Applied Mathematics Seminar, Fall 2015

## Mondays 3:55 PM - 5:00 PM, LCB 219

• This seminar can be taken for credit: Students can get 1-3 credits by registering to the Applied Math Seminar class Math 7875 Section 010 for Fall 2015. Students should talk to the seminar organizer before taking it for a credit. Grading is based on attendance and giving a talk by presenting an applied-mathematics paper (not necessarily your own). Student talks will be appropriately labeled to distinguish them from visitor talks. The seminar organizer is available to review your slides, for dry-runs etc.
• Please direct questions or comments about the seminar (or its class) to Yekaterina Epshteyn (epshteyn (at) math.utah.edu)
• Talks are announced through the applied-math mailing list. Please ask the seminar organizer for information about how to subscribe to this list.

August 24 (Welcome Back!)
Speaker: Andrej Cherkaev, Department of Mathematics, University of Utah
Title: Optimal Multicomponent Composites: Amazing 3d Structures and hint for new bounds.
Abstract:I will review the latest results concerning optimal multicomponent 2D- and 3D composite structures. These structures are minimizing sequences of a variational problem with a multiwell Lagrangian; their energy represents a relaxed energy of an optimal composite or a quasiconvex envelope of the Lagrangian. Analysis of the fields in optimal structures provides hints for modification of a lower bound for the relaxed energy.

September 14
Speaker: Jack Xin, Department of Mathematics, University of California, Irvine
Title: Minimizing the Difference of L1 and L2 Norms and Applications
Abstract:L1 norm minimization is a widely used convex method for enforcing sparsity in signal recovery and model selection. In this talk, we introduce a non-convex Lipschitz continuous function, the difference of L1 and L2 norms (DL12), and discuss its sparsity promoting properties. Using examples in compressed sensing and imaging, we show that there can be plenty of gain beyond L1 by minimizing DL12 at a moderate level of additional computation via the difference of convex function algorithms. We shall draw a connection of DL12 with penalty functions in statistics and machine learning.

September 18 (Student Talk). Note, Room JWB 333
Speaker: Vira Babenko, Department of Mathematics, University of Utah
Title: Numerical Analysis of Set-Valued and Fuzzy-Valued functions - A Unified Approach and Applications.
Abstract:A wide variety of questions from social, economic, physical, and biological sciences can be formulated using functions with values that are fuzzy sets or sets in finite or infinite dimensional spaces. Set-valued and fuzzy-valued functions attract attention of many researchers and allow them to look at numerous problems from a new point of view and provide them with new tools, ideas and results. In this talk we consider a generalized concept of such functions, that of functions with values in L-spaces. This class of functions encompasses set-valued and fuzzy functions as special cases which allows us to investigate them from a common point of view. We will discuss several problems of Approximation Theory and Numerical Analysis for functions with values in L-spaces. In particular, we will present numerical methods of solution of Fredholm and Volterra integral equations for such functions.

September 21
Speaker: Michael Meylan, School of Mathematical and Physical Sciences, The University of Newcastle, Australia
Title: Wave - Ice interaction, field measurements, laboratory experiments, and mathematical models
Abstract: The attenuation and scattering of waves by sea ice is a complex process and the current state of our knowledge is quite limited. This in turn makes it difficult to make even the most basic predictions of wave induced melting or to forecast the wave state in the frozen ocean. The key process which we need to model is the interaction of ocean waves with a single ice floe (or small groups of floes). However, we only have field measurements of large scale wave attenuation (over hundreds of ice floes) and it is actually not obvious how to scale from single floe models to multiple floe problems. Therefore the models are lacking validation at both the large and small scale. In a recent series of experiments performed in a wavetank we have tried to validate and test the range of applicability of our numerical models. I will present results and comparisons from these experiments and discuss their implications for accurate modelling of wave-ice interactions.

September 25. Note, Room LCB 215
Speaker: Owen Miller, Department of Mathematics, MIT
Title: Photonic Design: Reaching the Limits of Light-Matter Interactions
Abstract: Photonic devices are emerging for an increasing variety of technological applications, ranging from sensors to solar cells. In three areas - photovoltaics, nanoparticle scattering, and radiative heat transfer - I will show how large-scale computational optimization and rigorous analytical frameworks enable rapid search through large design spaces, and spur discovery of fundamental limits to interactions between light and matter.

