Convergence Analysis of an Adaptive Sparse Quadrature for High Dimensional Integration with Gaussian Random Variables
In this work we analyze and demonstrate the dimension independent convergence property of an adaptive sparse quadrature for high/infinite dimensional numerical integration problems with Gaussian random variables. We construct the adaptive sparse quadrature by tensorization of univariate quadrature in an admissible index set by a greedy algorithm. Several univariate quadrature rules, including Gauss Hermite rule, transformed Gauss Kronrod Patterson rule, and Genz Keister rule are investigated. The best $N$ term algebraic convergence rate $N^ s} is obtained under certain assumptions on the exactness and the boundedness of the univariate quadrature rules as well as the regularity of the parametric map with respect to the high/infinite dimensional Gaussian distributed parameters. The rate $s$ is shown to be dependent only on a sparsity parameter that controls the regularity and in particular independent of the number of parameter dimensions. Examples of nonlinear parametric function and parametric partial differential equations (PDE) are provided to verify the regularity assumption. Numerical experiments are performed on the integration for infinite dimensional parametric function, parametric PDE, and parametric Bayesian inversion to demonstrate the dimension independent convergence of the adaptive sparse quadrature errors, which is faster than that of Monte Carlo quadrature errors for the test problems.