Applied Mathematics Seminar, Fall 2016

## Mondays 4:00 PM - 5:00 PM, LCB 222

• 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 2016. 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 22 (Welcome Back and Group Photo!)
Speaker: Davit Harutyunyan, Department of Mathematics, The University of Utah
Title: Quantitative Wulff and Brunn-Minkowski inequalities for convex sets
Abstract: In this lecture we revisit the anisotropic isoperimetric (Wulff) and the Brunn-Minkowski inequalities for convex sets. The best know constant C(n)=Cn^8.5 depending on the space dimension n in both inequalities is due to Figalli, Maggi and Pratelli, 2010. We improve that constant to Cn^6 for convex sets and even better in some cases. We also conjecture, that the best constant in both inequalities must be of the form Cn^2, i.e., quadratic in n. The tools are the Brenier's mapping from the theory of optimal mass transportation combined with new sharp geometric-arithmetic mean and some algebraic inequalities plus a trace estimate by Figalli, Maggi and Pratelli.

September 26
Speaker: Jared Whitehead, Department of Mathematics, BYU
Title: Variations in the heat transport of Rayleigh Benard convection
Abstract: Rayleigh Benard convection is the canonical system where a fluid is heated from below and/or cooled from above, yielding an unstably driven system. When this temperature difference is sufficiently large, buoyancy effects induce convective motion within the fluid. As this driving force increases, the flow becomes turbulent. The fundamental question asked of this system is how the volume averaged heat transport depends on the driving force and material properties of the fluid. After reviewing the sometimes contradictory experimental evidence, we consider how different boundary conditions, heat sources, and variations in the kinematic properties of an incompressible fluid affect the heat transport in the turbulent, convective regime using rigorous upper bounds derived via variational techniques.

October 17
Speaker: Alexander Kurganov, Department of Mathematics, Tulane University
Title: CENTRAL-UPWIND SCHEMES FOR SHALLOW WATER MODELS
Abstract: In the first part of the talk, I will describe a general framework for designing finite-volume methods (both upwind and central) for hyperbolic systems of conservation laws. I will focus on Riemann-problem-solver-free non-oscillatory central schemes and, in particular, on central-upwind schemes that belong to the class of central schemes, but has some upwind features that help to reduce the amount of numerical diffusion typically present in staggered central schemes such as, for example, the first-order Lax-Friedrichs and second-order Nessyahu-Tadmor scheme. In the second part of the talk, I will discuss how central-upwind schemes can be extended to hyperbolic systems of balance laws, such as the Saint-Venant system and related shallow water models. The main difficulty in this extension is preserving a delicate balance between the flux and source terms. This is especially important in many practical situations, in which the solutions to be captured are (relatively) small perturbations of steady-state solutions. The other crucial point is preserving positivity of the computed water depth (and/or other quantities, which are supposed to remain nonnegative). I will present a general approach of designing well-balanced positivity preserving central-upwind schemes and illustrate their performance on a number of shallow water models.

October 24
Speaker: Noa Kraitzman, Department of Mathematics, The University of Utah
Title: Bifurcation and Competitive Evolution of Network Morphologies in the strong Functionalized Cahn-Hilliard (FCH) Free Energy
Abstract: The FCH is a higher-order free energy for blends of amphiphillic polymers and solvent which balances solvation energy of ionic groups against elastic energy of the underlying polymer backbone. Its gradient flows describe the formation of solvent network structures which are essential to ionic conduction in polymer membranes. The FCH possesses stable, coexisting network morphologies and we characterize their geometric evolution, bifurcation and competition through a center-stable manifold reduction which encompasses a broad class of coexisting network morphologies. The stability of the different networks is characterized by the meandering and pearling modes associated to the linearized system. For the $H^{-1}$ gradient flow of the FCH energy, using functional analysis and asymptotic methods, we drive a sharp-interface geometric motion which couples the flow of co-dimension 1 and 2 network morphologies, through the far-field chemical potential. In particular, we derive expressions for the pearling and meander eigenvalues for a class of far-from-self-intersection co- dimension 1 and 2 networks, and show that the linearization is uniformly elliptic off of the associated center stable space.

