Fast summation by interval clustering for an evolution equation with memory

TL;DR

This paper introduces an efficient algorithm for solving fractional diffusion equations that significantly reduces computational complexity from quadratic to near-linear time by employing interval clustering techniques.

Contribution

It presents a novel fast summation method based on interval clustering to efficiently handle the memory term in fractional diffusion equations.

Findings

Reduced computational complexity to O(MN log N)

Lowered active memory requirements to O(M log N)

Demonstrated effectiveness through numerical experiments

Abstract

We solve a fractional diffusion equation using a piecewise-constant, discontinuous Galerkin method in time combined with a continuous, piecewise-linear finite element method in space. If there are $N$ time levels and $M$ spatial degrees of freedom, then a direct implementation of this method requires $O(N^2M)$ operations and $O(NM)$ active memory locations, owing to the presence of a memory term: at each time step, the discrete evolution equation involves a sum over \emph{all} previous time levels. We show how the computational cost can be reduced to $O(MN\log N)$ operations and $O(M\log N)$ active memory locations.