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A multilevel Monte Carlo method for computing failure probabilities
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.
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
Place, publisher, year, edition, pages
2016. Vol. 4, 312-330 p.
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-262259DOI: 10.1137/140984294OAI: oai:DiVA.org:uu-262259DiVA: diva2:853139
Available from: 2016-04-05 Created: 2015-09-11 Last updated: 2017-12-04Bibliographically approved
In thesis
1. Multiscale and multilevel methods for porous media flow problems
Open this publication in new window or tab >>Multiscale and multilevel methods for porous media flow problems
2015 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

We consider two problems encountered in simulation of fluid flow through porous media. In macroscopic models based on Darcy's law, the permeability field appears as data.

The first problem is that the permeability field generally is not entirely known. We consider forward propagation of uncertainty from the permeability field to a quantity of interest. We focus on computing p-quantiles and failure probabilities of the quantity of interest. We propose and analyze improved standard and multilevel Monte Carlo methods that use computable error bounds for the quantity of interest. We show that substantial reductions in computational costs are possible by the proposed approaches.

The second problem is fine scale variations of the permeability field. The permeability often varies on a scale much smaller than that of the computational domain. For standard discretization methods, these fine scale variations need to be resolved by the mesh for the methods to yield accurate solutions. We analyze and prove convergence of a multiscale method based on the Raviart–Thomas finite element. In this approach, a low-dimensional multiscale space based on a coarse mesh is constructed from a set of independent fine scale patch problems. The low-dimensional space can be used to yield accurate solutions without resolving the fine scale.

Place, publisher, year, edition, pages
Uppsala University, 2015
Series
Information technology licentiate theses: Licentiate theses from the Department of Information Technology, ISSN 1404-5117 ; 2015-003
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
urn:nbn:se:uu:diva-262276 (URN)
Supervisors
Available from: 2015-09-09 Created: 2015-09-11 Last updated: 2017-08-31Bibliographically approved
2. Multiscale Methods and Uncertainty Quantification
Open this publication in new window or tab >>Multiscale Methods and Uncertainty Quantification
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
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:nbn:se:uu:diva-262354 (URN)978-91-554-9336-3 (ISBN)
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
3. Numerical Methods for Darcy Flow Problems with Rough and Uncertain Data
Open this publication in new window or tab >>Numerical Methods for Darcy Flow Problems with Rough and Uncertain Data
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
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:nbn:se:uu:diva-318589 (URN)978-91-554-9872-6 (ISBN)
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

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Elfverson, DanielHellman, FredrikMålqvist, Axel

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