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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

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 & 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.  Google Scholar

[2]

T. Arbogast and J. L. Bona, Methods of Applied Mathematics, University of Texas at Austin, 2008. Lecture Notes in Applied Mathematics. Google Scholar

[3]

P. Blanchard and E. Brüning, Mathematical Methods in Physics, Birhäuser Verlag, 2003. doi: 10.1007/978-1-4612-0049-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

_______, Analysis of the Hessian for inverse scattering problems. Part II: Inverse medium scattering of acoustic waves, Inverse Problems, 28 (2012), 055002. Google Scholar

[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). Google Scholar

[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.  Google Scholar

[9]

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, John Wiley & Sons, 1983.  Google Scholar

[10]

________, Inverse Acoustic and Electromagnetic Scattering, Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, Heidelberg, New-York, Tokyo, second ed., 1998. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

________, Compact gradient tracking in shape optimization, Computational Optimization and Applications, 39 (2008), 297-318. Google Scholar

[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.  Google Scholar

[15]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, second (Classics in Mathematics) ed., 2001.  Google Scholar

[16]

K. Kreutz-Delgado, The Complex Gradient Operator and the CR-calculus, Tech. Report UCSD-ECE275CG-S2009v1.0, University of California, San Diego, 2009. Google Scholar

[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.  Google Scholar

[18]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambidge University Press, 2000.  Google Scholar

[19]

J. Nocedal and S. J. Wright, Numerical Optimization, Springer Verlag, Berlin, Heidelberg, New York, second ed., 2006.  Google Scholar

[20]

A. Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation, SIAM, Philadelphia, PA, 2005. doi: 10.1137/1.9780898717921.  Google Scholar

[21]

E. Zeidler, Nonlinear Functional Analysis and Its Applications I: Fixed Point Theorems, Springer Verlag, Berlin, Heidelberg, New-York, 1986.  Google Scholar

show all references

References:
[1]

A. Björk, Numerical Methods for Least Squares Problems, SIAM, Philadelphia, PA, 1996. doi: 10.1137/1.9781611971484.  Google Scholar

[2]

T. Arbogast and J. L. Bona, Methods of Applied Mathematics, University of Texas at Austin, 2008. Lecture Notes in Applied Mathematics. Google Scholar

[3]

P. Blanchard and E. Brüning, Mathematical Methods in Physics, Birhäuser Verlag, 2003. doi: 10.1007/978-1-4612-0049-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

_______, Analysis of the Hessian for inverse scattering problems. Part II: Inverse medium scattering of acoustic waves, Inverse Problems, 28 (2012), 055002. Google Scholar

[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). Google Scholar

[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.  Google Scholar

[9]

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, John Wiley & Sons, 1983.  Google Scholar

[10]

________, Inverse Acoustic and Electromagnetic Scattering, Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, Heidelberg, New-York, Tokyo, second ed., 1998. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

________, Compact gradient tracking in shape optimization, Computational Optimization and Applications, 39 (2008), 297-318. Google Scholar

[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.  Google Scholar

[15]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, second (Classics in Mathematics) ed., 2001.  Google Scholar

[16]

K. Kreutz-Delgado, The Complex Gradient Operator and the CR-calculus, Tech. Report UCSD-ECE275CG-S2009v1.0, University of California, San Diego, 2009. Google Scholar

[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.  Google Scholar

[18]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambidge University Press, 2000.  Google Scholar

[19]

J. Nocedal and S. J. Wright, Numerical Optimization, Springer Verlag, Berlin, Heidelberg, New York, second ed., 2006.  Google Scholar

[20]

A. Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation, SIAM, Philadelphia, PA, 2005. doi: 10.1137/1.9780898717921.  Google Scholar

[21]

E. Zeidler, Nonlinear Functional Analysis and Its Applications I: Fixed Point Theorems, Springer Verlag, Berlin, Heidelberg, New-York, 1986.  Google Scholar

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