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Koskela, Antti
Publications (2 of 2) Show all publications
Koskela, A. & Jarlebring, E. (2019). On a generalization of neumann series of bessel functions using Hessenberg matrices and matrix exponentials. In: European Conference on Numerical Mathematics and Advanced Applications, ENUMATH 2017: . Paper presented at European Conference on Numerical Mathematics and Advanced Applications, ENUMATH 2017, Voss, Norway, 25 September 2017 through 29 September 2017 (pp. 205-214). Springer, 126
Open this publication in new window or tab >>On a generalization of neumann series of bessel functions using Hessenberg matrices and matrix exponentials
2019 (English)In: European Conference on Numerical Mathematics and Advanced Applications, ENUMATH 2017, Springer, 2019, Vol. 126, p. 205-214Conference paper, Published paper (Refereed)
Abstract [en]

The Neumann expansion of Bessel functions (of integer order) of a function g: ℂ→ ℂ corresponds to representing g as a linear combination of basis functions φ0, φ1, …, i.e., g(s)=∑ℓ=0 ∞wℓφℓ(s), where φi(s) = Ji(s), i = 0, …, are the Bessel functions. In this work, we study an expansion for a more general class of basis functions. More precisely, we assume that the basis functions satisfy an infinite dimensional linear ordinary differential equation associated with a Hessenberg matrix, motivated by the fact that these basis functions occur in certain iterative methods. A procedure to compute the basis functions as well as the coefficients is proposed. Theoretical properties of the expansion are studied. We illustrate that non-standard basis functions can give faster convergence than the Bessel functions.

Place, publisher, year, edition, pages
Springer, 2019
Series
Lecture Notes in Computational Science and Engineering, ISSN 1439-7358 ; 126
National Category
Computer Sciences
Identifiers
urn:nbn:se:kth:diva-241801 (URN)10.1007/978-3-319-96415-7_17 (DOI)2-s2.0-85060038484 (Scopus ID)9783319964140 (ISBN)
Conference
European Conference on Numerical Mathematics and Advanced Applications, ENUMATH 2017, Voss, Norway, 25 September 2017 through 29 September 2017
Note

QC 20190125

Available from: 2019-01-25 Created: 2019-01-25 Last updated: 2019-01-25Bibliographically approved
Koskela, A., Jarlebring, E. & Hochstenbach, M. E. (2016). Krylov approximation of linear odes with polynomial parameterization. SIAM Journal on Matrix Analysis and Applications, 37(2), 519-538
Open this publication in new window or tab >>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
Keywords
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:nbn:se:kth:diva-195542 (URN)10.1137/15M1032831 (DOI)000386450400002 ()2-s2.0-84976888718 (Scopus ID)
Note

QC 20161121

Available from: 2016-11-21 Created: 2016-11-03 Last updated: 2017-11-29Bibliographically approved
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