**
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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