A low-rank approach to the computation of path integrals

We present a method for solving the reaction-diffusion equation with general potential in free space. It is based on the approximation of the Feynman-Kac formula by a sequence of convolutions on sequentially diminishing grids. For computation of the convolutions we propose a fast algorithm based on the low-rank approximation of the Hankel matrices. The algorithm has complexity of O(nrMlog⁡M+nr2M) flops and requires O(Mr) floating-point numbers in memory, where n is the dimension of the integral, r≪n, and M is the mesh size in one dimension. The presented technique can be generalized to the higher-order diffusion processes.