Seminar in Numerical Analysis: Wim Vanroose (University of Antwerpen)

Efficient multigrid calculation of the far field map of the Helmholtz equation

Many imaging systems such phase contrast tomography, internal reflection microscopy or reaction microscopes measure the far or near field of the scattered wave. We present an efficient and scalable method to calculate the far- and near field of a Helmholtz equation describing a given object. The far and near field solution can be written as an integral of the Greens function multiplied by the solution of the Helmholtz equation with absorbing boundary conditions. By deforming the contour of the integral we only require the numerical solution of the Helmholtz equation along a complex valued contour.  We show that Helmholtz equation along this contour is equivalent to a complex shifted Laplacian that can be solved efficiently by multigrid. This results in an scalable method to calculated the far and near field integral. We discuss this numerical method, show its applicability, scalability and discuss its limitations.

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