| 1. |
- Wahlsten, Markus, 1986-, et al.
(författare)
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An efficient hybrid method for uncertainty quantification
- 2018
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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| 2. |
- Wahlsten, Markus, 1986-, et al.
(författare)
-
Robust boundary conditions for stochastic incompletely parabolic systems of equations
- 2018
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Ingår i: Journal of Computational Physics. - : Elsevier BV. - 0021-9991 .- 1090-2716. ; 371, s. 192-213
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 3. |
- Wahlsten, Markus, 1986-, et al.
(författare)
-
Stochastic Galerkin Projection and Numerical Integration for Stochastic Investigations of the Viscous Burgers’ Equation
- 2018
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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| 4. |
- Wahlsten, Markus, et al.
(författare)
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The effect of uncertain geometries on advection–diffusion of scalar quantities
- 2018
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Ingår i: BIT Numerical Mathematics. - : Springer. - 0006-3835 .- 1572-9125. ; 58:2, s. 509-529
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 5. |
- Wahlsten, Markus, 1986-
(författare)
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Uncertainty quantification for wave propagation and flow problems with random data
- 2018
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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