On the fast computation of high dimensional volume potentials
This work provides a scalable computational tool for high-dimensional potential problems, which are common in physics and engineering, but the method is an incremental improvement over existing approximate approximation techniques.
The paper proposes a fast, high-order method for approximating volume potentials that scales linearly with dimension, achieving O(h^8) accuracy for the Newton potential in dimensions up to 200,000.
A fast method of an arbitrary high order for approximating volume potentials is proposed, which is effective also in high dimensional cases. Basis functions introduced in the theory of approximate approximations are used. Results of numerical experiments, which show approximation order O(h^8) for the Newton potential in high dimensions, for example, for n= 200 000, are provided. The computation time scales linearly in the space dimension. New one-dimensional integral representations with separable integrands of the potentials of advection-diffusion and heat equations are obtained.