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Uncertainty quantification for wave propagation and flow problems with random data
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis we study partial differential equations with random inputs. The effects that different boundary conditions with random data and uncertain geometries have on the solution are analyzed. Further, comparisons and couplings between different uncertainty quantification methods are performed. The numerical simulations are based on provably strongly stable finite difference formulations based on summation-by-parts operators and a weak implementation of boundary and interface conditions.

The first part of this thesis treats the construction of variance reducing boundary conditions. It is shown how the variance of the solution can be manipulated by the choice of boundary conditions, and a close relation between the variance of the solution and the energy estimate is established. The technique is studied on both a purely hyperbolic system as well as an incompletely parabolic system of equations. The applications considered are the Euler, Maxwell's, and Navier--Stokes equations.

The second part focuses on the effect of uncertain geometry on the solution. We consider a two-dimensional advection-diffusion equation with a stochastically varying boundary. We transform the problem to a fixed domain where comparisons can be made. Numerical results are performed on a problem in heat transfer, where the frequency and amplitude of the prescribed uncertainty are varied.

The final part of the thesis is devoted to the comparison and coupling of different uncertainty quantification methods. An efficiency analysis is performed using the intrusive polynomial chaos expansion with stochastic Galerkin projection, and nonintrusive numerical integration. The techniques are compared using the non-linear viscous Burgers' equation. A provably stable coupling procedure for the two methods is also constructed. The general coupling procedure is exemplified using a hyperbolic system of equations.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2018. , p. 26
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1921
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-146966DOI: 10.3384/diss.diva-146966ISBN: 9789176853399 (print)OAI: oai:DiVA.org:liu-146966DiVA, id: diva2:1196053
Public defence
2018-05-04, Ada Lovelace, B-huset, Campus Valla, Linköping, 13:15 (English)
Opponent
Supervisors
Available from: 2018-04-09 Created: 2018-04-09 Last updated: 2018-04-09Bibliographically approved
List of papers
1. Variance reduction through robust design of boundary conditions for stochastic hyperbolic systems of equations
Open this publication in new window or tab >>Variance reduction through robust design of boundary conditions for stochastic hyperbolic systems of equations
2015 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 282, p. 1-22Article in journal (Refereed) Published
Abstract [en]

We consider a hyperbolic system with uncertainty in the boundary and initial data. Our aim is to show that different boundary conditions gives different convergence rates of the variance of the solution. This means that we can with the same knowledge of data get a more or less accurate description of the uncertainty in the solution. A variety of boundary conditions are compared and both analytical and numerical estimates of the variance of the solution is presented. As applications, we study the effect of this technique on Maxwell's equations as well as on a subsonic outflow boundary for the Euler equations.

Place, publisher, year, edition, pages
Elsevier, 2015
Keywords
Uncertainty quantification, hyperbolic system, initial boundary value problems, well posed, stability, boundary conditions, stochastic data, variance reduction, robust design, summation-by parts
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-111893 (URN)10.1016/j.jcp.2014.10.061 (DOI)000346430700001 ()
Available from: 2014-11-07 Created: 2014-11-07 Last updated: 2018-04-09
2. Robust Boundary Conditions for Stochastic Incompletely Parabolic Systems of Equations
Open this publication in new window or tab >>Robust Boundary Conditions for Stochastic Incompletely Parabolic Systems of Equations
2016 (English)Report (Other academic)
Abstract [en]

We study an incompletely parabolic system in three space dimensions with stochastic boundary and initial data. We show how the variance of the solution can be manipulated by the boundary conditions, while keeping the mean value of the solution unaffected. Estimates of the variance of the solution is presented both analytically and numerically. We exemplify the technique by applying it to an incompletely parabolic model problem, as well as the one-dimensional compressible Navier-Stokes equations.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2016. p. 39
Series
LiTH-MAT-R, ISSN 0348-2960 ; 19
Keywords
uncertainty quantifcation, incompletely parabolic system, initial boundary value problems, stochastic data, variance reduction, robust design
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-133365 (URN)LiTH-MAT-R--2016/19--SE (ISRN)
Available from: 2016-12-22 Created: 2016-12-22 Last updated: 2018-04-09Bibliographically approved
3. The effect of uncertain geometries on advection–diffusion of scalar quantities
Open this publication in new window or tab >>The effect of uncertain geometries on advection–diffusion of scalar quantities
2018 (English)In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 58, no 2, p. 509-529Article in journal (Refereed) Published
Abstract [en]

