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Multiscale Methods and Uncertainty Quantification
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.
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis we consider two great challenges in computer simulations of partial differential equations: multiscale data, varying over multiple scales in space and time, and data uncertainty, due to lack of or inexact measurements.

We develop a multiscale method based on a coarse scale correction, using localized fine scale computations. We prove that the error in the solution produced by the multiscale method decays independently of the fine scale variation in the data or the computational domain. We consider the following aspects of multiscale methods: continuous and discontinuous underlying numerical methods, adaptivity, convection-diffusion problems, Petrov-Galerkin formulation, and complex geometries.

For uncertainty quantification problems we consider the estimation of p-quantiles and failure probability. We use spatial a posteriori error estimates to develop and improve variance reduction techniques for Monte Carlo methods. We improve standard Monte Carlo methods for computing p-quantiles and multilevel Monte Carlo methods for computing failure probability.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2015. , 32 p.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1287
Keyword [en]
multiscale methods, finite element method, discontinuous Galerkin, Petrov-Galerkin, a priori, a posteriori, complex geometry, uncertainty quantification, multilevel Monte Carlo, failure probability
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
URN: urn:nbn:se:uu:diva-262354ISBN: 978-91-554-9336-3 (print)OAI: oai:DiVA.org:uu-262354DiVA: diva2:853534
Public defence
2015-10-30, Room 2446, Polacksbacken, Lägerhyddsvägen 2, Uppsala, 10:15 (English)
Opponent
Supervisors
Available from: 2015-10-08 Created: 2015-09-14 Last updated: 2015-10-12Bibliographically approved
List of papers
1. An adaptive discontinuous Galerkin multiscale method for elliptic problems
Open this publication in new window or tab >>An adaptive discontinuous Galerkin multiscale method for elliptic problems
2013 (English)In: Multiscale Modeling & simulation, ISSN 1540-3459, E-ISSN 1540-3467, Vol. 11, 747-765 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-200254 (URN)10.1137/120863162 (DOI)000325006000003 ()
Available from: 2013-08-01 Created: 2013-05-23 Last updated: 2017-12-06Bibliographically approved
2. Convergence of a discontinuous Galerkin multiscale method
Open this publication in new window or tab >>Convergence of a discontinuous Galerkin multiscale method
2013 (English)In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 51, 3351-3372 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-200256 (URN)10.1137/120900113 (DOI)000328903500017 ()
Available from: 2013-12-11 Created: 2013-05-23 Last updated: 2017-12-06Bibliographically approved
3. A discontinuous Galerkin multiscale method for convection–diffusion problems
Open this publication in new window or tab >>A discontinuous Galerkin multiscale method for convection–diffusion problems
2015 (English)In: Computing Research Repository, no 1509.03523Article in journal (Other academic) Submitted
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-262261 (URN)
Available from: 2015-09-11 Created: 2015-09-11 Last updated: 2015-10-12Bibliographically approved
4. On multiscale methods in Petrov–Galerkin formulation
Open this publication in new window or tab >>On multiscale methods in Petrov–Galerkin formulation
2015 (English)In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 131, 643-682 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-242930 (URN)10.1007/s00211-015-0703-z (DOI)000365081600002 ()
Available from: 2015-01-11 Created: 2015-02-03 Last updated: 2017-12-05Bibliographically approved
5. Multiscale methods for problems with complex geometry
Open this publication in new window or tab >>Multiscale methods for problems with complex geometry
2017 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 321, 103-123 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-262266 (URN)10.1016/j.cma.2017.03.023 (DOI)
Available from: 2017-03-31 Created: 2015-09-11 Last updated: 2017-05-10Bibliographically approved
6. Uncertainty quantification for approximate p-quantiles for physical models with stochastic inputs
Open this publication in new window or tab >>Uncertainty quantification for approximate p-quantiles for physical models with stochastic inputs
2014 (English)In: SIAM/ASA Journal on Uncertainty Quantification, ISSN 1560-7526, E-ISSN 2166-2525, Vol. 2, 826-850 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-242908 (URN)10.1137/140967039 (DOI)
Available from: 2014-12-23 Created: 2015-02-02 Last updated: 2017-12-05Bibliographically approved
7. A multilevel Monte Carlo method for computing failure probabilities
Open this publication in new window or tab >>A multilevel Monte Carlo method for computing failure probabilities
2016 (English)In: SIAM/ASA Journal on Uncertainty Quantification, ISSN 1560-7526, E-ISSN 2166-2525, Vol. 4, 312-330 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-262259 (URN)10.1137/140984294 (DOI)
Available from: 2016-04-05 Created: 2015-09-11 Last updated: 2017-12-04Bibliographically approved

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Elfverson, Daniel

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