Krylov approximation of ODEs with polynomial parameterization

TL;DR

This paper introduces a Krylov-based numerical method for efficiently solving parameter-dependent linear ODEs with matrix polynomial coefficients, enabling rapid computation across multiple parameters and time points.

Contribution

It presents a novel algorithm that parameterizes solutions of matrix polynomial ODEs, leveraging Krylov methods and Arnoldi's process, with proven superlinear convergence and error estimates.

Findings

Algorithm achieves efficient multi-parameter solutions.

Superlinear convergence is theoretically proven.

Illustrated with PDE discretization examples.

Abstract

We propose a new numerical method to solve linear ordinary differential equations of the type $\frac{\partial u}{\partial t}(t,\varepsilon) = A(\varepsilon) \, u(t,\varepsilon)$, where $A:\mathbb{C}\rightarrow\mathbb{C}^{n\times n}$ is a matrix polynomial with large and sparse matrix coefficients. The algorithm computes an explicit parameterization of approximations of $u(t,\varepsilon)$ such that approximations for many different values of $\varepsilon$ and $t$ can be obtained with a very small additional computational effort. The derivation of the algorithm is based on a reformulation of the parameterization as a linear parameter-free ordinary differential equation and on approximating the product of the matrix exponential and a vector with a Krylov method. The Krylov approximation is generated with Arnoldi's method and the structure of the coefficient matrix turns out to have an independence on the truncation parameter so that it can also be interpreted as Arnoldi's method applied to an infinite dimensional matrix. We prove the superlinear convergence of the algorithm and provide a posteriori error estimates to be used as termination criteria. The behavior of the algorithm is illustrated with examples stemming from spatial discretizations of partial differential equations.