Learned infinite elements for helioseismology

by Janosch Preuß
Doctoral thesis
Date of Examination:2021-12-02
Date of issue:2021-12-16
Advisor:Prof. Dr. Christoph Lehrenfeld
Referee:Prof. Dr. Christoph Lehrenfeld
Referee:Prof. Dr. Thorsten Hohage
Referee:Prof. Dr. Martin J. Gander
Persistent Address: http://dx.doi.org/10.53846/goediss-9011

 

 

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Abstract

English

This thesis presents efficient techniques for integrating the information contained in the Dirichlet-to-Neumann (DtN) map of time-harmonic waves propagating in a stratified medium into finite element discretizations. This task arises in the context of domain decomposition methods, e.g. when reducing a problem posed on an unbounded domain to a bounded computational domain on which the problem can then be discretized. Our focus is on stratified media like the Sun, that allow for strong reflection of waves and for which suitable methods are lacking. We present learned infinite elements as a possible approach to deal with such media utilizing the assumption of a separable geometry. In this case, the DtN map is separable, however, it remains a non-local operator with a dense matrix representation, which renders its direct use computationally inefficient. Therefore, we approximate the DtN only indirectly by adding additional degrees of freedom to the linear system in such a way that the Schur complement w.r.t. the latter provides an optimal approximation of DtN and sparsity of the linear system is preserved. This optimality is ensured via the solution of a small minimization problem, which incorporates solutions of one-dimensional time-harmonic wave equations and allows for great flexibility w.r.t. properties of the medium. In the first half of the thesis we provide an error analysis of the proposed method in a generic framework which demonstrates that exponentially fast convergence rates can be expected. Numerical experiments for the Helmholtz equation and an in-depth study on modelling the solar atmosphere with learned infinite elements demonstrate the high accuracy and flexibility of the proposed method in practical applications. In the second half of the thesis, the potential of learned infinite elements in the context of sweeping preconditioners for the efficient iterative solution of large linear systems is investigated. Even though learned infinite elements are very suitable for separable media, they can only be used for tiny perturbations thereof since the corresponding DtN maps turn out to be extremely sensitive to perturbations in the presence of strong reflections.
Keywords: Dirichlet-to-Neumann map; helioseismology; transparent boundary conditions; rational approximation; Helmholtz equation; sweeping preconditioners; non-linear least squares problem
 
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