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Applied Math. Seminar, Fall 2008

__ August 15 (Friday)__, LCB 222, 10:30am

Speaker: Peter B. Weichman, British Aerospace

__ August 18 (Monday)__, Location: LCB 219, 4:15pm

Speaker: Alexander Freidin, Institute of Problems in Mechanical Engineering Russian Academy of Sciences, St. Petersburg, Russia

**September 10 (Wed)**

Speaker: Fernando Guevara Vasquez, University of Utah

**Title: ** Edge illumination of extended targets

**Abstract: ** We use the singular value decomposition of the array
response matrix to image selectively the edges of extended
reflectors in a homogeneous medium. We show with numerical
simulations in an ultrasound regime, and analytically in the
Fraunhofer diffraction regime, that information about the
edges is contained in the singular vectors for singular
values that are intermediate between the large ones and
zero. Our results confirm well-known experimental
observations on the rank of the response matrix.

**October 1 (Wed)**

Speaker: Daniel Onofrei, University of Utah

**Title: ** Approximate cloaking for Helmholtz equation in the finite
frequency range. General theory and numerical results

**Abstract: ** In this talk we will discuss about the possibility of
cloaking materials from monochromatic EM guided waves or acoustics waves using
only nonsingular (regular) cloaks. Although perfect cloaking is impossible
using only regular materials, we will describe the procedure of building an
approximate cloak which achieves cloaking within a certain error independent of
the materials to be cloaked. Two central ideas behind our results are, the use
of a suitable nonsingular transformation of variables and the introduction of a
suitable conducting layer around the material to be cloaked in between the
material and the cloak. We will briefly introduce the main ideas and the
analytical results of our work and will be focused on presenting several
numerical results (for the two dimensional case) complimentary to our analysis,
to highlight the role of the conducting layer and the role of the cloak in the
cloaking process, and to show how the error in the approximate cloaking depends
on the conductivity in the layer. In all our numerical results extremely
singular materials (analytically described) to be cloaked will be considered.
We will also present the analytical arguments used to obtain such materials and
will numerically highlight their singular behavior. The numerical
exemplification of the approximate cloak for these materials and a given
incoming plane wave will be presented.

**October 8 (Wed)**

Speaker: Bacim Alali, University of Utah

**Title: ** Multiscale analysis of heterogeneous media in the Peridynamic formulation

**Abstract: ** We present a multiscale method for modeling the dynamics of fiber-reinforced composites using the peridynamic formulation, which is a nonlocal theory of continuum mechanics. The multiscale analysis delivers a multiscale numerical method that captures the dynamics at structural length scales while at the same time is capable of resolving the dynamics at the length scales of the fiber reinforcement.

__ October 20 (Mon 4:15pm)__ in LCB 115 (Computer lab)

Speaker: Nelson Beebe, University of Utah

**October 22 (Wed)**

Speaker: Masaki Iino, (Student talk) University of Utah

**Title: ** Optimal Stopping Rule and Applications

**Abstract: **
An optimal stopping problem is the problem of choosing some
action based on sequentially observed random variables in order to
maximize an expected payoff or to minimize an expected cost. In the area
of operations research, the action may be to replace a machine, hire a
secretary, or reorder stock, etc. In this talk, given the mathematical
definition of the optimal stopping rule, a few but very interesting
applications are introduced.

**October 29 (Wed)**

Speaker: Graeme Milton, University of Utah, Mathematics Department

**Title: ** Minimization variational principles for acoustics, elastodynamics, and electromagnetism in lossy inhomogeneous bodies at fixed frequency

**Abstract: ** The classical energy minimization principles of Dirichlet and
Thompson are extended as minimization principles to acoustics, elastodynamics
and electromagnetism in lossy inhomogeneous bodies at fixed frequency. This is
done by building upon ideas of Cherkaev and Gibiansky, who derived minimization
variational principles for quasistatics. In the absence of free current the
primary electromagnetic minimization variational principles have a minimum
which is the time-averaged electrical power dissipated in the body. The
variational principles provide constraints on the boundary values of the fields
when the moduli are known. Conversely, when the boundary values of the fields
have been measured, then they provide information about the values of the
moduli within the body. This should have application to electromagnetic
tomography. We also derive saddle point variational principles which correspond
to variational principles of Gurtin, Willis, and Borcea. This is joint work
with Pierre Seppecher and Guy Bouchitte.

