| Analysis of the Hessian for Inverse Scattering Problems. Part 3. Inverse Medium Scattering of Electromagnetic Waves in Three Dimensions |
Aug 2012 |
17 pages |
| Authors:
Tan Bui-Thanh; Omar Ghattas; TEXAS UNIV AT AUSTIN INST FOR COMPUTATIONAL ENGINEERING AND SCIENCES
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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 mis t. We derive and analyze the Hessian in both H older and Sobolev spaces. Using an integral equation approach based on Newton potential theory and compact embeddings in ... |
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| Adaptive Hessian-based Non-stationary Gaussian Process Response Surface Method for Probability Density Approximation with Application to Bayesian Solution of Large-scale Inverse Problems |
Oct 2011 |
38 pages |
| Authors:
Tan Bui-Thanh; Omar Ghattas; David Higdon; TEXAS UNIV AT AUSTIN INST FOR COMPUTATIONAL ENGINEERING AND SCIENCES
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| We develop an adaptive Hessian-based non-stationary Gaussian process response surface method to approximate a probability density function (pdf) that exploits its structure, in particular the Hessian of its negative logarithm. Of particular interest to us are pdfs that arise from the Bayesian solution of large-scale inverse problems, which imply very expensive-to-evaluate pdfs. The method can be considered as a piecewise adaptive Gaussian approximation in which a Gaussian tailored to the ... |
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| Analysis of the Hessian for Inverse Scattering Problems. Part 1: Inverse Shape Scattering of Acoustic Waves |
Jun 2011 |
40 pages |
| Authors:
Tan Bui-Thanh; Omar Ghattas; TEXAS UNIV AT AUSTIN INST FOR COMPUTATIONAL ENGINEERING AND SCIENCES
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| We derive expressions for the shape Hessian operator of the data misfit functional corresponding to the inverse problem of inferring the shape of a scatterer from reflected acoustic waves, using a Banach space setting and the Lagrangian approach. The shape Hessian is then analyzed in both Hoelder and Sobolev spaces. Using an integral equation approach and compact embeddings in Hoelder and Sobolev spaces, we show that the shape Hessian can ... |
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| Analysis of the Hessian for Inverse Scattering Problems. Part 2: Inverse Medium Scattering of Acoustic Waves |
Jun 2011 |
22 pages |
| Authors:
Tan Bui-Thanh; Omar Ghattas; TEXAS UNIV AT AUSTIN INST FOR COMPUTATIONAL ENGINEERING AND SCIENCES
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| We address the inverse problem for scattering of acoustic waves due to an inhomogeneous medium. We derive and analyze the Hessian in both H older and Sobolev spaces. Using an integral equation approach based on Newton potential theory and compact embeddings in Holder and Sobolev spaces, we show that the Hessian can be decomposed into two components, both of which are shown to be compact operators. Numerical examples are presented ... |
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| Analysis of an Hp-Non-conforming Discontinuous Galerkin Spectral Element Method for Wave |
Apr 2011 |
26 pages |
| Authors:
Tan Bui-Thanh; Omar Ghattas; TEXAS UNIV AT AUSTIN INST FOR COMPUTATIONAL ENGINEERING AND SCIENCES
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| We analyze the consistency, stability, and convergence of an hp discontinuous Galerkin spectral element method. The analysis is carried out simultaneously for acoustic, elastic coupled elastic-acoustic, and electromagnetic wave propagation. Our analytical results are developed for both conforming and non-conforming approximations on hexahedral meshes using either exact integration with Legendre-Gauss quadrature or inexact integration with Legendre-Gauss-Lobatto quadrature. A mortar-based non-conforming approximation is developed to treat both h and p non-conforming ... |
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| A PDE-Constrained Optimization Approach to Uncertainty Quantification in Inverse Problems, with Applications to Inverse Scattering |
28 Feb 2010 |
16 pages |
| Authors:
Omar Ghattas; Jucas Wilcox; Tan Bui-Thanh; JAMES MARTIN; George Biros; Stephanie Chaillet; TEXAS UNIV AT AUSTIN INST FOR COMPUTING SCIENCE AND COMPUTER APPLICATIONS
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| This project addressed the statistical inverse problem of reconstruction of an uncertain shape of a scatterer or properties of a medium from noisy observations of scattered wavefields. The Bayesian solution of this inverse problem yields a posterior pdf, requiring the solution of the forward wave equation to evaluate the probability of any point in parameter space. The standard approach is to sample this pdf via an MCMC method and then ... |
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