Stochastic Discontinuous Galerkin Methods for PDE-Based Models with Random Coefficients

Date

2023-6-14

Author

Çiloğlu, Pelin

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Uncertainty, such as uncertain parameters, arises from many complex physical systems in engineering and science, e.g., fluid dynamics, heat transfer, chemically reacting systems, underwater pollution, radiation transport, and oil field reservoirs. It is well known that these systems can be modeled by partial differential equations (PDEs) with random input data. However, the information available on the input data is very limited, which causes a high level of uncertainty in approximating the solution to these problems. Therefore, the idea of uncertainty quantification (UQ) has become a powerful tool to model such physical problems in the last decade. In this thesis, the aim is the development, analysis, and application of stochastic discontinuous Galerkin method for partial differential equation (PDE)--based models with random coefficients. As a model, we first focus on the single convection diffusion equation containing uncertainty. To identify the random coefficients, we use the well–known technique Karhunen Loève (KL) expansion. Stochastic Galerkin (SG) approach, turning the original problem containing uncertainties into a large system of deterministic problems, is applied to discretize the stochastic domain, while a discontinuous Galerkin method is preferred for the spatial discretization due to its better convergence behaviour for convection dominated PDEs. A priori and a posteriori error estimates are also derived. SG method generally results in a large coupled system of linear equations, the solution of which is computationally difficult to compute using standard solvers. Therefore, we provide low-rank iterative solvers for efficient computing of such solutions, which compute low-rank approximations to the solutions of those systems. Moreover, to overcome boundary and/or interior layers, localized regions where the derivative of the solution is large, an efficient adaptive algorithm is presented for the numerical solution of the parametric convection diffusion equations. On the other hand, certain parameters of a model are needed to be optimized in order to reach the desired target, for instance, the location where the oil is inserted into the medium, the temperature of a melting/heating process, or the shape of the aircraft wings. Therefore, we extend our findings to optimization problems and consider optimal control problems governed by convection diffusion equations involving random inputs.

Subject Keywords

PDE-constrained optimization, uncertainty quantification, stochastic discontinuous Galerkin, error estimates, low-rank approximation, convection diffusion equation with random coefficients, adaptive finite elements

URI

https://hdl.handle.net/11511/104248

Collections

Graduate School of Applied Mathematics, Thesis