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 ( n r M log ? M + n r 2 M ) flops and requires O ( M r ) 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.

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