Krylov approximation of linear odes with polynomial parameterization
2016 (English)In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 37, no 2, p. 519-538Article in journal (Refereed) Published
Abstract [en]
We propose a new numerical method to solve linear ordinary differential equations of the type δu/δt (t, ϵ) = A(ϵ) u(t,ϵ), where A: C → Cn×n is a matrix polynomial with large and sparse matrix coefficients. The algorithm computes an explicit parameterization of approximations of u(t, ϵ) such that approximations for many different values of ϵ 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 be independent of the truncation parameter so that it can also be interpreted as Arnoldi's method applied to an infinite dimensional matrix. We prove the super linear 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.
Place, publisher, year, edition, pages
Society for Industrial and Applied Mathematics Publications , 2016. Vol. 37, no 2, p. 519-538
Keywords [en]
Arnoldi's method, Exponential integrators, Frechet derivatives, Krylov methods, Matrix exponential, Matrix functions, Model order reduction, Parameterized ordinary differential equations, Algorithms, Approximation algorithms, Matrix algebra, Numerical methods, Parameter estimation, Parameterization, Polynomial approximation, Arnoldi's methods, Frechet derivative, Krylov method, Matrix exponentials, Parameterized, Ordinary differential equations
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-195542DOI: 10.1137/15M1032831ISI: 000386450400002Scopus ID: 2-s2.0-84976888718OAI: oai:DiVA.org:kth-195542DiVA, id: diva2:1048634
Note
QC 20161121
2016-11-212016-11-032022-06-27Bibliographically approved