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An optimization problem based on a Bayesian approach for the 2D Helmholtz equation. (English) Zbl 1451.35261
Summary: Elastography is a ill-posed inverse problem that aims at recovering the Lamé and density of the domain of interest from finite number of observations, we consider as model the Helmoltz equation. We present an implementation for solving the Helmholtz inverse problem in two dimensions via an optimization problem based on Bayesian approach. In addition, the accuracy of the method is also investigated with respect to the amount of information taken from the generalized Hermitian Eigenvalue problem and by comparing the maximum a posterior estimate to the true parameter distribution in simulated experiments.
MSC:
35R30 Inverse problems for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J25 Boundary value problems for second-order elliptic equations
65N21 Numerical methods for inverse problems for boundary value problems involving PDEs
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[1] Andrews, H.C., Patterson, C.L.: Singular value decompositions and digital image processing. In: IEEE Trans. on Communications, pp. 425-432 (1976)
[2] Aki, K.; Richards, PG, Quantitative Seismology (1980), San Francisco: W. H. Freeman and Co., San Francisco
[3] Bercoff, J.; Tanter, M.; Fink, M., Supersonic shear imaging: a new technique for soft tissue elasticity mapping, IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 51, 4, 396-409 (2004)
[4] Bui-Thanh, T.; Ghattas, O.; Martin, J.; Stadler, G., A computational framework for infinite-dimensional bayesian inverse problems. Part I: the linearized case, with application to global seismic inversion, SIAM J. Sci. Comput., 5, A2494-A2523 (2013) · Zbl 1287.35087
[5] Bui-Thanh, T.; Gnuyen, QP, FEM-Based discretization-invariant MCMC methods for PDE-constrained bayesian inverse problems, AIMS J. Inverse Probl. Imaging, 10, 943-975 (2016) · Zbl 1348.65013
[6] Chaber, B.; Szmurlo, R.; Starzynski, J., Solution of the complex-valued helmholtz equation using a dedicated finite element solver, Prz. Elektrotech., 93, 171-174 (2017)
[7] Cotter, SL; Roberts, GO; Stuart, AM; White, D., MCMC methods for functions: modifying old algorithms to make them faster, Stat. Sci., 28, 424-446 (2013) · Zbl 1331.62132
[8] Dashti, M.; Stuart, AM, Uncertainty quantification and weak approximation of an elliptic inverse problem, SIAM J. Numer. Anal., 49, 6, 2524-2542 (2011) · Zbl 1234.35309
[9] Dembo, RS; Eisenstat, SC; Steihaug, T., Inexact Newton methods, SIAM J. Numer. Anal., 19, 400-408 (1982) · Zbl 0478.65030
[10] Dembo, RS; Steihaug, T., Truncated Newton algorithms for large-scale optimization, Math. Program., 26, 190-212 (1983) · Zbl 0523.90078
[11] Dominguez, N.; Gibiat, V., Non-destructive imaging using the time domain topological energy method, Ultrasonics, 50, 367-372 (2010)
[12] Eckart, G.; Young, G., The approximation of one matrix by another of lower rank, Psychutnetiku, 1, 211-218 (1936) · JFM 62.1075.02
[13] Hager, WW, Updating the inverse of a matrix, SIAM Rev., 31, 221-239 (2017) · Zbl 0671.65018
[14] Halko, N.; Martinsson, PG; Tropp, JA, Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions, SIAM Rev., 53, 217-288 (2011) · Zbl 1269.65043
[15] Kaipio, J.; Somersalo, E., Statistical and Computational Inverse Problems (2005), New York: Springer, New York · Zbl 1068.65022
[16] Nash, SG, A survey of truncated-Newton methods, J. Comput. Appl. Math., 124, 45-59 (2017) · Zbl 0969.65054
[17] Nocedal, J.; Wright, SJ, Numerical Optimization (2006), Berlin: Springer, Berlin
[18] Pierce, NA; Giles, MB, An introduction to the adjoint approach to design, Flow, Turbulence and Combustion, 65, 393-415 (2000) · Zbl 0996.76023
[19] Stuart, AM, Inverse problems: a Bayesian perspective, Acta Number, 19, 451-559 (2010) · Zbl 1242.65142
[20] Van Houten, EEW; Miga, MI; Weaver, JB; Kennedy, FE; Paulsen, KD, Three-dimensional subzone-based reconstruction algorithm for MR elastography, Magn. Reson. Med., 41, 827-37 (2001)
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