The two dimensional advection–diffusion equation in a stochastically varyinggeometry is considered. The varying domain is transformed into a fixed one andthe numerical solution is computed using a high-order finite difference formulationon summation-by-parts form with weakly imposed boundary conditions. Statistics ofthe solution are computed non-intrusively using quadrature rules given by the probabilitydensity function of the random variable. As a quality control, we prove that thecontinuous problem is strongly well-posed, that the semi-discrete problem is stronglystable and verify the accuracy of the scheme. The technique is applied to a heat transferproblem in incompressible flow. Statistical properties such as confidence intervals andvariance of the solution in terms of two functionals are computed and discussed. Weshow that there is a decreasing sensitivity to geometric uncertainty as we graduallylower the frequency and amplitude of the randomness. The results are less sensitiveto variations in the correlation length of the geometry.

Place, publisher, year, edition, pages
Springer, 2018
Keywords
Incompressible flow, Advection–diffusion, Uncertainty quantification, Uncertain geometry, Boundary conditions, Parabolic problems, Variable coefficient, Temperature field, Heat transfer
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-140135 (URN)10.1007/s10543-017-0676-7 (DOI)000432718100012 ()
Note

Funding agencies: European Union [ACP3-GA-2013-605036]

Available from: 2017-08-31 Created: 2017-08-31 Last updated: 2018-06-04
4. Stochastic Galerkin Projection and Numerical Integration for Stochastic Investigations of the Viscous Burgers’ Equation
Open this publication in new window or tab >>Stochastic Galerkin Projection and Numerical Integration for Stochastic Investigations of the Viscous Burgers’ Equation
2018 (English)Report (Other academic)
Abstract [en]

We consider a stochastic analysis of the non-linear viscous Burgers’ equation and focus on the comparison between intrusive and non-intrusive uncer- tainty quantification methods. The intrusive approach uses a combination of polynomial chaos and stochastic Galerkin projection. The non-intrusive method uses numerical integration by combining quadrature rules and the probability density functions of the prescribed uncertainties. The two methods are applied to a provably stable formulation of the viscous Burgers’ equation, and compared. As measures of comparison: variance size, computational efficiency and accuracy are used.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2018. p. 14
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2018:2
Keywords
Uncertainty quantification, stochastic data, polynomial chaos, stochastic Galerkin, intrusive methods, non-intrusive methods, Burgers’ equation
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-146038 (URN)LiTH-MAT-R--2018/02--SE (ISRN)
Available from: 2018-03-22 Created: 2018-03-22 Last updated: 2018-04-09Bibliographically approved
5. An efficient hybrid method for uncertainty quantification
Open this publication in new window or tab >>An efficient hybrid method for uncertainty quantification
2018 (English)Report (Other academic)
Abstract [en]

A technique for coupling an intrusive and non-intrusive uncertainty quantification method is proposed. The intrusive approach uses a combination of polynomial chaos and stochastic Galerkin projection. The non-intrusive method uses numerical integration by combining quadrature rules and the probability density functions of the prescribed uncertainties. A strongly stable coupling procedure between the two methods at an interface is constructed. The efficiency of the hybrid method is exemplified using a hyperbolic system of equations, and verified by numerical experiments.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2018. p. 20
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2018:3
Keywords
Uncertainty quantification, numerical integration, stochastic Galerkin, polynomial chaos, coupling, projection operator
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-146036 (URN)LiTH-MAT-R--2018/03--SE (ISRN)
Available from: 2018-03-22 Created: 2018-03-22 Last updated: 2018-04-09Bibliographically approved

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