### 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 language | English (US) |
---|---|

Article number | 125016 |

Journal | Inverse Problems |

Volume | 33 |

Issue number | 12 |

DOIs | |

State | Published - Nov 30 2017 |

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### 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

}

*Inverse Problems*, vol. 33, no. 12, 125016. https://doi.org/10.1088/1361-6420/aa9909

**A general approach to regularizing inverse problems with regional data using Slepian wavelets.** / Michel, Volker; Simons, Frederik Jozef.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A general approach to regularizing inverse problems with regional data using Slepian wavelets

AU - Michel, Volker

AU - Simons, Frederik Jozef

PY - 2017/11/30

Y1 - 2017/11/30

N2 - 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.

AB - 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.

KW - Slepian function

KW - ill-posed problem

KW - inverse problem

KW - regional data

KW - regularization

KW - singular-value decomposition

KW - wavelet

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U2 - https://doi.org/10.1088/1361-6420/aa9909

DO - https://doi.org/10.1088/1361-6420/aa9909

M3 - Article

VL - 33

JO - Inverse Problems

JF - Inverse Problems

SN - 0266-5611

IS - 12

M1 - 125016

ER -