# American Institute of Mathematical Sciences

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

 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.
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, (1996). doi: 10.1137/1.9781611971484. [2] T. Arbogast and J. L. Bona, Methods of Applied Mathematics,, University of Texas at Austin, (2008). [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, (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). 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). [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. 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, (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. 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. doi: 10.1007/s00211-006-0005-6. [13] ________, Compact gradient tracking in shape optimization,, Computational Optimization and Applications, 39 (2008), 297. [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. doi: 10.1137/090780717. [15] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer-Verlag, (2001). [16] K. Kreutz-Delgado, The Complex Gradient Operator and the CR-calculus,, Tech. Report UCSD-ECE275CG-S2009v1.0, (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). 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, (2006). [20] A. Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation,, SIAM, (2005). doi: 10.1137/1.9780898717921. [21] E. Zeidler, Nonlinear Functional Analysis and Its Applications I: Fixed Point Theorems,, Springer Verlag, (1986).

show all references

##### References:
 [1] A. Björk, Numerical Methods for Least Squares Problems,, SIAM, (1996). doi: 10.1137/1.9781611971484. [2] T. Arbogast and J. L. Bona, Methods of Applied Mathematics,, University of Texas at Austin, (2008). [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, (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). 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). [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. 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, (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. 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. doi: 10.1007/s00211-006-0005-6. [13] ________, Compact gradient tracking in shape optimization,, Computational Optimization and Applications, 39 (2008), 297. [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. doi: 10.1137/090780717. [15] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer-Verlag, (2001). [16] K. Kreutz-Delgado, The Complex Gradient Operator and the CR-calculus,, Tech. Report UCSD-ECE275CG-S2009v1.0, (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). 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, (2006). [20] A. Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation,, SIAM, (2005). doi: 10.1137/1.9780898717921. [21] E. Zeidler, Nonlinear Functional Analysis and Its Applications I: Fixed Point Theorems,, Springer Verlag, (1986).
 [1] Leonardo Marazzi. Inverse scattering on conformally compact manifolds. Inverse Problems & Imaging, 2009, 3 (3) : 537-550. doi: 10.3934/ipi.2009.3.537 [2] Abdallah El Hamidi, Aziz Hamdouni, Marwan Saleh. On eigenelements sensitivity for compact self-adjoint operators and applications. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 445-455. doi: 10.3934/dcdss.2016006 [3] Frederic Weidling, Thorsten Hohage. Variational source conditions and stability estimates for inverse electromagnetic medium scattering problems. Inverse Problems & Imaging, 2017, 11 (1) : 203-220. doi: 10.3934/ipi.2017010 [4] Adán J. Corcho. Ill-Posedness for the Benney system. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 965-972. doi: 10.3934/dcds.2006.15.965 [5] Pei Yean Lee, John B Moore. Gauss-Newton-on-manifold for pose estimation. Journal of Industrial & Management Optimization, 2005, 1 (4) : 565-587. doi: 10.3934/jimo.2005.1.565 [6] Bernadette N. Hahn. Dynamic linear inverse problems with moderate movements of the object: Ill-posedness and regularization. Inverse Problems & Imaging, 2015, 9 (2) : 395-413. doi: 10.3934/ipi.2015.9.395 [7] Jérôme Bertrand. Prescription of Gauss curvature on compact hyperbolic orbifolds. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1269-1284. doi: 10.3934/dcds.2014.34.1269 [8] Xavier Carvajal, Mahendra Panthee. On ill-posedness for the generalized BBM equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4565-4576. doi: 10.3934/dcds.2014.34.4565 [9] Mahendra Panthee. On the ill-posedness result for the BBM equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 253-259. doi: 10.3934/dcds.2011.30.253 [10] Alexei Rybkin. On the boundary control approach to inverse spectral and scattering theory for Schrödinger operators. Inverse Problems & Imaging, 2009, 3 (1) : 139-149. doi: 10.3934/ipi.2009.3.139 [11] Masaru Ikehata. The enclosure method for inverse obstacle scattering using a single electromagnetic wave in time domain. Inverse Problems & Imaging, 2016, 10 (1) : 131-163. doi: 10.3934/ipi.2016.10.131 [12] Piero D'Ancona, Mamoru Okamoto. Blowup and ill-posedness results for a Dirac equation without gauge invariance. Evolution Equations & Control Theory, 2016, 5 (2) : 225-234. doi: 10.3934/eect.2016002 [13] In-Jee Jeong, Benoit Pausader. Discrete Schrödinger equation and ill-posedness for the Euler equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 281-293. doi: 10.3934/dcds.2017012 [14] G. Fonseca, G. Rodríguez-Blanco, W. Sandoval. Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1327-1341. doi: 10.3934/cpaa.2015.14.1327 [15] Yannis Angelopoulos. Well-posedness and ill-posedness results for the Novikov-Veselov equation. Communications on Pure & Applied Analysis, 2016, 15 (3) : 727-760. doi: 10.3934/cpaa.2016.15.727 [16] Yonggeun Cho, Gyeongha Hwang, Soonsik Kwon, Sanghyuk Lee. Well-posedness and ill-posedness for the cubic fractional Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2863-2880. doi: 10.3934/dcds.2015.35.2863 [17] Christodoulos E. Athanasiadis, Vassilios Sevroglou, Konstantinos I. Skourogiannis. The inverse electromagnetic scattering problem by a mixed impedance screen in chiral media. Inverse Problems & Imaging, 2015, 9 (4) : 951-970. doi: 10.3934/ipi.2015.9.951 [18] Eric P. Choate, Hong Zhou. Optimization of electromagnetic wave propagation through a liquid crystal layer. Discrete & Continuous Dynamical Systems - S, 2015, 8 (2) : 303-312. doi: 10.3934/dcdss.2015.8.303 [19] Jean-François Crouzet. 3D coded aperture imaging, ill-posedness and link with incomplete data radon transform. Inverse Problems & Imaging, 2011, 5 (2) : 341-353. doi: 10.3934/ipi.2011.5.341 [20] Tsukasa Iwabuchi, Kota Uriya. Ill-posedness for the quadratic nonlinear Schrödinger equation with nonlinearity $|u|^2$. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1395-1405. doi: 10.3934/cpaa.2015.14.1395

2016 Impact Factor: 1.094