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Hongkai Zhao : Approximate Separability of Greens Function for Helmholtz Equation in the High Frequency Limit

Approximate separable representations of Green’s functions for differential operators is a basic and important question in the analysis of differential equations, the development of efficient numerical algorithms and imaging. Being able to approximate a Green’s function as a sum with few separable terms is equivalent to low rank properties of corresponding numerical solution operators. This will allow for matrix compression and fast solution techniques. Green's functions for coercive elliptic differential operators have been shown to be highly separable and the resulting low rank property for discretized system was explored to develop efficient numerical algorithms. However, the case of Helmholtz equation in the high frequency limit is more challenging both mathematically and numerically. We introduce new tools based on the study of relation between two Green’s functions with different source points and a tight dimension estimate for the best linear subspace approximating a set of almost orthogonal vectors to prove new lower bounds for the number of terms in the representation for the Green's function for Helmholtz operator in the high frequency limit. Upper bounds are also derived. We give explicit sharp estimates for cases that are common in practice and present numerical examples. This is a joint work with Bjorn Engquist.

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