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Numerical Methods for Darcy Flow Problems with Rough and Uncertain Data
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)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

We address two computational challenges for numerical simulations of Darcy flow problems: rough and uncertain data. The rapidly varying and possibly high contrast permeability coefficient for the pressure equation in Darcy flow problems generally leads to irregular solutions, which in turn make standard solution techniques perform poorly. We study methods for numerical homogenization based on localized computations. Regarding the challenge of uncertain data, we consider the problem of forward propagation of uncertainty through a numerical model. More specifically, we consider methods for estimating the failure probability, or a point estimate of the cumulative distribution function (cdf) of a scalar output from the model.

The issue of rough coefficients is discussed in Papers I–III by analyzing three aspects of the localized orthogonal decomposition (LOD) method. In Paper I, we define an interpolation operator that makes the localization error independent of the contrast of the coefficient. The conditions for its applicability are studied. In Paper II, we consider time-dependent coefficients and derive computable error indicators that are used to adaptively update the multiscale space. In Paper III, we derive a priori error bounds for the LOD method based on the Raviart–Thomas finite element.

The topic of uncertain data is discussed in Papers IV–VI. The main contribution is the selective refinement algorithm, proposed in Paper IV for estimating quantiles, and further developed in Paper V for point evaluation of the cdf. Selective refinement makes use of a hierarchy of numerical approximations of the model and exploits computable error bounds for the random model output to reduce the cost complexity. It is applied in combination with Monte Carlo and multilevel Monte Carlo methods to reduce the overall cost. In Paper VI we quantify the gains from applying selective refinement to a two-phase Darcy flow problem.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2017. , 41 p.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1495
Keyword [en]
numerical homogenization, multiscale methods, rough coefficients, high contrast coefficients, mixed finite elements, cdf estimation, multilevel Monte Carlo methods, Darcy flow problems
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
URN: urn:nbn:se:uu:diva-318589ISBN: 978-91-554-9872-6 (print)OAI: oai:DiVA.org:uu-318589DiVA: diva2:1085068
Public defence
2017-05-19, ITC 2446, Lägerhyddsvägen 2, Uppsala, 10:15 (English)
Opponent
Supervisors
Available from: 2017-04-26 Created: 2017-03-27 Last updated: 2017-06-28
List of papers
1. Contrast independent localization of multiscale problems
Open this publication in new window or tab >>Contrast independent localization of multiscale problems
2016 (English)In: Computing Research Repository, no 1610.07398Article in journal (Other academic) Submitted
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-318587 (URN)
Available from: 2017-03-23 Created: 2017-03-26 Last updated: 2017-06-30Bibliographically approved
2. Numerical homogenization of time-dependent diffusion
Open this publication in new window or tab >>Numerical homogenization of time-dependent diffusion
2017 (English)In: Computing Research Repository, no 1703.08857Article in journal (Other academic) Submitted
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-318588 (URN)
Available from: 2017-03-26 Created: 2017-03-26 Last updated: 2017-06-30Bibliographically approved
3. Multiscale mixed finite elements
Open this publication in new window or tab >>Multiscale mixed finite elements
2016 (English)In: Discrete and Continuous Dynamical Systems. Series S, ISSN 1937-1632, E-ISSN 1937-1179, Vol. 9, 1269-1298 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-262258 (URN)10.3934/dcdss.2016051 (DOI)000387662300002 ()
Available from: 2016-10-15 Created: 2015-09-11 Last updated: 2017-03-27Bibliographically approved
4. 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: 2017-03-27Bibliographically approved
5. 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: 2017-03-27Bibliographically approved
6. Multilevel Monte Carlo methods for computing failure probability of porous media flow systems
Open this publication in new window or tab >>Multilevel Monte Carlo methods for computing failure probability of porous media flow systems
2016 (English)In: Advances in Water Resources, ISSN 0309-1708, E-ISSN 1872-9657, Vol. 94, 498-509 p.Article in journal (Refereed) Published
National Category
Computational Mathematics Oceanography, Hydrology, Water Resources
Identifiers
urn:nbn:se:uu:diva-298476 (URN)10.1016/j.advwatres.2016.06.007 (DOI)000381529000037 ()
Available from: 2016-06-15 Created: 2016-07-05 Last updated: 2017-03-27Bibliographically approved

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