October 31
Speaker: Michael Ryvkin, The Iby and Aladar Fleischman Faculty of Engineering, Tel Aviv University
Title: ANALYSIS OF NON-PERIODIC STRESS STATE IN PERIODIC MATERIALS. Applications to fracture and optimization.
Abstract: Many man-made materials have a periodic microstructure, periodic materials are widely met also in nature. Study of overall elastic properties of such materials and their optimization is a well-developed topic. The corresponding problems are characterized by a periodic stress state, however, in many cases of interest the stress state in periodic material is non-periodic. The non-periodicity can result from a non-periodic applied loading, from presence of cracks, inclusions and other flaws, and from the finite dimensions of the sample to be considered.
In these cases a direct numerical simulation is computationally expensive due to a large number of degrees of freedom to be involved, but reducing the analysis domain to a single repetitive cell is not straightforward. This goal is achieved by applying the discrete Fourier transform, casted as the representative cell method. As a result, one has to resolve a number of representative(repetitive) cell problems in the transforms space and can obtain the sought elastic field by the inverse transformation. The important feature of these problems is that they are independent and, consequently, can be treated by the use of parallel computing. It is shown how to plug-in the method into an efficient multiscale analysis scheme for arbitrary shaped sample of periodic material.
The suggested approach is employed for the fracture analysis of beam lattices: two-dimensional honeycombs and spatial open cell Kelvin foam, both cracks nucleation and propagation problems are addressed. Solid periodically voided and composite materials with flaws are considered as well, the optimal parameter combinations maximizing the fracture toughness are determined.

November 7
Speaker: Orly Alter, Departments of Bioengineering and Human Genetics, The University of Utah
Title: Cancer Diagnostics and Prognostics from Comparative Spectral Decompositions of Patient-Matched Genomic Profiles
Abstract: I will, first, briefly review our matrix and tensor modeling of large-scale molecular biological data, which, as we demonstrated, can be used to correctly predict previously unknown physical, cellular, and evolutionary mechanisms that govern the activity of DNA and RNA. Second, I will describe our recent generalized singular value decomposition (GSVD) and tensor GSVD comparisons of the genomes of tumor and normal cells from the same sets of astrocytoma brain and, separately, ovarian cancer patients, which uncovered patterns of DNA copy-number alterations that are correlated with a patient's survival and response to treatment. Third, I will present our higher-order GSVD, the only mathematical framework that can create a single coherent model from, i.e., simultaneously find similarities and dissimilarities across multiple two-dimensional datasets, by extending the GSVD from two to more than two matrices.

November 14
Speaker: Lajos Horvath, Department of Mathematics, The University of Utah
Title: Eigenvalue analysis of large dimensional matrices
Abstract: Testing for stability in linear factor models has become an important topic in both the statistics and econometrics research communities. The available methodologies address testing for changes in the mean/linear trend, or testing for breaks in the covariance structure by checking for the constancy of common factor loadings. In such cases when an external shock induces a change to the stochas tic structure of high dimensional data, it is unclear whether the change would be reflected in the mean, the covariance structure, or both. We develop a test for structural stability of linear factor models that is based on monitoring for changes in the largest eigenvalue of the sample covariance matrix. The asymptotic distribu tion of the proposed test statistic is established under the null hypothesis that the mean and covariance structure of the cross sectional units remain stable during the observation period. We show that the test is consistent assuming common breaks in the mean or factor loadings. These results are investigated by means of a Monte Carlo simulation study, and their usefulness is demonstrated with an application to U.S. treasury yield curve data, in which some interesting features of the 2007-2008 subprime crisis are illuminated.
Joint work with Gregory Rice.

November 21. Note starting time is 4:15pm.
Speaker: Vianey Villamizar, Department of Mathematics, BYU
Title: Exact Local Absorbing Boundary Conditions for Acoustic Waves in terms of Farfield Expansions
Abstract: We devise a new high order local absorbing boundary condition (ABC) for radiating problems and scattering of time-harmonic acoustic waves from obstacles of arbitrary shape. By introducing an artificial boundary $S$ enclosing the scatterer, the original unbounded domain $\Omega$ is decomposed into a bounded computational domain $\Omega^-$ and an exterior unbounded domain $\Omega^+$. Then, we define interface conditions at the artificial boundary $S$ , from truncated versions of the well-known Wilcox and Karp farfield expansion representations of the exact solution in the exterior region $\Omega^+$. As a result, we obtain a new local absorbing boundary condition (ABC) for a bounded problem on $\Omega^-$, which effectively accounts for the outgoing behavior of the scattered field. Contrary to the low order absorbing conditions previously defined, the order of the error induced by this ABC can easily match the order of the numerical method in $\Omega^-$. We accomplish this by simply adding as many terms as needed to the truncated farfield expansions of Wilcox or Karp. The convergence of these expansions guarantees that the order of approximation of the new ABC can be increased arbitrarily without having to enlarge the radius of the artificial boundary. We include numerical results in two and three dimensions which demonstrate the improved accuracy and simplicity of this new formulation when compared to other absorbing boundary conditions.
Collaborators: Sebastian Acosta and Blake Dastrup

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