Inverse point source location with the Helmholtz equation on a bounded domain

Konstantin Pieper; Bao Quoc Tang; Philip Trautmann; Daniel Walter

doi:10.1007/s10589-020-00205-y

Inverse point source location with the Helmholtz equation on a bounded domain

Konstantin Pieper, Bao Quoc Tang, Philip Trautmann, Daniel Walter

Data Analysis and Machine Learning

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

The problem of recovering acoustic sources, more specifically monopoles, from point-wise measurements of the corresponding acoustic pressure at a limited number of frequencies is addressed. To this purpose, a family of sparse optimization problems in measure space in combination with the Helmholtz equation on a bounded domain is considered. A weighted norm with unbounded weight near the observation points is incorporated into the formulation. Optimality conditions and conditions for recovery in the small noise case are discussed, which motivates concrete choices of the weight. The numerical realization is based on an accelerated conditional gradient method in measure space and a finite element discretization.

Original language	English
Pages (from-to)	213-249
Number of pages	37
Journal	Computational Optimization and Applications
Volume	77
Issue number	1
DOIs	https://doi.org/10.1007/s10589-020-00205-y
State	Published - Sep 1 2020

Funding

The authors gratefully acknowledge support through the International Research Training Group IGDK 1754, funded by the German Science Foundation (DFG) and the Austrian Science Fund (FWF). K. Pieper acknowledges funding by the U.S. Air Force Office of Scientific Research grant FA9550-15-1-0001. This manuscript has also been supported, in part, by UT-Battelle, LLC, under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. Accordingly, the U.S. Government retains a non-exclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes. P. Trautmann gratefully acknowledges the support by the ERC advanced grant 668998 (OCLOC) under the EU’s H2020 research program. D. Walter acknowledges support from the TopMath Graduate Center of TUM Graduate School and from the TopMath Program at the Elite Network of Bavaria. The authors gratefully acknowledge support through the International Research Training Group IGDK 1754, funded by the German Science Foundation (DFG) and the Austrian Science Fund (FWF). K. Pieper acknowledges funding by the U.S. Air Force Office of Scientific Research grant FA9550-15-1-0001. This manuscript has also been supported, in part, by UT-Battelle, LLC, under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. Accordingly, the U.S. Government retains a non-exclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes. P. Trautmann gratefully acknowledges the support by the ERC advanced grant 668998 (OCLOC) under the EU?s H2020 research program. D. Walter acknowledges support from the TopMath Graduate Center of TUM Graduate School and from the TopMath Program at the Elite Network of Bavaria.

Funders	Funder number
EU?s H2020
EU’s H2020
German Science Foundation
UT-Battelle, LLC
U.S. Department of Energy
Air Force Office of Scientific Research	FA9550-15-1-0001
UT-Battelle	DE-AC05-00OR22725
Engineering Research Centers
European Research Council	668998
Deutsche Forschungsgemeinschaft
Austrian Science Fund
Graduate School, Technische Universität München

Keywords

Helmholtz equation
Inverse source location
PDE-constrained optimization
Sparsity

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10.1007/s10589-020-00205-y

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abstract = "The problem of recovering acoustic sources, more specifically monopoles, from point-wise measurements of the corresponding acoustic pressure at a limited number of frequencies is addressed. To this purpose, a family of sparse optimization problems in measure space in combination with the Helmholtz equation on a bounded domain is considered. A weighted norm with unbounded weight near the observation points is incorporated into the formulation. Optimality conditions and conditions for recovery in the small noise case are discussed, which motivates concrete choices of the weight. The numerical realization is based on an accelerated conditional gradient method in measure space and a finite element discretization.",

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Inverse point source location with the Helmholtz equation on a bounded domain

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