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Fall 2012 Seminars

Wednesday, September 12, 2012
Mathematical Sciences Colloquium
Speaker: Guillaume Bal, Columbia University
Title: Hybrid Inverse Problems
Location: Amos Eaton 214
Time: 4:00 - 5:00 pm

Abstract: Several coupled-physics medical imaging modalities, such as Photo-acoustic tomography or Transient elastography, have been proposed and analyzed recently to obtain high contrast, high resolution, reconstructions of constitutive properties of tissues. These inverse problems, called hybrid, coupled-physics, or multi-wave inverse problems, typically involve two steps. The first step is an inverse boundary value problem, which provides internal information about the parameters. The second step, called the quantitative step, aims to reconstruct the parameters from knowledge of the internal information obtained during the first step. This talk will review several recent results of uniqueness, stability, explicit reconstruction procedures and numerical algorithms obtained for the second step.

 

Thursday, September 6, 2012
Joint Inverse Problems/Mathematical Sciences Colloquium Seminar
Speaker: Maarten V. de Hoop, Purdue University
Title: Local analysis of the inverse boundary value problem for the Helmholtz equation and iterative reconstruction
Location: Sage 3101
Time: 4:00 - 5:00 pm

Abstract:

We consider time-harmonic seismic waves, described by the Helmholtz equation, and view the Dirichlet-to-Neumann map on the earth's surface as the data. We establish conditional Lipschitz stability for the inverse boundary value problem. The stability is obtained for models of the form of linear combinations of piecewise constant functions, naturally admitting the presence of certain conormal singularities. The dimension is determined by the number of linearly independent combinations. The stability constant grows exponentially with the dimension of this model space. We also include attenuation.

We consider then the nonlinear Landweber iteration and obtain a convergence result, assuming Lipschitz stability. Specifically, we obtain a radius of convergence, which depends on the above mentioned stability constant, and a convergence rate. Essentially, the radius reduces rapidly with the dimension of the model space, and hence compressive approximations become an important component of the analysis and procedure. We finally introduce a convergent nonlinear projected steepest descent iteration for the case of conditional Lipschitz stability.

To mitigate the growth of the stability constant with dimension on the one hand, and the approximations by sparse model representations with associated errors on the other hand, we introduce a multi-level approach with an associated condition on stability constants and on the approximation errors between neighboring levels to guarantee convergence.

We briefly summarize the computational techniques we developed leading to a massively parallel structured direct Helmholtz solver. These include a parallel Hierarchically SemiSeparable (HSS) matrix compression, factorization, and solution approach. We will show some numerical examples from reflection seismology.

Joint research with E. Beretta, L. Qiu, O. Scherzer, X.S. Li. S. Wang and J. Xia


 

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