A greedy algorithm for nonlinear inverse problems with an application to nonlinear inverse gravimetry

Kontak, Max; Michel, Volker

doi:10.1007/s13137-018-0110-6

Cited by 4 publications

(4 citation statements)

References 45 publications

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“…A Tikhonov-based regularization method, the so-called regularized functional matching pursuit algorithm [11,26,30], and its enhancement, the regularized orthogonal functional matching pursuit algorithm [29,39], will be used in a forthcoming work to solve these two problems numerically. Based on these methods, there exists a recently developed algorithm called the regularized weak functional matching pursuit algorithm which could improve and accelerate the numerical calculations of the inverse MEG and inverse EEG problem, see [20,21].…”

Section: Helmholtz Decomposition For Eegmentioning

confidence: 99%

Electro-magnetoencephalography for a spherical multiple-shell model: novel integral operators with singular-value decompositions

Leweke¹,

Michel²,

Fokas³

2020

Inverse Problems

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The inverse magnetoencephalography and electroencephalography problems for spherical models have been extensively discussed in the literature. Using the spherical multiple-shell model, we derive novel vector-valued and singularity-free integral equations for both problems based on the quasi-static Maxwell's equations. These equations are solved via a Fourier series expansion. We call this procedure the Edmonds approach, since an orthonormal system based on the Edmonds-vector-spherical harmonics is used for the Fourier series. Employing the associated singular-value decomposition, we provide a complete answer to the non-uniqueness question of these two problems: only the harmonic part of the solenoidal component of the neuronal current is visible to the simultaneous use of magnetoencephalography and electroencephalography. The remaining components of the current are invisible to both techniques. We state Picard's condition for the existence of a solution and derive an explicit formula for the best-approximate solution of the neuronal current. In comparison to previous approaches, the Edmonds approach requires the fewest a-priori assumptions on the neuronal current. Finally, we show that the results obtained by means of the Edmonds approach are consistent with results derived earlier via the Helmholtz decomposition.

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Section: Helmholtz Decomposition For Eegmentioning

confidence: 99%

Electro-magnetoencephalography for a spherical multiple-shell model: novel integral operators with singular-value decompositions

Leweke¹,

Michel²,

Fokas³

2020

Inverse Problems

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“…For practical purposes, we will give a short overview of the main principles of the RFMP algorithm, which is a regularization method for ill-posed linear problems, next. For more details on any IPMP, see, for example, [6,7,8,9,19,20,21,30,31,32,48]. For theoretical discussions of the IPMPs, we refer to this literature.…”

Section: Basic Principles Of Linear Ill-posed Inverse Problemsmentioning

confidence: 99%

“…Therefore, sophisticated algorithms need to be used to solve the problem of the downward continuation of satellite data. Previous studies showed that the (Regularized) Functional Matching Pursuit ((R)FMP), the (Regularized) Orthogonal Functional Matching Pursuit ((R)OFMP) as well as the latest (Regularized) Weak Functional Matching Pursuit ((R)WFMP) are possible approaches for this and other inverse problems, see, for instance, [3,6,7,8,9,17,19,21,20,29,30,31,32,48]. In the sequel, we will write Inverse Problem Matching Pursuit (IPMP) if we refer to either one of the mentioned algorithms.…”

Section: Introductionmentioning

confidence: 99%

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A first approach to learning a best basis for gravitational field modelling

Michel

Schneider

2020

Int J Geomath

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Gravitational field modelling is an important tool for inferring past and present dynamic processes of the Earth. Functions on the sphere such as the gravitational potential are usually expanded in terms of either spherical harmonics or radial basis functions (RBFs). The (Regularized) Functional Matching Pursuit ((R)FMP) and its variants use an overcomplete dictionary of diverse trial functions to build a best basis as a sparse subset of the dictionary and compute a model, for instance, of the gravity field, in this best basis. Thus, one advantage is that the dictionary may contain spherical harmonics and RBFs. Moreover, these methods represent a possibility to obtain an approximative and stable solution of an ill-posed inverse problem, such as the downward continuation of gravitational data from the satellite orbit to the Earth's surface, but also other inverse problems in geomathematics and medical imaging. A remaining drawback is that in practice, the dictionary has to be finite and, so far, could only be chosen by rule of thumb or trial-and-error. In this paper, we develop a strategy for automatically choosing a dictionary by a novel learning approach. We utilize a nonlinear constrained optimization problem to determine best-fitting RBFs (Abel-Poisson kernels). For this, we use the Ipopt software package with an HSL subroutine. Details of the algorithm are explained and first numerical results are shown.

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A matching pursuit approach to the geophysical inverse problem of seismic traveltime tomography under the ray theory approximation

Schneider,

Michel,

Sigloch

et al. 2024

Geophysical Journal International

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Summary Seismic travel time tomography is a geophysical imaging method to infer the 3-D interior structure of the solid Earth. Most commonly formulated as a linearized inverse problem, it maps differences between observed and expected wave travel times to interior regions where waves propagate faster or slower than the expected average. The Earth’s interior is typically parametrized by a single kind of localized basis function. Here we present an alternative approach that uses matching pursuits on large dictionaries of basis functions. Within the past decade the (Learning) Inverse Problem Matching Pursuits ((L)IPMPs) have been developed. They combine global and local trial functions. An approximation is built in a so-called best basis, chosen iteratively from an intentionally overcomplete set or dictionary. In each iteration, the choice for the next best basis element reduces the Tikhonov–Phillips functional. This is in contrast to classical methods that use either global or local basis functions. The LIPMPs have proven their applicability in inverse problems like the downward continuation of the gravitational potential as well as the MEG-/EEG-problem from medical imaging. Here, we remodel the Learning Regularized Functional Matching Pursuit (LRFMP), which is one of the LIPMPs, for travel time tomography in a ray theoretical setting. In particular, we introduce the operator, some possible trial functions and the regularization. We show a numerical proof of concept for artificial travel time delays obtained from a contrived model for velocity differences. The corresponding code is available online.

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A greedy algorithm for nonlinear inverse problems with an application to nonlinear inverse gravimetry

Cited by 4 publications

References 45 publications

Electro-magnetoencephalography for a spherical multiple-shell model: novel integral operators with singular-value decompositions

Electro-magnetoencephalography for a spherical multiple-shell model: novel integral operators with singular-value decompositions

A first approach to learning a best basis for gravitational field modelling

A matching pursuit approach to the geophysical inverse problem of seismic traveltime tomography under the ray theory approximation

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