A FAST COLLOCATION METHOD FOR SOLVING STOCHASTIC INTEGRAL EQUATIONS

摘要

Based on sparse grid multiscale piecewise polynomial bases, we develop a fast collocation method for solving Fredholm integral equations of the second kind with stochastic loading terms. It is proved that the proposed method preserves the optimal rate of convergence and has linear (up to a logarithmic factor) computational complexity. The reduction in computational costs come from three aspects: the adoption of the collocation principle, the use of the sparse grid for the solution space, and the employment of a truncation strategy for the coefficient matrix of the resulting linear system. Numerical experiments confirm the theoretical results on approximation accuracy and computational efficiency of the proposed method.