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.
An adaptive radial basis function approach for efficiently solving multidimensional spatiotemporal integrodifferential equations
Nuffield School of Medicine, University of Oxford（牛津大学纳菲尔德医学院）；Department of Mathematics, University of Houston（休斯顿大学数学系）；School of Mathematics, University of Birmingham（伯明翰大学数学学院）
MATLAB-Based Layerwise Self-Adaptive Physics-Informed Neural Network in Applications to Multidimensional Coupled Burgers' Equations with High Reynolds Numbers
Utah Valley University（犹他山谷大学）；Westminster University（韦敏斯特大学）
Fast multidimensional convolution in low-rank formats via cross approximation
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究所）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Institute of Numerical Mathematics, Russian Academy of Sciences（俄罗斯科学院数值数学研究所）
Space-time adaptive ADER-DG finite element method with LST-DG predictor and a posteriori sub-cell ADER-WENO finite-volume limiting for multidimensional detonation waves simulation
