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International Journal of Computer Mathematics Volume 70, 1998 - Issue 2
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Original Articles

Numerical methods for elliptic inverse problems

Jun Zou Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong
Pages 211-232 | Received 04 Nov 1997, Published online: 19 Mar 2007

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Lihua Wang, Zhen Wang, Zhihao Qian, Yukui Gao & Yueting Zhou. (2018) Direct collocation method for identifying the initial conditions in the inverse wave problem using radial basis functions. Inverse Problems in Science and Engineering 26:12, pages 1695-1727.
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Joachim Gwinner. (2018) On two-coefficient identification in elliptic variational inequalities. Optimization 67:7, pages 1017-1030.
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M.M. Doyley, B. Jadamba, A.A. Khan, M. Sama & B. Winkler. (2014) A New Energy Inversion for Parameter Identification in Saddle Point Problems with an Application to the Elasticity Imaging Inverse Problem of Predicting Tumor Location. Numerical Functional Analysis and Optimization 35:7-9, pages 984-1017.
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Dinh Nho Hào & Tran Nhan Tam Quyen. (2014) Finite element methods for coefficient identification in an elliptic equation. Applicable Analysis 93:7, pages 1533-1566.
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M. S. Gockenbach, B. Jadamba & A. A. Khan. (2008) Equation error approach for elliptic inverse problems with an application to the identification of Lamé parameters. Inverse Problems in Science and Engineering 16:3, pages 349-367.
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Siyu Cen, Bangti Jin, Qimeng Quan & Zhi Zhou. (2023) Hybrid neural-network FEM approximation of diffusion coefficient in elliptic and parabolic Problems. IMA Journal of Numerical Analysis.
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Abir Benyoucef, Leila Alem & Lahcène Chorfi. (2022) Uniqueness and stability of parameter identification in elliptic boundary value problem. Journal of Innovative Applied Mathematics and Computational Sciences 2:2, pages 30-37.
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Jens Berg & Kaj Nyström. (2021) Neural networks as smooth priors for inverse problems for PDEs. Journal of Computational Mathematics and Data Science 1, pages 100008.
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Wenyi Tian, Xiaoming Yuan & Hangrui Yue. (2021) An ADMM-Newton-CNN numerical approach to a TV model for identifying discontinuous diffusion coefficients in elliptic equations: convex case with gradient observations. Inverse Problems 37:8, pages 085004.
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Bangti Jin & Zhi Zhou. (2021) Error Analysis of Finite Element Approximations of Diffusion Coefficient Identification for Elliptic and Parabolic Problems. SIAM Journal on Numerical Analysis 59:1, pages 119-142.
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Tran Nhan Tam Quyen. (2020) Determining two coefficients in diffuse optical tomography with incomplete and noisy Cauchy data. Inverse Problems 36:9, pages 095011.
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Huan Liu, Rommel Real, Xiliang Lu, Xianzheng Jia & Qinian Jin. (2020) Heuristic discrepancy principle for variational regularization of inverse problems. Inverse Problems 36:7, pages 075013.
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A E Artyukhov & V I Sklabinskiy. (2017) Application of vortex three-phase separators for improving the reliability of pump and compressor stations of hydrocarbon processing plants. IOP Conference Series: Materials Science and Engineering 233, pages 012014.
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Pham Quy Muoi. (2015) Sparsity Regularization of the Diffusion Coefficient Identification Problem: Well-Posedness and Convergence Rates. Bulletin of the Malaysian Mathematical Sciences Society 39:3, pages 1145-1164.
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Pham Quy Muoi, Dinh Nho Hào, Peter Maass & Michael Pidcock. (2016) Descent gradient methods for nonsmooth minimization problems in ill-posed problems. Journal of Computational and Applied Mathematics 298, pages 105-122.
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B. Jadamba, A. A. Khan, G. Rus, M. Sama & B. Winkler. (2014) A New Convex Inversion Framework for Parameter Identification in Saddle Point Problems with an Application to the Elasticity Imaging Inverse Problem of Predicting Tumor Location. SIAM Journal on Applied Mathematics 74:5, pages 1486-1510.
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E. Crossen, M.S. Gockenbach, B. Jadamba, A.A. Khan & B. Winkler. (2014) An equation error approach for the elasticity imaging inverse problem for predicting tumor location. Computers & Mathematics with Applications 67:1, pages 122-135.
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Jan Hegemann, Alejandro Cantarero, Casey L. Richardson & Joseph M. Teran. (2013) An Explicit Update Scheme for Inverse Parameter and Interface Estimation of Piecewise Constant Coefficients in Linear Elliptic PDEs. SIAM Journal on Scientific Computing 35:2, pages A1098-A1119.
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Dinh Nho Hào & Tran Nhan Tam Quyen. (2012) Convergence rates for total variation regularization of coefficient identification problems in elliptic equations II. Journal of Mathematical Analysis and Applications 388:1, pages 593-616.
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Dinh Nho Hào & Tran Nhan Tam Quyen. (2011) Convergence rates for Tikhonov regularization of a two-coefficient identification problem in an elliptic boundary value problem. Numerische Mathematik 120:1, pages 45-77.
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Dinh Nho Hào & Tran Nhan Tam Quyen. (2011) Convergence rates for total variation regularization of coefficient identification problems in elliptic equations I. Inverse Problems 27:7, pages 075008.
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Dinh Nho Hào & Tran Nhan Tam Quyen. (2010) Convergence rates for Tikhonov regularization of coefficient identification problems in Laplace-type equations. Inverse Problems 26:12, pages 125014.
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Mark S. Gockenbach & Akhtar A. Khan. (2008) An Abstract Framework for Elliptic Inverse Problems: Part 2. An Augmented Lagrangian Approach. Mathematics and Mechanics of Solids 14:6, pages 517-539.
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B. Jadamba, A.A. Khan & F. Raciti. (2008) On the inverse problem of identifying Lamé coefficients in linear elasticity. Computers & Mathematics with Applications 56:2, pages 431-443.
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A.A. Khan & B.D. Rouhani. (2007) Iterative regularization for elliptic inverse problems. Computers & Mathematics with Applications 54:6, pages 850-860.
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Mark S. Gockenbach & Akhtar A. Khan. (2005) An Abstract Framework for Elliptic Inverse Problems: Part 1. An Output Least-Squares Approach. Mathematics and Mechanics of Solids 12:3, pages 259-276.
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