Optimization algorithm for reconstructing interface changes of a conductivity inclusion from modal measurements

Habib Ammari, Elena Beretta, Elisa Francini, Hyeonbae Kang, Mikyoung Lim

    Research output: Contribution to journalArticle

    Abstract

    In this paper, we propose an original and promising optimization approach for reconstructing interface changes of a conductivity inclusion from measurements of eigenvalues and eigenfunctions associated with the transmission problem for the Laplacian. Based on a rigorous asymptotic analysis, we derive an asymptotic formula for the perturbations in the modal measurements that are due to small changes in the interface of the inclusion. Using fine gradient estimates, we carefully estimate the error term in this asymptotic formula. We then provide a key dual identity which naturally yields to the formulation of the proposed optimization problem. The viability of our reconstruction approach is documented by a variety of numerical results. The resolution limit of our algorithm is also highlighted.

    Original languageEnglish (US)
    Pages (from-to)1757-1777
    Number of pages21
    JournalMathematics of Computation
    Volume79
    Issue number271
    DOIs
    StatePublished - Jul 1 2010

    Fingerprint

    Asymptotic Formula
    Conductivity
    Optimization Algorithm
    Inclusion
    Gradient Estimate
    Transmission Problem
    Asymptotic analysis
    Eigenvalues and Eigenfunctions
    Error term
    Viability
    Asymptotic Analysis
    Eigenvalues and eigenfunctions
    Optimization Problem
    Perturbation
    Numerical Results
    Optimization
    Formulation
    Estimate

    Keywords

    • Asymptotic expansion
    • Optimization problem
    • Reconstruction algorithm
    • Shape reconstruction
    • Vibration analysis

    ASJC Scopus subject areas

    • Algebra and Number Theory
    • Computational Mathematics
    • Applied Mathematics

    Cite this

    Optimization algorithm for reconstructing interface changes of a conductivity inclusion from modal measurements. / Ammari, Habib; Beretta, Elena; Francini, Elisa; Kang, Hyeonbae; Lim, Mikyoung.

    In: Mathematics of Computation, Vol. 79, No. 271, 01.07.2010, p. 1757-1777.

    Research output: Contribution to journalArticle

    Ammari, Habib ; Beretta, Elena ; Francini, Elisa ; Kang, Hyeonbae ; Lim, Mikyoung. / Optimization algorithm for reconstructing interface changes of a conductivity inclusion from modal measurements. In: Mathematics of Computation. 2010 ; Vol. 79, No. 271. pp. 1757-1777.
    @article{ccebf42b01a34086b48ad95947a4168d,
    title = "Optimization algorithm for reconstructing interface changes of a conductivity inclusion from modal measurements",
    abstract = "In this paper, we propose an original and promising optimization approach for reconstructing interface changes of a conductivity inclusion from measurements of eigenvalues and eigenfunctions associated with the transmission problem for the Laplacian. Based on a rigorous asymptotic analysis, we derive an asymptotic formula for the perturbations in the modal measurements that are due to small changes in the interface of the inclusion. Using fine gradient estimates, we carefully estimate the error term in this asymptotic formula. We then provide a key dual identity which naturally yields to the formulation of the proposed optimization problem. The viability of our reconstruction approach is documented by a variety of numerical results. The resolution limit of our algorithm is also highlighted.",
    keywords = "Asymptotic expansion, Optimization problem, Reconstruction algorithm, Shape reconstruction, Vibration analysis",
    author = "Habib Ammari and Elena Beretta and Elisa Francini and Hyeonbae Kang and Mikyoung Lim",
    year = "2010",
    month = "7",
    day = "1",
    doi = "10.1090/S0025-5718-10-02344-6",
    language = "English (US)",
    volume = "79",
    pages = "1757--1777",
    journal = "Mathematics of Computation",
    issn = "0025-5718",
    publisher = "American Mathematical Society",
    number = "271",

    }

    TY - JOUR

    T1 - Optimization algorithm for reconstructing interface changes of a conductivity inclusion from modal measurements

    AU - Ammari, Habib

    AU - Beretta, Elena

    AU - Francini, Elisa

    AU - Kang, Hyeonbae

    AU - Lim, Mikyoung

    PY - 2010/7/1

    Y1 - 2010/7/1

    N2 - In this paper, we propose an original and promising optimization approach for reconstructing interface changes of a conductivity inclusion from measurements of eigenvalues and eigenfunctions associated with the transmission problem for the Laplacian. Based on a rigorous asymptotic analysis, we derive an asymptotic formula for the perturbations in the modal measurements that are due to small changes in the interface of the inclusion. Using fine gradient estimates, we carefully estimate the error term in this asymptotic formula. We then provide a key dual identity which naturally yields to the formulation of the proposed optimization problem. The viability of our reconstruction approach is documented by a variety of numerical results. The resolution limit of our algorithm is also highlighted.

    AB - In this paper, we propose an original and promising optimization approach for reconstructing interface changes of a conductivity inclusion from measurements of eigenvalues and eigenfunctions associated with the transmission problem for the Laplacian. Based on a rigorous asymptotic analysis, we derive an asymptotic formula for the perturbations in the modal measurements that are due to small changes in the interface of the inclusion. Using fine gradient estimates, we carefully estimate the error term in this asymptotic formula. We then provide a key dual identity which naturally yields to the formulation of the proposed optimization problem. The viability of our reconstruction approach is documented by a variety of numerical results. The resolution limit of our algorithm is also highlighted.

    KW - Asymptotic expansion

    KW - Optimization problem

    KW - Reconstruction algorithm

    KW - Shape reconstruction

    KW - Vibration analysis

    UR - http://www.scopus.com/inward/record.url?scp=77955898990&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=77955898990&partnerID=8YFLogxK

    U2 - 10.1090/S0025-5718-10-02344-6

    DO - 10.1090/S0025-5718-10-02344-6

    M3 - Article

    VL - 79

    SP - 1757

    EP - 1777

    JO - Mathematics of Computation

    JF - Mathematics of Computation

    SN - 0025-5718

    IS - 271

    ER -