M. S. Litsarev

M. S. Litsarev, I. V. Oseledets

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 $\mathcal{O}(nr M \log M + nr^2 M)$ flops and requires $\mathcal{O}(M r)$ floating-point numbers in memory, where $n$ is the dimension of the integral, $r \ll n$, and $M$ is the mesh size in one dimension. The presented technique can be generalized to the higher-order diffusion processes.

M. S. Litsarev, I. V. Oseledets

An efficient technique based on low-rank separated approximations is proposed for computation of three-dimensional integrals arising in the energy deposition model that describes ion-atomic collisions. Direct tensor-product quadrature requires grids of size $4000^3$ which is unacceptable. Moreover, several of such integrals have to be computed simultaneously for different values of parameters. To reduce the complexity, we use the structure of the integrand and apply numerical linear algebra techniques for the construction of low-rank approximation. The resulting algorithm is $10^3$ faster than spectral quadratures in spherical coordinates used in the original DEPOSIT code. The approach can be generalized to other multidimensional problems in physics.