Minisymposia Abstracts

Speaker: Fernando Guevara Vasquez
University of Utah
Title: Uncertainty quantification in resistor network inversion
Abstract: We present a method for finding the electrical conductivity in a domain from electrical measurements at the boundary. Our method consists of two steps. In the first step we find a resistor network that fits the data and then we estimate the conductivity from the resistors by interpreting the network as a finite volumes discretization of the problem. We show through a Monte Carlo study that our discretization of the conductivity reduces the uncertainty in the reconstructions, as compared to a conventional discretization.

Speaker: Vladimir Druskin
Schlumberger-Doll Research
Title: Matrix S-fraction approach for multidimensional inverse spectral problems
Abstract: The multidimensional time domain problems of electromagnetic and seismic exploration can be formulated as inverse spectral problems with multi-input/multi-output (MIMO) data. We consider model reduction approaches to obtain low order approximation of these MIMO data and construct an equivalent block-tridiagonal system with the help of the Stieltjes truncated continuous fractions with matrix coefficients. Then we develop finite-difference interpretation of this system by extending the 1-D optimal grid framework of Borcea et al (2005) and Guevara Vasquez (2006). Efficacy of the new approach is demonstrated on 2D numerical examples.

Speaker: Mikhail Zaslavsky
Schlumberger-Doll Research
Title: On combining model reduction and Gauss-Newton algorithms for solution of inverse problems
Abstract: We suggest an approach to speed up the Gauss-Newton solution of inverse PDE problems by minimizing the number of forward problem calls. The acceleration is based on effective incorporation of the information from the previous iteration via a reduced order model (ROM). It is designed with the help of Galerkin and pseudo-Galerkin methods for self-adjoint and complex symmetric problems respectively. The constructed ROM generates effective multivariate rational interpolation matching the forward solutions and the Jacobians from the previous iterations. To speed up the convergence, the cost-free multi-linear search was developed. Numerical examples for the inverse conductivity problem for the 3D Maxwell system show significant acceleration.

Speaker: Thomas Hagstrom
Southern Methodist University
Title: Approximate Radiation Boundary Conditions for Time-Dependent Waves
Abstract: As most interesting problems in wave propagation are posed on effectively unbounded domains, their efficient numerical solution requires the imposition of accurate near-field radiation boundary conditions. We will present our recent work on optimal local radiation boundary condition sequences for models which are homogeneous and isotropic outside of the computational domain. Here we have found that the key requirement to obtain long-time accuracy is to construct conditions which are accurate for both the propagating and the evanescent spectrum. We will also discuss the difficulties which arise when trying to treat more complex systems which are anisotropic or inhomogeneous and suggest some approaches which we believe hold promise.

Speaker: Alexander Mamonov
University of Texas, Austin
Title: Resistor Networks and Optimal Grids for Electrical Impedance Tomography with Partial Boundary Measurements
Abstract: We present methods to solve the partial data Electrical Impedance Tomography (EIT) problem numerically. Our methods regularize the problem by using sparse representations of the unknown conductivity on adaptive finite volume grids known as the optimal grids. The discretized problem is reduced to solving the discrete inverse problems for resistor networks. Two distinct approaches implementing this strategy are presented.
The first approach uses the results for the full data EIT with circular resistor networks. The optimal grids for such networks are essentially one dimensional objects, which can be computed explicitly. We solve the partial data problem by reducing it to the full data case using the theory of extremal quasiconformal mappings.
The second approach is based on pyramidal resistor networks. The optimal grids in this case are computed using the sensitivity analysis of both the continuum and the discrete EIT problems.
Numerical results show two main advantages of our approaches compared to the traditional optimization-based methods. First, the inversion based on resistor networks is much faster than any iterative algorithm. Second, we are able to reconstruct the conductivities of ultra high contrast, which usually presents a challenge to inversion methods.

Speaker: Shari Moskow
Drexel University
Title: Optimal Grids for Anisotropic Problems
Abstract: Spectral convergence of optimal grids for anisotropic problems is both numerically observed and explained. For elliptic problems, the gridding algorithm is reduced to a Stieltjes rational approximation on an interval of a line in the complex plane instead of the real axis as in the isotropic case. We show rigorously why this occurs for a semi- infinite and bounded interval. We then extend the gridding algorithm to hyperbolic problems on bounded domains. For the propagative modes, the problem is reduced to a rational approximation on an interval of the negative real semiaxis, similarly to in the isotropic case. For the wave problem we present numerical examples in 2-D anisotropic media. joint work with Vladimir Druskin and Sergey Asvadurov.

Speaker: Elena Cherkaev
University of Utah
Title: Inverse problem for the structure of composite materials
Abstract: The talk discusses inverse homogenization problem which is a problem of deriving information about the microgeometry of a two-component composite material from measurements of its effective properties. Constrained Pade approximations are used to reconstruct the spectral measure of the corresponding differential operator from the analytic Stieltjes representation of the effective complex permittivity. This representation relates the n-point correlation functions of the microstructure to the moments of the spectral measure, which contains all information about the microgeometry. The problem of identification of the spectral function from effective measurements known in an interval of frequency has a unique solution, however the problem is ill-posed. Reconstruction of the spectral measure reduces the problem to an unusual type of inverse spectral problem, which has a unique solution only in particular classes of composite structures. Generally, characterization of microgeometry is non-unique, however, some geometric information can be infered.

Speaker: Yalchin Efendiev
Texas A&M University
Title: Multiscale model reduction techniques for flows in heterogeneous porous media
Abstract: The development of numerical algorithms for simulations of flow processes in large-scale highly heterogeneous porous formations is challenging because properties of natural geologic porous formations (e.g., permeability) display high variability and complex spatial correlation structures which can span a hierarchy of length scales. It is usually necessary to resolve a wide range of length and time scales, which can be prohibitively expensive, in order to obtain accurate predictions of the flow, mechanical deformation, and transport processes under investigation. In practice, some types of coarsening (or upscaling) of the detailed model are usually performed before the model can be used to simulate complex processes. Many approaches have been developed and applied successfully when a scale separation adequately describes the spatial variability of the subsurface properties (e.g., permeability) that have bounded variations. The quality of these approaches deteriorates for complex heterogeneities without scale separation and high contrast. In this talk, I will describe multiscale model reduction techniques that can be used to systematically reduce the degrees of freedoms of fine-scale simulations and discuss applications to preconditioners. Numerical results will be presented that show that one can improve the accuracy of multiscale methods by systematically adding new coarse basis functions and obtain contrast-independent preconditioners for complex heterogeneities.

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