Why is the high frequency Helmholtz equation difficult to solve? (Hongkai Zhao, University of California)

24.09.2019 14:00

The Helmholtz equation is a linear partial differential equation modeling time harmonic wave propagation. It is notoriously difficult to solve numerically when the frequency is high. I will present our recent study (joint with Engquist) on approximate separability of the Green's function for the Helmholtz equation in high frequency limit. Computationally, being able to approximate a Green’s function by a sum with few separable terms is equivalent to the existence of low rank approximation of the discretized systems which can be utilized for developing fast solvers. Our study is based on geometric characterization of two Green's functions with different source locations. We provide both lower and upper bounds for the number of terms needed for a separable approximation of the Green’s function of the Helmholtz equation in terms of the frequency. Sharpness and implications of these bounds will be shown for computation setups that are commonly used in practice. I will also make a comparison with coercive elliptic differential equations with rough coefficients in divergence form.

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Room 623, Séminaire d'analyse numérique

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