Approximate Separability of Green's Function for High Frequency Helmholtz Equations

Abstract

Abstract : Approximate separable representations of Green's functions for differential operators is a basic and an important aspect in the analysis of differential equations and in the development of efficient numerical algorithms for solving them. Being able to approximate a Green's function as a sum with few separable terms is equivalent to the existence of low rank approximation of corresponding discretized system. This property can be explored for matrix compression and efficient numerical algorithms. Green's functions for coercive elliptic differential operators in divergence form have been shown to be highly separable and low rank approximation for their discretized systems has been utilized to develop efficient numerical algorithms. The case of Helmholtz equation in the high frequency limit is more challenging both mathematically and numerically. In this work, we develop a new approach to study approximate separability for the Green's function of Helmholtz equation in the high frequency limit based on an explicit characterization of the relation between two Green's functions and a tight dimension estimate for the best linear subspace approximating a set of almost orthogonal vectors. We derive both lower bounds and upper bounds and show their sharpness for cases that are commonly used in practice.