A general approach to regularizing inverse problems with regional data using Slepian wavelets

Volker Michel, Frederik Jozef Simons

Research output: Contribution to journalArticle

Abstract

Slepian functions are orthogonal function systems that live on subdomains (for example, geographical regions on the Earth's surface, or bandlimited portions of the entire spectrum). They have been firmly established as a useful tool for the synthesis and analysis of localized (concentrated or confined) signals, and for the modeling and inversion of noise-contaminated data that are only regionally available or only of regional interest. In this paper, we consider a general abstract setup for inverse problems represented by a linear and compact operator between Hilbert spaces with a known singular-value decomposition (svd). In practice, such an svd is often only given for the case of a global expansion of the data (e.g. on the whole sphere) but not for regional data distributions. We show that, in either case, Slepian functions (associated to an arbitrarily prescribed region and the given compact operator) can be determined and applied to construct a regularization for the ill-posed regional inverse problem. Moreover, we describe an algorithm for constructing the Slepian basis via an algebraic eigenvalue problem. The obtained Slepian functions can be used to derive an svd for the combination of the regionalizing projection and the compact operator. As a result, standard regularization techniques relying on a known svd become applicable also to those inverse problems where the data are regionally given only. In particular, wavelet-based multiscale techniques can be used. An example for the latter case is elaborated theoretically and tested on two synthetic numerical examples.

Original languageEnglish (US)
Article number125016
JournalInverse Problems
Volume33
Issue number12
DOIs
StatePublished - Nov 30 2017

Fingerprint

Singular value decomposition
Inverse problems
Inverse Problem
Wavelets
Compact Operator
Geographical regions
Orthogonal functions
Orthogonal Functions
Regularization Technique
Hilbert spaces
Data Distribution
Linear Operator
Eigenvalue Problem
Mathematical operators
Regularization
Inversion
Hilbert space
Earth (planet)
Entire
Projection

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Signal Processing
  • Applied Mathematics
  • Computer Science Applications
  • Mathematical Physics

Keywords

  • Slepian function
  • ill-posed problem
  • inverse problem
  • regional data
  • regularization
  • singular-value decomposition
  • wavelet

Cite this

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abstract = "Slepian functions are orthogonal function systems that live on subdomains (for example, geographical regions on the Earth's surface, or bandlimited portions of the entire spectrum). They have been firmly established as a useful tool for the synthesis and analysis of localized (concentrated or confined) signals, and for the modeling and inversion of noise-contaminated data that are only regionally available or only of regional interest. In this paper, we consider a general abstract setup for inverse problems represented by a linear and compact operator between Hilbert spaces with a known singular-value decomposition (svd). In practice, such an svd is often only given for the case of a global expansion of the data (e.g. on the whole sphere) but not for regional data distributions. We show that, in either case, Slepian functions (associated to an arbitrarily prescribed region and the given compact operator) can be determined and applied to construct a regularization for the ill-posed regional inverse problem. Moreover, we describe an algorithm for constructing the Slepian basis via an algebraic eigenvalue problem. The obtained Slepian functions can be used to derive an svd for the combination of the regionalizing projection and the compact operator. As a result, standard regularization techniques relying on a known svd become applicable also to those inverse problems where the data are regionally given only. In particular, wavelet-based multiscale techniques can be used. An example for the latter case is elaborated theoretically and tested on two synthetic numerical examples.",
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A general approach to regularizing inverse problems with regional data using Slepian wavelets. / Michel, Volker; Simons, Frederik Jozef.

In: Inverse Problems, Vol. 33, No. 12, 125016, 30.11.2017.

Research output: Contribution to journalArticle

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