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Analysis of the Hessian for inverse scattering problems. Part III: Inverse medium scattering of electromagnetic waves in three dimensions
1. | Department of Aerospace Engineering and Engineering Mechanics, Institute for Computational Engineering & Sciences, The University of Texas at Austin, Austin, TX 78712, United States |
2. | Institute for Computational Engineering & Sciences, Jackson School of Geosciences, and Department of Mechanical Engineering, The University of Texas at Austin, Austin, TX 78712, United States |
References:
[1] |
A. Björk, Numerical Methods for Least Squares Problems, SIAM, Philadelphia, PA, 1996.
doi: 10.1137/1.9781611971484. |
[2] |
T. Arbogast and J. L. Bona, Methods of Applied Mathematics, University of Texas at Austin, 2008. Lecture Notes in Applied Mathematics. |
[3] |
P. Blanchard and E. Brüning, Mathematical Methods in Physics, Birhäuser Verlag, 2003.
doi: 10.1007/978-1-4612-0049-9. |
[4] |
T. Bui-Thanh, C. Burstedde, O. Ghattas, J. Martin, G. Stadler and L. C. Wilcox, Extreme-scale UQ for Bayesian inverse problems governed by PDEs, in SC12: Proceedings of the International Conference for High Performance, Computing, Networking, Storage and Analysis, 2012.
doi: 10.1109/SC.2012.56. |
[5] |
T. Bui-Thanh and O. Ghattas, Analysis of the Hessian for inverse scattering problems. Part I: Inverse shape scattering of acoustic waves, Inverse Problems, 28 (2012), 055001, 32 pp.
doi: 10.1088/0266-5611/28/5/055001. |
[6] |
_______, Analysis of the Hessian for inverse scattering problems. Part II: Inverse medium scattering of acoustic waves, Inverse Problems, 28 (2012), 055002. |
[7] |
T. Bui-Thanh, O. Ghattas and D. Higdon, Adaptive Hessian-based non-stationary Gaussian process response surface method for probability density approximation with application to Bayesian solution of large-scale inverse problems, Submitted to SIAM Journal on Scientific Computing, (2011). |
[8] |
S. Chaillat and G. Biros, FaIMS: A fast algorithm for the inverse medium problem with multiple frequencies and multiple sources for the scalar Helmholtz equations, Under Review, 231 (2012), 4403-4421.
doi: 10.1016/j.jcp.2012.02.006. |
[9] |
D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, John Wiley & Sons, 1983. |
[10] |
________, Inverse Acoustic and Electromagnetic Scattering, Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, Heidelberg, New-York, Tokyo, second ed., 1998. |
[11] |
L. Demanet, P. -D. Ltourneau, N. Boumal, H. Calandra, J. Chiu and S. Snelson, Matrix probing: A randomized preconditioner for the wave-equation Hessian, Applied and Computational Harmonic Analysis, 32 (2012), 155-168
doi: 10.1016/j.acha.2011.03.006. |
[12] |
K. Eppler and H. Harbrecht, Coupling of FEM-BEM in shape optimization, Numerische Mathematik, 104 (2006), 47-68.
doi: 10.1007/s00211-006-0005-6. |
[13] |
________, Compact gradient tracking in shape optimization, Computational Optimization and Applications, 39 (2008), 297-318. |
[14] |
H. P. Flath, L. C. Wilcox, V. Akçelik, J. Hill, B. van Bloemen Waanders and O. Ghattas, Fast algorithms for Bayesian uncertainty quantification in large-scale linear inverse problems based on low-rank partial Hessian approximations, SIAM Journal on Scientific Computing, 33 (2011), 407-432.
doi: 10.1137/090780717. |
[15] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, second (Classics in Mathematics) ed., 2001. |
[16] |
K. Kreutz-Delgado, The Complex Gradient Operator and the CR-calculus, Tech. Report UCSD-ECE275CG-S2009v1.0, University of California, San Diego, 2009. |
[17] |
J. Martin, L. C. Wilcox, C. Burstedde and O. Ghattas, A stochastic Newton MCMC method for large-scale statistical inverse problems with application to seismic inversion, SIAM Journal on Scientific Computing, 34 (2012), A1460-A1487.
doi: 10.1137/110845598. |
[18] |
W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambidge University Press, 2000. |
[19] |
J. Nocedal and S. J. Wright, Numerical Optimization, Springer Verlag, Berlin, Heidelberg, New York, second ed., 2006. |
[20] |
A. Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation, SIAM, Philadelphia, PA, 2005.
