Volume 28, Issue 4
Parameter Identification in Uncertain Scalar Conservation Laws Discretized with the Discontinuous Stochastic Galerkin Scheme

Louisa SchlachterClaudia Totzeck

Commun. Comput. Phys., 28 (2020), pp. 1585-1608.

Published online: 2020-08

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  • Abstract

We study an identification problem which estimates the parameters of the underlying random distribution for uncertain scalar conservation laws. The hyperbolic equations are discretized with the so-called discontinuous stochastic Galerkin method, i.e., using a spatial discontinuous Galerkin scheme and a Multielement stochastic Galerkin ansatz in the random space. We assume an uncertain flux or uncertain initial conditions and that a data set of an observed solution is given. The uncertainty is assumed to be uniformly distributed on an unknown interval and we focus on identifying the correct endpoints of this interval. The first-order optimality conditions from the discontinuous stochastic Galerkin discretization are computed on the time-continuous level. Then, we solve the resulting semi-discrete forward and backward schemes with the Runge-Kutta method. To illustrate the feasibility of the approach, we apply the method to a stochastic advection and a stochastic equation of Burgers' type. The results show that the method is able to identify the distribution parameters of the random variable in the uncertain differential equation even if discontinuities are present.

  • Keywords

Uncertainty quantification, polynomial chaos, stochastic Galerkin, multielement, discontinuous Galerkin, parameter identification, optimization.

  • AMS Subject Headings

35L60, 35Q62, 37L65, 49K20, 65M08, 65M60

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-28-1585, author = {Schlachter , Louisa and Totzeck , Claudia}, title = {Parameter Identification in Uncertain Scalar Conservation Laws Discretized with the Discontinuous Stochastic Galerkin Scheme}, journal = {Communications in Computational Physics}, year = {2020}, volume = {28}, number = {4}, pages = {1585--1608}, abstract = {

We study an identification problem which estimates the parameters of the underlying random distribution for uncertain scalar conservation laws. The hyperbolic equations are discretized with the so-called discontinuous stochastic Galerkin method, i.e., using a spatial discontinuous Galerkin scheme and a Multielement stochastic Galerkin ansatz in the random space. We assume an uncertain flux or uncertain initial conditions and that a data set of an observed solution is given. The uncertainty is assumed to be uniformly distributed on an unknown interval and we focus on identifying the correct endpoints of this interval. The first-order optimality conditions from the discontinuous stochastic Galerkin discretization are computed on the time-continuous level. Then, we solve the resulting semi-discrete forward and backward schemes with the Runge-Kutta method. To illustrate the feasibility of the approach, we apply the method to a stochastic advection and a stochastic equation of Burgers' type. The results show that the method is able to identify the distribution parameters of the random variable in the uncertain differential equation even if discontinuities are present.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2019-0221}, url = {http://global-sci.org/intro/article_detail/cicp/18112.html} }
TY - JOUR T1 - Parameter Identification in Uncertain Scalar Conservation Laws Discretized with the Discontinuous Stochastic Galerkin Scheme AU - Schlachter , Louisa AU - Totzeck , Claudia JO - Communications in Computational Physics VL - 4 SP - 1585 EP - 1608 PY - 2020 DA - 2020/08 SN - 28 DO - http://doi.org/10.4208/cicp.OA-2019-0221 UR - https://global-sci.org/intro/article_detail/cicp/18112.html KW - Uncertainty quantification, polynomial chaos, stochastic Galerkin, multielement, discontinuous Galerkin, parameter identification, optimization. AB -

We study an identification problem which estimates the parameters of the underlying random distribution for uncertain scalar conservation laws. The hyperbolic equations are discretized with the so-called discontinuous stochastic Galerkin method, i.e., using a spatial discontinuous Galerkin scheme and a Multielement stochastic Galerkin ansatz in the random space. We assume an uncertain flux or uncertain initial conditions and that a data set of an observed solution is given. The uncertainty is assumed to be uniformly distributed on an unknown interval and we focus on identifying the correct endpoints of this interval. The first-order optimality conditions from the discontinuous stochastic Galerkin discretization are computed on the time-continuous level. Then, we solve the resulting semi-discrete forward and backward schemes with the Runge-Kutta method. To illustrate the feasibility of the approach, we apply the method to a stochastic advection and a stochastic equation of Burgers' type. The results show that the method is able to identify the distribution parameters of the random variable in the uncertain differential equation even if discontinuities are present.

Louisa Schlachter & Claudia Totzeck. (2020). Parameter Identification in Uncertain Scalar Conservation Laws Discretized with the Discontinuous Stochastic Galerkin Scheme. Communications in Computational Physics. 28 (4). 1585-1608. doi:10.4208/cicp.OA-2019-0221
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