Multigrid methods for wave scattering problems governed by the Helmholtz and Schrödinger equation.
22 May 2015
UAntwerp, Campus Middelheim, Room G0.10 - Middelheimlaan 1 - 2020 Antwerp
PhD defence Siegfried Cools - Faculty of Science, Department of Mathematics and Computer Science
The propagation and scattering of waves appears in a large number of contemporary physical and chemical applications among which acoustics, electromagnetics and quantum-mechanics. Moreover, the wave character of various types of vibration and radiation is widely exploited in modern imaging techniques. In seismology and computed tomography, information about the inner structure of an object (such as the earth or the human body) is obtained by measuring the scattered wave pattern at a certain distance to the object of interest.
The numerical simulation of state-of-the-art wave scattering problems in three (or even higher) spatial dimensions is a substantial mathematical problem, in which the development, analysis and improvement of mathematical methods for the solution of the time-independent Helmholtz and Schrödinger equation play a crucial part. The fast convergence of iterative schemes like Krylov subspace methods and multigrid methods for large and high-dimensional problems is particularly useful in this setting.
The research performed in this thesis is situated within the search for efficient solution methods for indefinite Helmholtz equations and the closely related problem of the efficient calculation of the near- and far field scattered wave patterns at medium to large distances from the object. The latter problem is particularly interesting for applications in the area of inverse problems such as medical imaging and seismology.
The results in this study are primarily based on the properties of the damped Helmholtz problem (complex shifted Laplacian), where the discretization is defined on a complex valued spatial domain. This perturbed problem strongly resembles the original Helmholtz problem; however, contrary to the latter the complex-valued problem can be very efficiently inverted using iterative methods. One of the main research questions which we aim to answer in this thesis concerns the applicability of the favorable properties of the damped Helmholtz problem for the efficient solution of the original Helmholtz problem, and for the calculation of related wave patterns such as the far field map.
In 2013, a panel of judges at the 16th Copper Mountain Conference accredited Siegfried Cools with the annual Student Paper Award for the original and innovative research reported in the paper 'An efficient Multigrid calculation of the Far field map for Helmholtz problems', SIAM Journal on Scientific Computing, 36 (3): pp. B367-B395, July 2014.