**November 12 (Wed)**

Speaker: Denis Ridzal, Sandia National Laboratories

**Title: ** Scientific Discovery via Advanced Discretization and Optimization Methods

**Abstract: **Recent advances in the development, analysis, and implementation of compatible discretization and embedded optimization methods are expanding and redefining the nature of questions that can be answered by scientific computing. From the point of view of engineering design, this shift has prompted the fundamental distinction between the optimization of a modest number of parameters within conventional mathematical models, typically the realm of black-box methods, and the discovery of radically new, often counterintuitive design patterns, enabled by recent research in function-space and topology optimization.

This talk reviews several numerical methods and tools essential to promoting the science of computing to a means of scientific discovery. A unique aspect of our work is the focus on their theoretical and practical merging into integrated discovery environments. This process brings about new and unexpected mathematical and software challenges that drive our research and call for unorthodox approaches.

Numerical methods for the solution of partial differential equations (PDEs) are seen as a foundation of the discovery loop. We will show how our recent theoretical insights in the area of compatible (or mimetic) PDE discretizations have helped guide the development of a next generation of software tools, which in turn prompted further mathematical questions and opened up new research directions.

At the same time, the need to solve optimization problems with very large design spaces (for example, those arising in function-space optimization) has motivated the development of optimization algorithms that dynamically manage convergence indicators such as linear solver tolerances. Unlike conventional algorithms, the self-governing optimization schemes can take advantage of very coarse linear representations of the original problem, thereby significantly reducing computational costs.

Finally, the theoretical merging of advanced discretization and optimization techniques, as well as their practical use, have helped us gain critical insight into their mathematical interaction. We present a key result indicating that the compatibility of a discretization with respect to a PDE need not imply stable and accurate solution of an optimization problem governed by that PDE, and offer alternatives.

**November 19 (Wed)**

Speaker: Andrej Cherkaev, University of Utah, Mathematics department

**Title: ** Localized polyconvexity and Bounds for effective properties of multimaterial conducting composites.

**Abstract: ** We deal with variational problems with nonconvex Lagrangians. The problems are relaxed by computing the quasiconvex envelopes of the Lagrangians. The bounds for the quasiconvex envelope are discussed. The lower bounds are obtained by modification of polyconvex envelope. The modified technique takes into account constraints on the bounded range of fields in optimal structures. The modified bounds are solutions of a formulated relaxed finite-dimensional constrained optimization problem (called localized polyconvex envelope).

The bounds allow for a solution of a long-stanging problem of exact bounds for the effective conductivity of an isotropic multimaterial composite. These bounds refine Hashin-Shtrikman and Nesi bounds in the region of parameters where the last ones are loose.

For three-material composites, bounds for effective conductivity are explicitely found. These bounds are exact. Three-material isotropic microstructures of extremal conductivity are determined and it is demonstrated that they realize the bounds for all values of parameters. Optimal structures are laminates of a finite rank. They vary with the volume fractions and experience two topological transitions: For large values of material of minimal conductivity, its subdomain percolates (is connected), for intermediate values of that fraction, no material forms a connected domain, and for small values of that fraction, the domain of intermediate material percolates.

In collaboration with Yuan Zhang, the results are now extended to anisotropic composites: A new bounds and optimal structures are determined for a special case when conductivity of one of the material is infinite.

**December 3 (Wed)**

Speaker: Guillaume Bal, Columbia University

**Title: ** Some convergence results in equations with random coefficients.

**Abstract: ** The theory of homogenization for equations with random coefficients is now quite well-developed. What is less studied is the theory for the correctors to homogenization, which asymptotically characterize the randomness in the solution of the equation and as such are important to quantify in many areas of applied sciences. I will present recent results in the theory of correctors for elliptic and parabolic problems and briefly mention how such correctors may be used to improve reconstructions in inverse problems.
Homogenized (deterministic effective medium) solutions are not the only possible limits for solutions of equations with highly oscillatory random coefficients as the correlation length in the medium converges to zero. When fluctuations are sufficiently large, the limit may take the form of a stochastic equation and stochastic partial differential equations (SPDE) are routinely used to model small scale random forcing. In the very specific setting of a parabolic equation with large, Gaussian, random potential, I will show the following result: in low spatial dimensions, the solution to the parabolic equation indeed converges to the solution of a SPDE, which however needs to be written in a (somewhat unconventional) Stratonovich form; in high spatial dimension, the solution to the parabolic equation converges to a homogenized (hence deterministic) equation and randomness appears as a central limit-type corrector. One of the possible corollaries for this result is that SPDE models may indeed be appropriate in low spatial dimensions but not necessarily in higher spatial dimensions.

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