doi: 10.1137/1.9780898717921. |
[21] |
E. Zeidler, Nonlinear Functional Analysis and Its Applications I: Fixed Point Theorems, Springer Verlag, Berlin, Heidelberg, New-York, 1986. |
show all references
References:
[1] |
A. Björk, Numerical Methods for Least Squares Problems, SIAM, Philadelphia, PA, 1996.
doi: 10.1137/1.9781611971484. |
[2] |
T. Arbogast and J. L. Bona, Methods of Applied Mathematics, University of Texas at Austin, 2008. Lecture Notes in Applied Mathematics. |
[3] |
P. Blanchard and E. Brüning, Mathematical Methods in Physics, Birhäuser Verlag, 2003.
doi: 10.1007/978-1-4612-0049-9. |
[4] |
T. Bui-Thanh, C. Burstedde, O. Ghattas, J. Martin, G. Stadler and L. C. Wilcox, Extreme-scale UQ for Bayesian inverse problems governed by PDEs, in SC12: Proceedings of the International Conference for High Performance, Computing, Networking, Storage and Analysis, 2012.
doi: 10.1109/SC.2012.56. |
[5] |
T. Bui-Thanh and O. Ghattas, Analysis of the Hessian for inverse scattering problems. Part I: Inverse shape scattering of acoustic waves, Inverse Problems, 28 (2012), 055001, 32 pp.
doi: 10.1088/0266-5611/28/5/055001. |
[6] |
_______, Analysis of the Hessian for inverse scattering problems. Part II: Inverse medium scattering of acoustic waves, Inverse Problems, 28 (2012), 055002. |
[7] |
T. Bui-Thanh, O. Ghattas and D. Higdon, Adaptive Hessian-based non-stationary Gaussian process response surface method for probability density approximation with application to Bayesian solution of large-scale inverse problems, Submitted to SIAM Journal on Scientific Computing, (2011). |
[8] |
S. Chaillat and G. Biros, FaIMS: A fast algorithm for the inverse medium problem with multiple frequencies and multiple sources for the scalar Helmholtz equations, Under Review, 231 (2012), 4403-4421.
doi: 10.1016/j.jcp.2012.02.006. |
[9] |
D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, John Wiley & Sons, 1983. |
[10] |
________, Inverse Acoustic and Electromagnetic Scattering, Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, Heidelberg, New-York, Tokyo, second ed., 1998. |
[11] |
L. Demanet, P. -D. Ltourneau, N. Boumal, H. Calandra, J. Chiu and S. Snelson, Matrix probing: A randomized preconditioner for the wave-equation Hessian, Applied and Computational Harmonic Analysis, 32 (2012), 155-168
doi: 10.1016/j.acha.2011.03.006. |
[12] |
K. Eppler and H. Harbrecht, Coupling of FEM-BEM in shape optimization, Numerische Mathematik, 104 (2006), 47-68.
doi: 10.1007/s00211-006-0005-6. |
[13] |
________, Compact gradient tracking in shape optimization, Computational Optimization and Applications, 39 (2008), 297-318. |
[14] |
H. P. Flath, L. C. Wilcox, V. Akçelik, J. Hill, B. van Bloemen Waanders and O. Ghattas, Fast algorithms for Bayesian uncertainty quantification in large-scale linear inverse problems based on low-rank partial Hessian approximations, SIAM Journal on Scientific Computing, 33 (2011), 407-432.
doi: 10.1137/090780717. |
[15] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, second (Classics in Mathematics) ed., 2001. |
[16] |
K. Kreutz-Delgado, The Complex Gradient Operator and the CR-calculus, Tech. Report UCSD-ECE275CG-S2009v1.0, University of California, San Diego, 2009. |
[17] |
J. Martin, L. C. Wilcox, C. Burstedde and O. Ghattas, A stochastic Newton MCMC method for large-scale statistical inverse problems with application to seismic inversion, SIAM Journal on Scientific Computing, 34 (2012), A1460-A1487.
doi: 10.1137/110845598. |
[18] |
W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambidge University Press, 2000. |
[19] |
J. Nocedal and S. J. Wright, Numerical Optimization, Springer Verlag, Berlin, Heidelberg, New York, second ed., 2006. |
[20] |
A. Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation, SIAM, Philadelphia, PA, 2005.
doi: 10.1137/1.9780898717921. |
[21] |
E. Zeidler, Nonlinear Functional Analysis and Its Applications I: Fixed Point Theorems, Springer Verlag, Berlin, Heidelberg, New-York, 1986. |
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