krylov methods for Stieltjes functions of Hermitian matrices

Given a limited amount of memory and a target accuracy, we propose and c...
07/07/2017

On Sound Relative Error Bounds for Floating-Point Arithmetic

State-of-the-art static analysis tools for verifying finite-precision co...
12/11/2020

DPG approximation of eigenvalue problems

In this paper, the discontinuous Petrov–Galerkin approximation of the La...
03/27/2019

Posteriori Probabilistic Bounds of Convex Scenario Programs with Validation Tests

Scenario programs have established themselves as efficient tools towards...