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Modulated plane wave methods for Helmholtz problems in heterogeneous media

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Inverse Problems

A major challenge in seismic imaging is full waveform inversion in the frequency domain. If an acoustic model is assumed the underlying problem formulation is a Helmholtz equation with varying speed of sound. Typically, in seismic applications the solution has many wavelengths across the computational domain, leading to very large linear systems after discretisation with standard finite element methods. Much progress has been achieved in recent years by the development of better preconditioners for the iterative solution of these linear systems. But the fundamental problem of requiring many degrees of freedom per wavelength for the discretisation remains.

For problems in homogeneous media, that is, spatially constant wave velocity, plane wave finite element methods have gained significant attention. The idea is that instead of polynomials on each element we use a linear combination of oscillatory plane wave solutions. These basis functions already oscillate with the right wavelength, leading to a significant reduction in the required number of unknowns. However, higher-order convergence is only achieved for problems with constant or piecewise constant media.

In this talk we discuss the use of modulated plane waves in heterogeneous media, products of low-degree polynomials and oscillatory plane wave solutions for a (local) average homogeneous medium. The idea is that high-order convergence in a varying medium is recovered due to the polynomial modulation of the plane waves. Wave directions are chosen based on information from raytracing or other fast solvers for the eikonal equation. This approach is related to the Amplitude FEM originally proposed by Giladi and Keller in 2001. However, for the assembly of the systems we will use a discontinuous Galerkin method, which allows a simple way of incorporating multiple phase information in one element. We will discuss the dependence of the element sizes on the wavelenth and the accuracy of the phase information, and present several examples that demonstrate the properties of modulated plane wave methods for heterogeneous media problems.

This talk is part of the Isaac Newton Institute Seminar Series series.

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