Discretization-invariant Bayesian inversion and Besov space priors

    Research output: Contribution to journalArticleScientificpeer-review

    Abstract

    "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."
    Original languageEnglish
    JournalInverse problems and imaging
    Volume3
    Issue number1
    Pages (from-to)87-122
    Number of pages36
    ISSN1930-8337
    DOIs
    Publication statusPublished - 2009
    MoE publication typeA1 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

    @article{1e236425af0248bbac07674f04c129eb,
    title = "Discretization-invariant Bayesian inversion and Besov space priors",
    abstract = "{"}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.{"}",
    keywords = "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",
    author = "Matti Lassas and Eero Saksman and Samuli Siltanen",
    year = "2009",
    doi = "10.3934/ipi.2009.3.87",
    language = "English",
    volume = "3",
    pages = "87--122",
    journal = "Inverse problems and imaging",
    issn = "1930-8337",
    publisher = "American Institute of Mathematical Sciences",
    number = "1",

    }

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

    In: Inverse problems and imaging, Vol. 3, No. 1, 2009, p. 87-122.

    Research output: Contribution to journalArticleScientificpeer-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 -