Exploiting residual information in the parameter choice for discrete ill-posed problems

Per Christian Hansen, Misha E. Kilmer, Rikke Høj Kjeldsen

    Research output: Contribution to journalJournal articleResearchpeer-review

    Abstract

    Most algorithms for choosing the regularization parameter in a discrete ill-posed problem are based on the norm of the residual vector. In this work we propose a different approach, where we seek to use all the information available in the residual vector. We present important relations between the residual components and the amount of information that is available in the noisy data, and we show how to use statistical tools and fast Fourier transforms to extract this information efficiently. This approach leads to a computationally inexpensive parameter-choice rule based on the normalized cumulative periodogram, which is particularly suited for large-scale problems.
    Original languageEnglish
    JournalBIT Numerical Mathematics
    Volume46
    Issue number1
    Pages (from-to)41-59
    ISSN0006-3835
    DOIs
    Publication statusPublished - 2006

    Cite this

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    title = "Exploiting residual information in the parameter choice for discrete ill-posed problems",
    abstract = "Most algorithms for choosing the regularization parameter in a discrete ill-posed problem are based on the norm of the residual vector. In this work we propose a different approach, where we seek to use all the information available in the residual vector. We present important relations between the residual components and the amount of information that is available in the noisy data, and we show how to use statistical tools and fast Fourier transforms to extract this information efficiently. This approach leads to a computationally inexpensive parameter-choice rule based on the normalized cumulative periodogram, which is particularly suited for large-scale problems.",
    author = "Hansen, {Per Christian} and Kilmer, {Misha E.} and Kjeldsen, {Rikke H{\o}j}",
    year = "2006",
    doi = "10.1007/s10543-006-0042-7",
    language = "English",
    volume = "46",
    pages = "41--59",
    journal = "BIT Numerical Mathematics",
    issn = "0006-3835",
    publisher = "Springer Netherlands",
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    Exploiting residual information in the parameter choice for discrete ill-posed problems. / Hansen, Per Christian; Kilmer, Misha E.; Kjeldsen, Rikke Høj.

    In: BIT Numerical Mathematics, Vol. 46, No. 1, 2006, p. 41-59.

    Research output: Contribution to journalJournal articleResearchpeer-review

    TY - JOUR

    T1 - Exploiting residual information in the parameter choice for discrete ill-posed problems

    AU - Hansen, Per Christian

    AU - Kilmer, Misha E.

    AU - Kjeldsen, Rikke Høj

    PY - 2006

    Y1 - 2006

    N2 - Most algorithms for choosing the regularization parameter in a discrete ill-posed problem are based on the norm of the residual vector. In this work we propose a different approach, where we seek to use all the information available in the residual vector. We present important relations between the residual components and the amount of information that is available in the noisy data, and we show how to use statistical tools and fast Fourier transforms to extract this information efficiently. This approach leads to a computationally inexpensive parameter-choice rule based on the normalized cumulative periodogram, which is particularly suited for large-scale problems.

    AB - Most algorithms for choosing the regularization parameter in a discrete ill-posed problem are based on the norm of the residual vector. In this work we propose a different approach, where we seek to use all the information available in the residual vector. We present important relations between the residual components and the amount of information that is available in the noisy data, and we show how to use statistical tools and fast Fourier transforms to extract this information efficiently. This approach leads to a computationally inexpensive parameter-choice rule based on the normalized cumulative periodogram, which is particularly suited for large-scale problems.

    U2 - 10.1007/s10543-006-0042-7

    DO - 10.1007/s10543-006-0042-7

    M3 - Journal article

    VL - 46

    SP - 41

    EP - 59

    JO - BIT Numerical Mathematics

    JF - BIT Numerical Mathematics

    SN - 0006-3835

    IS - 1

    ER -