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

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

doi:10.1007/s10589-020-00205-y

Cited by 16 publications

(35 citation statements)

References 33 publications

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“…In [9] the authors suggests to improve the GCG algorithm by performing several steps of a proximal gradient method for (1.5) starting from the current coefficients as initial guess. Acceleration of GCG by fully resolving the coefficient optimization problem (1.5) in each iteration of the method has been proposed in [6,25,52,61]. Alternatively to coefficient optimization, point moving strategies have been suggested.…”

Section: Accelerated Gcg Methodsmentioning

confidence: 99%

“…Note that the nonconvex (and also nonsmooth, if both coefficients and positions are optimized) point moving problem to be solved here is more computationally intensive than (1.5). Moreover, for sparse minimization problems associated to PDEs, often the kernel k is not given analytically and needs to be further approximated [12,39,40,50,52]. To solve (P) in practice, the operator K is replaced by an approximation employing finite elements.…”

Section: Accelerated Gcg Methodsmentioning

confidence: 99%

“…This then renders both problem formulations essentially equivalent. Our main motivation for studying these problems is given by applications in inverse source location [9,52], optimal control [15,30,39,40], or compressed sensing [3,10,24]: Here, (P) is of the simpler form: where u encodes a collection of vector valued signals originating from a number of source locations x ∈ Ω, and K models the signal that will be received by a measurement setup. The data vector y d contains (potentially noisy) observations obtained in practice, and the first term measures the misfit of the data to the response of the model.…”

Section: Introductionmentioning

confidence: 99%

“…More complicated models involve partial differential equations [9,13,52]. Here, Ku corresponds to (possibly pointwise) observations of the PDE solution corresponding to a source term u.…”

Section: Introductionmentioning

confidence: 99%

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Linear convergence of accelerated conditional gradient algorithms in spaces of measures

Pieper

Walter

2021

ESAIM: COCV

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A class of generalized conditional gradient algorithms for the solution of optimization problem in spaces of Radon measures is presented. The method iteratively inserts additional Dirac-delta functions and optimizes the corresponding coefficients. Under general assumptions, a sub-linear O(1/k) rate in the objective functional is obtained, which is sharp in most cases. To improve efficiency, one can fully resolve the finite-dimensional subproblems occurring in each iteration of the method. We provide an analysis for the resulting procedure: under a structural assumption on the optimal solution, a linear O (\zeta^k) convergence rate is obtained locally.

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Section: Accelerated Gcg Methodsmentioning

confidence: 99%

Section: Accelerated Gcg Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…More complicated models involve partial differential equations [9,13,52]. Here, Ku corresponds to (possibly pointwise) observations of the PDE solution corresponding to a source term u.…”

Section: Introductionmentioning

confidence: 99%

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Linear convergence of accelerated conditional gradient algorithms in spaces of measures

Pieper

Walter

2021

ESAIM: COCV

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“…The boundary of Ω is assumed to consist two parts, ∂Ω = Γ r ∪ Γ a , where Γ r is the sound hard part of the boundary and Γ a = ∂Ω\Γ r is the set of absorbing walls, see [32] or represents a nonreflecting boundary that is used to enable truncation of the computational domain, see, e.g., the classical reference [10] and the citing literature. Correspondingly, we impose the boundary conditions…”

Section: Localization Of Sound Sources From Microphone Array Measuremmentioning

confidence: 99%

Some application examples of minimization based formulations of inverse problems and their regularization

Huynh¹,

Kaltenbacher²

2021

IPI

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In this paper we extend a recent idea of formulating and regularizing inverse problems as minimization problems, so without using a forward operator, thus avoiding explicit evaluation of a parameter-to-state map. We do so by rephrasing three application examples in this minimization form, namely (a) electrical impedance tomography with the complete electrode model (b) identification of a nonlinear magnetic permeability from magnetic flux measurements (c) localization of sound sources from microphone array measurements. To establish convergence of the proposed regularization approach for these problems, we first of all extend the existing theory. In particular, we take advantage of the fact that observations are finite dimensional here, so that inversion of the noisy data can to some extent be done separately, using a right inverse of the observation operator. This new approach is actually applicable to a wide range of real world problems.

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