Georgia Institute of Technology College of Sciences

Approximate separable representation of the Green’s functions for differential operators is a fundamental question in the analysis of differential equations and development of efficient numerical algorithms. It can reveal intrinsic complexity, e.g., Kolmogorov n-width or degrees of freedom of the corresponding differential equation. Computationally, 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 the discretized system which can be explored for matrix compression and fast solution techniques such as in fast multiple method and direct matrix inverse solver. In this talk, we will mainly focus on Helmholtz equation in the high frequency limit for which we developed a new approach to study the approximate separability of Green’s function based on an geometric 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 and implications for computation setups that are commonly used in practice. We will also make comparisons with other types of differential operators such as coercive elliptic differential operator with rough coefficients in divergence form and hyperbolic differential operator. This is a joint work with Bjorn Engquist.