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Title: Sweeping preconditioners for Helmholtz problems using absorption
Author: Arter, Elizabeth
ISNI:       0000 0004 7967 8657
Awarding Body: University of Bath
Current Institution: University of Bath
Date of Award: 2019
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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.
Supervisor: Graham, Ivan ; Spence, Euan Sponsor: Schlumberger ; Engineering and Physical Sciences Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available