Abstract
The discretisation of boundary value problems for the Helmholtz equation (frequency domain wave equation) leads to linear systems that are non-self adjoint and highly indefinite. The iterative solution of these problems is difficult and much recent research has focussed on the construction of good preconditioners. This thesis examines the effect of adding absorption to sweeping-type preconditioners for Helmholtz problems. The application of interest for the Helmholtz problems in this thesis is seismic inversion.First we look at the beneficial effects of absorption on low-rank separable
expansions for the Hankel function, that provide valuable theoretical motivation
for sweeping-type preconditioners. We find that when absorption is included,
there are three ways in which benefits are seen: the quality of the separable
expansion increases, or the size of the domains for which the separable expansion is valid increases, or a lower rank may be sufficient to gain the same quality of separable expansion.
Next we focus on the effect of adding absorption to Schur complement matrices
arising in the construction of sweeping-type preconditioners. The theoretical and
numerical results show good agreement on the following points: the dependence
of the rank upon the quality of the approximation, the independence of the
rank from the wavenumber, the exponential improvement in the quality of the
approximation with absorption and the ranks remaining low for taller domains
when absorption is included.
Finally we look at the effect of adding absorption on the iteration counts of
several variants of sweeping preconditioners. We find that in some cases we see
improvements due to absorption and in others we do not. The performance of
the iterative method is highly dependent on the parameters used in both the
discretisation of the problem and the construction of the preconditioners.
| Date of Award | 19 Jun 2019 |
|---|---|
| Original language | English |
| Awarding Institution |
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| Sponsors | Schlumberger & Engineering and Physical Sciences Research Council |
| Supervisor | Ivan Graham (Supervisor) & Euan Spence (Supervisor) |