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  • 1.
    Nordström, Jan
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Wahlsten, Markus
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Robust Design of Initial Boundary Value Problems2019In: Uncertainty Management for Robust Industrial Design in Aeronautics: Findings and Best Practice Collected During UMRIDA, a Collaborative Research Project (2013–2016) Funded by the European Union / [ed] Hirsch, C.; Wunsch, D.; Szumbarski, J.; Łaniewski-Wołłk, Ł.; Pons-Prats, J., Springer, 2019, p. 463-478Chapter in book (Refereed)
    Abstract [en]

    We study hyperbolic and incompletely parabolic systems with stochastic boundary and initial data. Estimates of the variance of the solution are presented both analytically and numerically. It is shown that one can reduce the variance for a given input, with a specific choice of boundary condition. The technique is applied to the Maxwell, Euler, and Navier–Stokes equations. Numerical calculations corroborate the theoretical conclusions.

  • 2.
    Nordström, Jan
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Wahlsten, Markus
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Variance reduction through robust design of boundary conditions for stochastic hyperbolic systems of equations2014Report (Other academic)
    Abstract [en]

    We consider a hyperbolic system in one space dimension with uncertainty in the boundary and initial data. Our aim is to show that di erent 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 an application, we study the effect of this technique on a subsonic outow boundary for the Euler equations.

  • 3.
    Nordström, Jan
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Wahlsten, Markus
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Variance reduction through robust design of boundary conditions for stochastic hyperbolic systems of equations2015In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 282, p. 1-22Article in journal (Refereed)
    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.

  • 4.
    Nordström, Jan
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Wahlsten, Markus
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Nikkar, Samira
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
    Boundary Conditions for Hyperbolic Systems of Equations on Curved Domains2014Report (Other academic)
    Abstract [en]

    Our focus in this paper is on the fundamental system of partial differential equation with boundary conditions (the continuous problem) that all types of numerical methods must respect. First, a constant coefficient hyperbolic system of equations which turns into a variable coefficient system of equations by transforming to a non-cartesian domain is considered. We discuss possible formulations of time-dependent boundary conditions leading to well-posed or strongly well-posed problems. Next, we re-use the previous theoretical derivations for the problem with boundary conditions applied at the wrong position and/or with an incorrect normal (a typical result with a less than perfect mesh generator). Possible error sources are discussed and a crude error estimate is derived.

  • 5.
    Wahlsten, Markus
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Uncertainty quantification for wave propagation and flow problems with random data2018Doctoral 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.

  • 6.
    Wahlsten, Markus
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    An efficient hybrid method for uncertainty quantification2018Report (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.

  • 7.
    Wahlsten, Markus
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    An Investigation of Uncertainty due to Stochastically Varying Geometry: An Initial Study2015Conference paper (Other academic)
    Abstract [en]

    We study hyperbolic problems with uncertain stochastically varying geometries. Our aim is to investigate how the stochastically varying uncertainty in the geometry affects the solution of the partial differential equation in terms of the mean and variance of the solution. The problem considered is the two dimensional advection equation on a general domain, which is transformed using curvilinear coordinates to a unit square. The numerical solution is computed using a high order finite difference formulation on summation-by-parts form with weakly imposed boundary conditions. The statistics of the solution are computed nonintrusively using quadrature rules given by the probability density function of the random variable.

    We prove that the continuous problem is strongly well-posed and that the semi-discrete problem is strongly stable. Numerical calculations using the method of manufactured solution verify the accuracy of the scheme and the statistical properties of the solution are discussed.

  • 8.
    Wahlsten, Markus
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Correction: On Stochastic Investigation of Flow Problems Using the Viscous Burgers’ Equation as an Example (vol 81, p 1111, 2019)2019In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 81, no 2, p. 1118-1118Article in journal (Other academic)
  • 9.
    Wahlsten, Markus
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Galerkin projection and numerical integration for a stochastic investigation of the viscous Burgers equation: An initial attempt2019In: Numerical Mathematics and Advanced Applications ENUMATH 2017. ENUMATH 2017 / [ed] Florin Adrian Radu, Kundan Kumar, Inga Berre, Jan Martin Nordbotten, Iuliu Sorin Pop, Cham: Springer, 2019, Vol. 126, p. 1005-1013Chapter in book (Refereed)
    Abstract [en]

    We consider a stochastic analysis of the non-linear viscous Burgers’ equation and focus on the comparison between intrusive and non-intrusive uncertainty 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.

  • 10.
    Wahlsten, Markus
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    On Stochastic Investigation of Flow Problems Using the Viscous Burgers’ Equation as an Example2019In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 81, no 2, p. 1111-1117Article in journal (Refereed)
    Abstract [en]

    We consider a stochastic analysis of non-linear viscous fluid flow problems with smooth and sharp gradients in stochastic space. As a representative example we consider the viscous Burgers’ equation and compare two typical intrusive and non-intrusive uncertainty quantification methods. The specific intrusive approach uses a combination of polynomial chaos and stochastic Galerkin projection. The specific non-intrusive method uses numerical integration by combining quadrature rules and the probability density functions of the prescribed uncertainties. The two methods are compared in terms of error in the estimated variance, computational efficiency and accuracy. This comparison, although not general, provide insight into uncertainty quantification of problems with a combination of sharp and smooth variations in stochastic space. It suggests that combining intrusive and non-intrusive methods could be advantageous.

  • 11.
    Wahlsten, Markus
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Robust boundary conditions for stochastic incompletely parabolic systems of equations2018In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 371, p. 192-213Article in journal (Refereed)
    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.

    The full text will be freely available from 2020-05-21 14:42
  • 12.
    Wahlsten, Markus
    et al.
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Nordström, Jan
    Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
    Stochastic Galerkin Projection and Numerical Integration for Stochastic Investigations of the Viscous Burgers’ Equation2018Report (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.

1 - 12 of 12
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  • ieee
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  • fi-FI
  • nn-NO
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