Projection schemes for high-dimensional problems in stochastic structural dynamics

The focus of the present thesis is to formulate efficient schemes to solve high-dimensional stochastic ordinary differential equations (SODEs) encountered in stochastic structural dynamics. Most of the methods for the parametric uncertainty analysis of SODEs suffer from the curse of dimensionality. To alleviate it, we investigate a few different methods. Firstly, we formulate a Generalized Spectral Decomposition (GSD) method for linear SODEs. It is a stochastic Galerkin method proposed to alleviate the curse of dimensionality faced by the classical gPC-based stochastic Galerkin projection scheme. Numerical studies suggest that it is not straightforward to scale the GSD method to large-scale problems since it is a sequential iterative scheme and there is no clear guidance on how to optimize the number of expansion terms. Secondly, we propose an anchored ANOVA Petrov-Galerkin (AAPG) scheme for linear SODEs. The main idea of the AAPG scheme is to approximate the dynamic response using a Hoeffding functional ANOVA decomposition along with appropriate constraints to ensure the uniqueness of the decomposition. We show that when the test functions in the weighted residual form are chosen appropriately, the original high-dimensional stochastic problem can be decoupled into a sequence of low-dimensional stochastic subproblems that can be solved independently of each other. We also extend the AAPG scheme to nonlinear SODEs and reach the conclusion that the AAPG scheme holds significant potential to alleviate the curse of dimensionality for linear and nonlinear SODEs, as confirmed by numerical studies. Finally, we carry out a theoretical analysis of the AAPG scheme focusing on a priori error estimation and computational cost. This analysis motivates the formulation of an adaptive version of the AAPG scheme that enables additional improvements in computational efficiency by reducing the number of subproblems to be solved. Numerical studies on test problems with up to 80 stochastic degrees-of-freedom suggest that the adaptive AAPG can provide a similar level of accuracy as the original scheme while providing a substantial reduction in computational cost.Ph.D