Analysis of the Hessian for inverse scattering problems. Part III: Inverse medium scattering of electromagnetic waves in three dimensions

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November 2013, 7(4): 1139-1155. doi: 10.3934/ipi.2013.7.1139

Analysis of the Hessian for inverse scattering problems. Part III: Inverse medium scattering of electromagnetic waves in three dimensions

Tan Bui-Thanh ^1, and Omar Ghattas ^2,

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

Received November 2012 Revised September 2013 Published November 2013

Abstract
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Continuing our previous work [6, Inverse Problems, 2012, 28, 055002] and [5, Inverse Problems, 2012, 28, 055001], we address the ill-posedness of the inverse scattering problem of electromagnetic waves due to an inhomogeneous medium by studying the Hessian of the data misfit. We derive and analyze the Hessian in both Hölder and Sobolev spaces. Using an integral equation approach based on Newton potential theory and compact embeddings in Hölder and Sobolev spaces, we show that the Hessian can be decomposed into three components, all of which are shown to be compact operators. The implication of the compactness of the Hessian is that for small data noise and model error, the discrete Hessian can be approximated by a low-rank matrix. This in turn enables fast solution of an appropriately regularized inverse problem, as well as Gaussian-based quantification of uncertainty in the estimated inhomogeneity.

Keywords: electromagnetic wave propagation, adjoint, Gauss-Newton., compact operators, Hessian, Newton potential, Inverse medium scattering, ill-posedness, compact embeddings, Riesz-Fredholm theory.

Mathematics Subject Classification: Primary: 49N45, 49K20, 49K40, 49K30; Secondary: 49J50, 49N6.

Citation: Tan Bui-Thanh, Omar Ghattas. Analysis of the Hessian for inverse scattering problems. Part III: Inverse medium scattering of electromagnetic waves in three dimensions. Inverse Problems and Imaging, 2013, 7 (4) : 1139-1155. doi: 10.3934/ipi.2013.7.1139

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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