In photovoltaics, the famous ray-optical 4n^2 limit to absorption enhancement has for decades served as a critical design goal, and it motivated the use of quasi-random textures in commercial solar cells. I will show that at subwavelength scales, non-intuitive, computationally designed textures outperform random ones, and can closely approach the 4n^2 limit. Pivoting to metallic structures, where there has not been an analogous "4n^2" limit, I will show how energy-conservation principles lead to fundamental limits to the optical response of metals, answering a long-standing question about the tradeoff between resonant enhancement and material loss. The limits were stimulated by a computational discovery in nanoparticle optimization, where I will present theoretical designs and experimental measurements (by a collaborator) approaching the upper bounds of absorption and scattering. The limits can be extended to the emerging field of radiative heat transfer, where they suggest the possibility for periodic, nanostructured media to exhibit orders-of-magnitude improvement over previous designs.

October 19 (Student Talk and Ph.D defense)
Speaker: Predrag Krtolica, Department of Mathematics, University of Utah
Title: Compatibility Conditions in Discrete Structures and Application to Damage
Abstract: The work introduces compatibility conditions in discrete lattices and describes their properties. A connection between discrete and continuum compatibility conditions is made. The spread of damage in lattices is analyzed using compatibility conditions.

This talk is a part of the defense of PhD dissertation.

October 26
Speaker: Jeremy Marzuola, Department of Mathematics, University of North Carolina, Chapel Hill
Title: Morse/Maslov Indices for Elliptic Operators on General Domains
Abstract:With Graham Cox and Chris Jones, we first study second-order, self-adjoint elliptic operators on a smooth one-parameter family of domains without any assumptions on the symmetry. It will follow that the Morse index for the elliptic operator can be related to the Maslov index of an appropriately defined path in a symplectic Hilbert space defined on the boundary. Specifically, the Maslov index of the path we define relates the Morse index of the initial domain to the Morse index of the final domain. This is particularly useful when the domain can be taken to have arbitrarily small volume, because the spectral problem is particularly simple in that case. This generalizes previous results of Deng-Jones that were only available on star-shaped domains, or for Dirichlet boundary conditions. With the Morse index theorem in hand, we will also explore several higher dimensional applications in stability theory. Then, we will discuss recent results on manifold decompositions and applications of the Maslov index to elliptic operators on general domains, even without boundary.

This is work constitutes a large scale stability theory project undertaken with G. Cox, C. Jones, R. Marangell, A. Sukhtayev, and S. Sukhtaiev.

November 2
Speaker: Simon Lemaire, CERMICS Laboratory at École des Ponts ParisTech (Marne-la-Vallée, France)
Title: Hybrid High-Order methods for the arbitrary-order structure-preserving discretization of PDEs on polytopal meshes
Abstract: Hybrid High-Order (HHO) methods are discontinuous skeletal methods that enable the discretization of PDEs on general polygonal/polyhedral meshes. HHO methods are based on face- and cell-centered polynomial unknowns (hence the term hybrid), and allow for high-order discretizations. They offer several assets: their construction is dimension-independent, they are locally conservative, and they make the robust treatment of physical parameters possible in various situations (heterogeneous/anisotropic diffusion, quasi-incompressible linear elasticity, advection-dominated transport...). When compared to interior penalty DG methods, HHO methods are also appealing in terms of computational cost. They have now been tested on a wide variety of linear and nonlinear problems.

Joint work with Daniele A. Di Pietro and Alexandre Ern

November 9. Note, Time of this seminar is 4:05pm.
Speaker: Fernando Guevara Vasquez, Department of Mathematics, University of Utah
Title: Discrete conductivity and Schroedinger inverse problems
Abstract: We consider the problem of finding the electric properties of components in an electrical circuit (i.e. possibly complex edge weights in a graph) from measurements made at few accessible nodes. This is a discrete analogue to continuum inverse problems such as electrical impedance tomography and the inverse Schroedinger problem. We show that if the linearization to the discrete inverse problem about one set of weights is injective, then the weights are determined uniquely by the measurements, except for a zero measure set. This is without making any assumption on the topology of the graph. The proof borrows ideas from the Complex Geometric Optics method that has been used to show uniqueness for continuum inverse problems like the ones mentioned above.

November 16
Speaker: Davit Harutyunyan, Department of Mathematics, University of Utah
Title: Towards characterization of all $3\times3$ quasiconvex quadratic forms
Abstract: Given a quasiconvex quadratic form defined on the set of $d\times d$ matrices, $d\geq 3,$ we prove that if the determinant of its acoustic tensor is an irreducible extremal polynomial that is not identically zero, then the form itself is an extremal quasiconvex quadratic form, i.e. it loses its quasiconvexity whenever a convex quadratic form is subtracted from it. In the special case $d=3$ we do complete analysis in the case when the determinant of the acoustic tensor of the form is an extremal polynomial. Namely we prove the following results:

1) If the determinant of the acoustic tensor of the quadratic form is an extremal polynomial that is not a perfect square, then the form itself is an extremal quadratic form.

