Identification of discontinuous coefficients in elliptic problems using total variation regularization
Tony Chan*, Xue Cheng Tai
We propose several formulations for recovering discontinuous coefficients in elliptic problems by using total variation (TV) regularization. The motivation for using TV is its well-established ability to recover sharp discontinuities. We employ an augmented Lagrangian variational formulation for solving the output-least-squares inverse problem. In addition to the basic output-least-squares formulation, we introduce two new techniques for handling large observation errors. First, we use a filtering step to remove as much of the observation error as possible. Second, we introduce two extensions of the output-least-squares model; one model employs observations of the gradient of the state variable while the other uses the flux. Numerical experiments indicate that the combination of these two techniques enables us to successfully recover discontinuous coefficients even under large observation errors.
SIAM Journal on Scientific Computing
Inverse problems
Noise removal
Parameter estimation
Total variational norm
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Total Variation Regularization Mathematics 100%
Discontinuous Coefficients Mathematics 77%
Elliptic Problems Mathematics 59%
Observation Mathematics 42%
Output Mathematics 36%
Least Squares Mathematics 29%
Inverse problems Engineering & Materials Science 23%
Formulation Mathematics 20%
Chan, T., & Tai, X. C. (2003). Identification of discontinuous coefficients in elliptic problems using total variation regularization. SIAM Journal on Scientific Computing, 25(3), 881-904. https://doi.org/10.1137/S1064827599326020
Chan, Tony ; Tai, Xue Cheng. / Identification of discontinuous coefficients in elliptic problems using total variation regularization. In: SIAM Journal on Scientific Computing. 2003 ; Vol. 25, No. 3. pp. 881-904.
@article{6da064eb5bf74e0684746056c9499a6d,
title = "Identification of discontinuous coefficients in elliptic problems using total variation regularization",
abstract = "We propose several formulations for recovering discontinuous coefficients in elliptic problems by using total variation (TV) regularization. The motivation for using TV is its well-established ability to recover sharp discontinuities. We employ an augmented Lagrangian variational formulation for solving the output-least-squares inverse problem. In addition to the basic output-least-squares formulation, we introduce two new techniques for handling large observation errors. First, we use a filtering step to remove as much of the observation error as possible. Second, we introduce two extensions of the output-least-squares model; one model employs observations of the gradient of the state variable while the other uses the flux. Numerical experiments indicate that the combination of these two techniques enables us to successfully recover discontinuous coefficients even under large observation errors.",
keywords = "Elliptic, Inverse problems, Noise removal, Parameter estimation, Regularization, Total variational norm",
author = "Tony Chan and Tai, {Xue Cheng}",
journal = "SIAM Journal on Scientific Computing",
publisher = "Society for Industrial and Applied Mathematics Publications",
Chan, T & Tai, XC 2003, 'Identification of discontinuous coefficients in elliptic problems using total variation regularization', SIAM Journal on Scientific Computing, vol. 25, no. 3, pp. 881-904. https://doi.org/10.1137/S1064827599326020
Identification of discontinuous coefficients in elliptic problems using total variation regularization. / Chan, Tony; Tai, Xue Cheng.
In: SIAM Journal on Scientific Computing, Vol. 25, No. 3, 01.11.2003, p. 881-904.
T1 - Identification of discontinuous coefficients in elliptic problems using total variation regularization
AU - Chan, Tony
AU - Tai, Xue Cheng
N2 - We propose several formulations for recovering discontinuous coefficients in elliptic problems by using total variation (TV) regularization. The motivation for using TV is its well-established ability to recover sharp discontinuities. We employ an augmented Lagrangian variational formulation for solving the output-least-squares inverse problem. In addition to the basic output-least-squares formulation, we introduce two new techniques for handling large observation errors. First, we use a filtering step to remove as much of the observation error as possible. Second, we introduce two extensions of the output-least-squares model; one model employs observations of the gradient of the state variable while the other uses the flux. Numerical experiments indicate that the combination of these two techniques enables us to successfully recover discontinuous coefficients even under large observation errors.
AB - We propose several formulations for recovering discontinuous coefficients in elliptic problems by using total variation (TV) regularization. The motivation for using TV is its well-established ability to recover sharp discontinuities. We employ an augmented Lagrangian variational formulation for solving the output-least-squares inverse problem. In addition to the basic output-least-squares formulation, we introduce two new techniques for handling large observation errors. First, we use a filtering step to remove as much of the observation error as possible. Second, we introduce two extensions of the output-least-squares model; one model employs observations of the gradient of the state variable while the other uses the flux. Numerical experiments indicate that the combination of these two techniques enables us to successfully recover discontinuous coefficients even under large observation errors.
KW - Elliptic
KW - Inverse problems
KW - Noise removal
KW - Parameter estimation
KW - Regularization
KW - Total variational norm
U2 - 10.1137/S1064827599326020
DO - 10.1137/S1064827599326020
JO - SIAM Journal on Scientific Computing
JF - SIAM Journal on Scientific Computing
Chan T, Tai XC. Identification of discontinuous coefficients in elliptic problems using total variation regularization. SIAM Journal on Scientific Computing. 2003 Nov 1;25(3):881-904. https://doi.org/10.1137/S1064827599326020