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 SchlumbergerDoll Research 
Title:  Matrix Sfraction 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
multiinput/multioutput (MIMO) data. We consider model reduction approaches
to obtain low order approximation of these MIMO data and construct an
equivalent blocktridiagonal system with the help of the Stieltjes truncated
continuous fractions with matrix coefficients. Then we develop
finitedifference interpretation of this system by extending the 1D 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 SchlumbergerDoll Research 
Title:  On combining model reduction and GaussNewton algorithms for solution of inverse problems 
Abstract: 
We suggest an approach to speed up the GaussNewton 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 pseudoGalerkin methods for selfadjoint 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 costfree
multilinear 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 TimeDependent Waves 
Abstract: 
As most interesting problems in wave propagation are posed on effectively
unbounded domains, their efficient numerical solution requires the
imposition of accurate nearfield 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 longtime
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 optimizationbased 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 2D 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 twocomponent 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 npoint 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 illposed. 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 nonunique,
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
largescale 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 finescale 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 contrastindependent preconditioners for complex
heterogeneities.

Please address administrative questions to aipc@math.tamu.edu. Scientific questions should be addressed to the chair of the Scientific Program Committee: rundell AT math.tamu.edu