2) If the determinant of the acoustic tensor of the quadratic form is identically zero, then the form is either extremal or polyconvex.

3) If the determinant of the acoustic tensor of the quadratic form is a perfect square, then the form is either polyconvex or rank-one plus an extremal, that has a zero acoustic tensor determinant.

We also prove the following Krein-Millman property: Any quasiconvex quadratic form in $d$ variables is a sum of an extremal and a polyconvex forms. Here we use the notion of extremality introduced by Milton. We conjecture, that if the determinant of the acoustic tensor of the quadratic form is not an extremal polynomial, then the form is not an extremal either. This results are expected to be very useful for deriving optimal bounds in the theory of composite materials.

Joint work with Graeme Milton

November 20. Note, Room JWB 333
Speaker: Aaron Welters, Department of Mathematical Sciences, Florida Institute of Technology
Title: Toward a Theory of Broadband Absorption Suppression in Magnetic Composites
Abstract: A major problem with magnetic materials in application is they naturally have high losses in a wide frequency range of interest (e.g., Faraday rotation using ferromagnets in optical frequencies). Composites can inherit significantly altered properties from those of their components. Does this apply to losses and magnetic properties? How can broadband absorption suppression in magneticdielectric composites be achieved? In this talk, we will discuss the development of a theory of broadband absorption suppression in magnetic composites based on a Lagrangian and Hamiltonian approach from classical mechanics. Using a Lagrangian framework, we introduce a model for two-component linear systems with a high-loss and a lossless component in order to study the interplay of dissipation (losses) and gyroscopy (magnetism) as well as the dominant mechanisms of energy loss. New results towards answering these questions will be discussed related to modal dichotomy and selective overdamping phenomena in gyroscopic-dissipative systems. The potential of selective overdamping for significant broadband absorption suppression is explored via examples involving electric circuits with gyrators and resistors.

This is joint work with Alex Figotin (UCI).

November 30
Speaker: Francois Monard, Department of Mathematics, University of Michigan
Title: Reconstruction methods for coupled-physics inverse problems
Abstract: In this talk, we will review a general reconstruction approach for some inverse parameter-reconstruction problems in elliptic PDEs (namely, conductivity and elasticity), from knowledge of internal functionals. Such problems are motivated by the field of coupled-physics (or hybrid) inverse problems, coupling high-contrast and high-resolution imaging methods with the aim to derive future imaging methods with both qualities mentioned. Typically, such reconstruction approaches consist in deriving algebraic reconstruction formulas, which are valid under some compatibility conditions satisfied by the measurements, namely, some rank maximality conditions of the PDE solutions generating the measurements. A second step is then to show that such conditions can be fulfilled by some solutions, which can be, in some cases, generated from explicit boundary prescriptions. Such inversion algorithms are then proved to be much more stable (on the Hilbert scale) than their inverse boundary value problems counterpart, and in some cases, injective where the latter problems are not. The stability statement above is then the mathematical confirmation that the resolution accessible on reconstructed parameters in these models is tremendously improved and presents great practical potential.

The works presented involve various collaborations with Guillaume Bal, Gunther Uhlmann, Chenxi Guo, Sebastien Imperiale, Cedric Bellis, Eric Bonnetier and Faouzi Triki.

December 3, joint Stochastics/Applied Math/Math Bio Seminar. Note Time is 9am - 10am. SW 132
Speaker: Sean Lawley, Department of Mathematics, University of Utah
Title: Randomly switching ODEs, PDEs, and SDEs: Mathematical analysis and biological insight
Abstract: Prompted by diverse biological applications, we consider ODEs with randomly switching right-hand sides and PDEs and SDEs with randomly switching boundary conditions. In this talk, I will describe the tools for analyzing these systems and show how they can answer questions in biochemistry, physiology, neuroscience, and cellular transport. Special attention will be given to establishing mathematical connections between these three classes of stochastic processes.

Seminar organizer: Yekaterina Epshteyn (epshteyn (at) math.utah.edu).

155 South 1400 East, Room 233, Salt Lake City, UT 84112-0090, T:+1 801 581 6851, F:+1 801 581 4148