Adaptive Methods Exploring Intrinsic Sparse Structures of Stochastic Partial Differential Equations
Many physical and engineering problems involving uncertainty enjoy certain low-dimensional structures, e.g., in the sense of Karhunen-Loeve expansions (KLEs), which in turn indicate the existence of reduced-order models and better formulations for efficient numerical simulations. In this thesis, we...
| Summary: | Many physical and engineering problems involving uncertainty enjoy certain low-dimensional structures, e.g., in the sense of Karhunen-Loeve expansions (KLEs), which in turn indicate the existence of reduced-order models and better formulations for efficient numerical simulations. In this thesis, we target a class of time-dependent stochastic partial differential equations whose solutions enjoy such structures at any time and propose a new methodology (DyBO) to derive equivalent systems whose solutions closely follow KL expansions of the original stochastic solutions. KL expansions are known to be the most compact representations of stochastic processes in an L<sup>2</sup> sense. Our methods explore such sparsity and offer great computational benefits compared to other popular generic methods, such as traditional Monte Carlo (MC), generalized Polynomial Chaos (gPC) method, and generalized Stochastic Collocation (gSC) method. Such benefits are demonstrated through various numerical examples ranging from spatially one-dimensional examples, such as stochastic Burgers' equations and stochastic transport equations to spatially two-dimensional examples, such as stochastic flows in 2D unit square. Parallelization is also discussed, aiming toward future industrial-scale applications. In addition to numerical examples, theoretical aspects of DyBO are also carefully analyzed, such as preservation of bi-orthogonality, error propagation, and computational complexity. Based on theoretical analysis, strategies are proposed to overcome difficulties in numerical implementations, such as eigenvalue crossing and adaptively adding or removing mode pairs. The effectiveness of the proposed strategies is numerically verified. Generalization to a system of SPDEs is considered as well in the thesis, and its success is demonstrated by applications to stochastic Boussinesq convection problems. Other generalizations, such as generalized stochastic collocation formulation of DyBO method, are also discussed. |
|---|