Approximate Separability of the Green's Function of the Helmholtz Equation in the High Frequency Limit

The minimum number of terms that are needed in a separable approximation for a Green's function reveals the intrinsic complexity of the solution space of the underlying differential equation. It also has implications for whether low‐rank structures exist in the linear system after numerical discretization. The Green's function for a coercive elliptic differential operator in divergence form was shown to be highly separable [2], and efficient numerical algorithms exploiting low‐rank structures of the discretized systems were developed. In this work, a new approach to study the approximate separability of the Green's function of the Helmholtz equation in the high‐frequency limit is developed. We show (1) lower bounds based on an explicit characterization of the correlation between two Green's functions and a tight dimension estimate for the best linear subspace to approximate a set of decorrelated Green's functions, (2) upper bounds based on constructing specific separable approximations, and (3) sharpness of these bounds for a few case studies of practical interest.