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A least squares radial basis function partition of unity method for solving PDEs
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
2017 (English)In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 39, A2538-A2563 p.Article in journal (Refereed) Published
Place, publisher, year, edition, pages
2017. Vol. 39, A2538-A2563 p.
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-316488DOI: 10.1137/17M1118087OAI: oai:DiVA.org:uu-316488DiVA: diva2:1077950
Projects
eSSENCE
Available from: 2017-11-09 Created: 2017-03-01 Last updated: 2017-11-23Bibliographically approved
In thesis
1. Localised Radial Basis Function Methods for Partial Differential Equations
Open this publication in new window or tab >>Localised Radial Basis Function Methods for Partial Differential Equations
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Radial basis function methods exhibit several very attractive properties such as a high order convergence of the approximated solution and flexibility to the domain geometry. However the method in its classical formulation becomes impractical for problems with relatively large numbers of degrees of freedom due to the ill-conditioning and dense structure of coefficient matrix. To overcome the latter issue we employ a localisation technique, namely a partition of unity method, while the former issue was previously addressed by several authors and was of less concern in this thesis.

In this thesis we develop radial basis function partition of unity methods for partial differential equations arising in financial mathematics and glaciology. In the applications of financial mathematics we focus on pricing multi-asset equity and credit derivatives whose models involve several stochastic factors. We demonstrate that localised radial basis function methods are very effective and well-suited for financial applications thanks to the high order approximation properties that allow for the reduction of storage and computational requirements, which is crucial in multi-dimensional problems to cope with the curse of dimensionality. In the glaciology application we in the first place make use of the meshfree nature of the methods and their flexibility with respect to the irregular geometries of ice sheets and glaciers. Also, we exploit the fact that radial basis function methods are stated in strong form, which is advantageous for approximating velocity fields of non-Newtonian viscous liquids such as ice, since it allows to avoid a full coefficient matrix reassembly within the nonlinear iteration.

In addition to the applied problems we develop a least squares radial basis function partition of unity method that is robust with respect to the node layout. The method allows for scaling to problem sizes of a few hundred thousand nodes without encountering the issue of large condition numbers of the coefficient matrix. This property is enabled by the possibility to control the coefficient matrix condition number by the rate of oversampling and the mode of refinement.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2018. 54 p.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1600
Keyword
Radial basis function, Partition of unity, Computational finance, Option pricing, Credit default swap, Glaciology, Fluid dynamics, Non-Newtonian flow, Anisotropic RBF
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
urn:nbn:se:uu:diva-332715 (URN)978-91-513-0157-0 (ISBN)
Public defence
2018-01-19, ITC 2446, Polacksbacken, Lägerhyddsvägen 2, Uppsala, 10:15 (English)
Opponent
Supervisors
Available from: 2017-12-14 Created: 2017-11-21 Last updated: 2017-12-14

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Larsson, ElisabethShcherbakov, Victor

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