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Localised Radial Basis Function Methods for Partial Differential Equations
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.
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. , p. 54
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1600
Keywords [en]
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: urn:nbn:se:uu:diva-332715ISBN: 978-91-513-0157-0 (print)OAI: oai:DiVA.org:uu-332715DiVA, id: diva2:1159181
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: 2018-03-08
List of papers
1. Radial basis function partition of unity methods for pricing vanilla basket options
Open this publication in new window or tab >>Radial basis function partition of unity methods for pricing vanilla basket options
2016 (English)In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 71, p. 185-200Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-272085 (URN)10.1016/j.camwa.2015.11.007 (DOI)000369455000012 ()
Projects
eSSENCE
Available from: 2015-12-03 Created: 2016-01-11 Last updated: 2017-11-30Bibliographically approved
2. BENCHOP—The BENCHmarking project in Option Pricing
Open this publication in new window or tab >>BENCHOP—The BENCHmarking project in Option Pricing
Show others...
2015 (English)In: International Journal of Computer Mathematics, ISSN 0020-7160, E-ISSN 1029-0265, Vol. 92, p. 2361-2379Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-260897 (URN)10.1080/00207160.2015.1072172 (DOI)000363753800003 ()
Projects
eSSENCE
Available from: 2015-09-21 Created: 2015-08-25 Last updated: 2018-08-21Bibliographically approved
3. Radial basis function partition of unity operator splitting method for pricing multi-asset American options
Open this publication in new window or tab >>Radial basis function partition of unity operator splitting method for pricing multi-asset American options
2016 (English)In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 56, p. 1401-1423Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-284299 (URN)10.1007/s10543-016-0616-y (DOI)000388968500012 ()
Projects
eSSENCE
Available from: 2016-04-08 Created: 2016-04-16 Last updated: 2017-11-30Bibliographically approved
4. Pricing derivatives under multiple stochastic factors by localized radial basis function methods
Open this publication in new window or tab >>Pricing derivatives under multiple stochastic factors by localized radial basis function methods
2017 (English)In: Computing Research Repository, no 1711.09852Article in journal (Other academic) Submitted
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-333468 (URN)
Projects
eSSENCE
Available from: 2017-11-27 Created: 2017-11-14 Last updated: 2018-08-22Bibliographically approved
5. Inuence of jump-at-default in IR and FX on Quanto CDS prices
Open this publication in new window or tab >>Inuence of jump-at-default in IR and FX on Quanto CDS prices
(English)Manuscript (preprint) (Other academic)
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-333467 (URN)
Available from: 2017-11-13 Created: 2017-11-13 Last updated: 2017-11-21
6. A meshfree approach to non-Newtonian free surface ice flow: Application to the Haut Glacier d'Arolla
Open this publication in new window or tab >>A meshfree approach to non-Newtonian free surface ice flow: Application to the Haut Glacier d'Arolla
2017 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 330, p. 633-649Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-310706 (URN)10.1016/j.jcp.2016.10.045 (DOI)000394408900034 ()
Projects
eSSENCE
Available from: 2016-10-24 Created: 2016-12-19 Last updated: 2017-11-21Bibliographically approved
7. Anisotropic radial basis function methods for continental size ice sheet simulations
Open this publication in new window or tab >>Anisotropic radial basis function methods for continental size ice sheet simulations
2018 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 372, p. 161-177Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-333469 (URN)10.1016/j.jcp.2018.06.020 (DOI)
Projects
eSSENCE
Available from: 2018-06-15 Created: 2017-11-14 Last updated: 2018-10-12Bibliographically approved
8. A least squares radial basis function partition of unity method for solving PDEs
Open this publication in new window or tab >>A least squares radial basis function partition of unity method for solving PDEs
2017 (English)In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 39, p. A2538-A2563Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-316488 (URN)10.1137/17M1118087 (DOI)000418659900017 ()
Projects
eSSENCE
Available from: 2017-11-09 Created: 2017-03-01 Last updated: 2018-06-16Bibliographically approved

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