### Abstract

Original language | English |
---|---|

Journal | Inverse problems and imaging |

Volume | 3 |

Issue number | 1 |

Pages (from-to) | 87-122 |

Number of pages | 36 |

ISSN | 1930-8337 |

DOIs | |

Publication status | Published - 2009 |

MoE publication type | A1 Journal article-refereed |

### Fields of Science

- Inverse problem
- statistical inversion
- Bayesian inversion
- discretization invariance
- reconstruction
- wavelet
- Besov space
- CHAIN MONTE-CARLO
- X-RAY TOMOGRAPHY
- STATISTICAL INVERSION
- STOCHASTIC INVERSION
- RECONSTRUCTION
- CONVERGENCE
- MODELS
- RADIOTOMOGRAPHY
- REGULARIZATION
- RADIOGRAPHS
- 111 Mathematics

### Cite this

}

**Discretization-invariant Bayesian inversion and Besov space priors.** / Lassas, Matti; Saksman, Eero; Siltanen, Samuli.

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

TY - JOUR

T1 - Discretization-invariant Bayesian inversion and Besov space priors

AU - Lassas, Matti

AU - Saksman, Eero

AU - Siltanen, Samuli

PY - 2009

Y1 - 2009

N2 - "Bayesian solution of an inverse problem for indirect measurement M = AU + epsilon is considered, where U is a function on a domain of R-d. Here A is a smoothing linear operator and epsilon is Gaussian white noise. The data is a realization m(k) of the random variable M-k = P(k)AU + P-k epsilon, where P-k is a linear, finite dimensional operator related to measurement device. To allow computerized inversion, the unknown is discretized as U-n = TnU, where T-n is a finite dimensional projection, leading to the computational measurement model M-kn = P(k)AU(n) + P-k epsilon. Bayes formula gives then the posterior distribution pi(kn)(u(n) vertical bar m(kn)) similar to Pi(n)(u(n)) exp(-1/2 parallel to m(kn) - PkAun parallel to(2)(2)) in R-d, and the mean u(kn) := integral u(n) pi(kn)(u(n) vertical bar m(k)) du(n) is considered as the reconstruction of U. We discuss a systematic way of choosing prior distributions Pi(n) for all n >= n(0) > 0 by achieving them as projections of a distribution in a infinite-dimensional limit case. Such choice of prior distributions is discretization-invariant in the sense that Pi(n) represent the same a priori information for all n and that the mean u(kn) converges to a limit estimate as k, n -> infinity. Gaussian smoothness priors and wavelet-based Besov space priors are shown to be discretization invariant. In particular, Bayesian inversion in dimension two with B-11(1) prior is related to penalizing the l(1) norm of the wavelet coefficients of U."

AB - "Bayesian solution of an inverse problem for indirect measurement M = AU + epsilon is considered, where U is a function on a domain of R-d. Here A is a smoothing linear operator and epsilon is Gaussian white noise. The data is a realization m(k) of the random variable M-k = P(k)AU + P-k epsilon, where P-k is a linear, finite dimensional operator related to measurement device. To allow computerized inversion, the unknown is discretized as U-n = TnU, where T-n is a finite dimensional projection, leading to the computational measurement model M-kn = P(k)AU(n) + P-k epsilon. Bayes formula gives then the posterior distribution pi(kn)(u(n) vertical bar m(kn)) similar to Pi(n)(u(n)) exp(-1/2 parallel to m(kn) - PkAun parallel to(2)(2)) in R-d, and the mean u(kn) := integral u(n) pi(kn)(u(n) vertical bar m(k)) du(n) is considered as the reconstruction of U. We discuss a systematic way of choosing prior distributions Pi(n) for all n >= n(0) > 0 by achieving them as projections of a distribution in a infinite-dimensional limit case. Such choice of prior distributions is discretization-invariant in the sense that Pi(n) represent the same a priori information for all n and that the mean u(kn) converges to a limit estimate as k, n -> infinity. Gaussian smoothness priors and wavelet-based Besov space priors are shown to be discretization invariant. In particular, Bayesian inversion in dimension two with B-11(1) prior is related to penalizing the l(1) norm of the wavelet coefficients of U."

KW - Inverse problem

KW - statistical inversion

KW - Bayesian inversion

KW - discretization invariance

KW - reconstruction

KW - wavelet

KW - Besov space

KW - CHAIN MONTE-CARLO

KW - X-RAY TOMOGRAPHY

KW - STATISTICAL INVERSION

KW - STOCHASTIC INVERSION

KW - RECONSTRUCTION

KW - CONVERGENCE

KW - MODELS

KW - RADIOTOMOGRAPHY

KW - REGULARIZATION

KW - RADIOGRAPHS

KW - 111 Mathematics

U2 - 10.3934/ipi.2009.3.87

DO - 10.3934/ipi.2009.3.87

M3 - Article

VL - 3

SP - 87

EP - 122

JO - Inverse problems and imaging

JF - Inverse problems and imaging

SN - 1930-8337

IS - 1

ER -