Iterative Helmholtz Solver: Towards more accuracy and scalability of the deflation based solver
Vandana Dwarka

Supervisor: Kees Vuik

Site of the project: TU Delft

start of the project: July 2016

In November 2016 the Interim Thesis has appeared and a presentation has been given.

The Master project has been finished in September 2017 by the completion of the Masters Thesis and a final presentation has been given.

For working address etc. we refer to our alumnipage.

Summary of the master project:
Helmholtz problems are use in many applications as there are Sonar research, biomedical imaging and seismics. In order to have detailed images there is a strong need to use higher wavenumbers and thus finer grids. Both lead to an increase in computing time.

Using the well known Complex Shifted Laplace (CSLP) preconditioner the number of required iterations increases only linearly with the wavenumber and it is independent of the size of the grid. Combining this solver with Deflation has lead to the MLKM and ADEF/CLSP method which is also independent of the wavenumber for medium-size and large wavenumbers. For very large wavenumbers it appears that the ADEF/CLSP again shows a slow increase in the number of iterations and the method is not scalable.

In this master thesis we will investigate the reason for this behaviour and look for a better version of the ADEF/CLSP method in order to make it scalable. Another issue which will be studied is the effect of pollution on the solution and the performance of the solvers.

After a literature study, we will start by a detailed study of the close to zero eigenvalues which seems to be the cause of the non-scalability. The optimal choice of the shift parameter in the ADEF/CLSP method will also be looked into. If time permits the effect of pollution on the quality of the solution will also be investigated.

Some literature:

Y.A. Erlangga and C.W. Oosterlee and C. Vuik
A Novel Multigrid Based Preconditioner For Heterogeneous Helmholtz Problems
SIAM J. Sci. Comput.,27, pp. 1471-1492, 2006

A.H. Sheikh and D. Lahaye and C. Vuik
On the convergence of shifted Laplace preconditioner combined with multilevel deflation
Numerical Linear Algebra with Applications, 20, pp. 645-662, 2013

A.H. Sheikh and D. Lahaye and L. Garcia Ramos and R. Nabben and C. Vuik
Accelerating the shifted Laplace preconditioner for the Helmholtz equation by multilevel deflation
Journal of Computational Physics, 322, pp. 473-490, 2016

Contact information: Kees Vuik

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