Fast Adaptive Algorithms in the Non-Standard Form for Multidimensional Problems
This work provides an alternative to the Fast Multipole Method for a wide class of kernels, with the advantage of easy extension to higher dimensions, benefiting computational scientists solving integral equations.
The paper presents a fast adaptive multiresolution algorithm for applying integral operators with radially symmetric kernels in 1D, 2D, and 3D, achieving controllable accuracy at a cost comparable to the Fast Multipole Method (FMM). The algorithm is demonstrated for Poisson and Schrödinger equations in 3D.
We present a fast, adaptive multiresolution algorithm for applying integral operators with a wide class of radially symmetric kernels in dimensions one, two and three. This algorithm is made efficient by the use of separated representations of the kernel. We discuss operators of the class $(-Δ+μ^{2}I)^{-α}$, where $μ\geq0$ and $0<α<3/2$, and illustrate the algorithm for the Poisson and Schrödinger equations in dimension three. The same algorithm may be used for all operators with radially symmetric kernels approximated as a weighted sum of Gaussians, making it applicable across multiple fields by reusing a single implementation. This fast algorithm provides controllable accuracy at a reasonable cost, comparable to that of the Fast Multipole Method (FMM). It differs from the FMM by the type of approximation used to represent kernels and has an advantage of being easily extendable to higher dimensions.