Applied Mathematics Seminar, Fall 2017

## Mondays 4:00 PM - 5:00 PM, LCB 215

• 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 2017. 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.

September 11
Speaker: Ornella Mattei, Department of Mathematics, University of Utah
Title: Field patterns: A new type of wave
Abstract: Field patterns are a new type of wave propagating in one-dimensional linear media with moduli that vary both in space and time. Specifically, the geometry of these space-time materials is commensurate with the slope of the characteristic lines so that a disturbance does not generate a complicate cascade of subsequent disturbances, but rather concentrates on a periodic space-time pattern, that we call field pattern. Field patterns present spectacularly novel features. One of the most interesting ones is the appearance of a wave generated from an instantaneous source, whose amplitude, unlike a conventional wake, does not tend to zero away from the wave front. Furthermore, very interestingly, the band structure associated with these special space-time geometries is infinitely degenerate: associated with each point on the dispersion diagram is an infinite space of Bloch functions, a basis for which are generalized functions each concentrated on a field pattern.

September 15. Note Room is LCB 222. Time is Friday September 15 4pm - 5pm.
Speaker: Roland Pulch, Institute for Mathematics and Informatics, University of Greifswald
Title: Model order reduction for linear dynamical systems with quadratic outputs
Abstract: We investigate model order reduction (MOR) for linear dynamical systems, where a quadratic output is de ned as a quantity of interest. The system can be transformed into a linear dynamical system with many linear outputs. MOR is feasible by the method of balanced truncation, but suffers from the large number of outputs. Alternatively, we derive an equivalent quadratic-bilinear system with a single linear output. The properties of the quadratic-bilinear system are analyzed. We examine an MOR by the technique of balanced truncation, where a stabilization of the system is required. Therein, the solution of quadratic Lyapunov equations is traced back to the solution of linear Lyapunov equations. We present numerical results for several test examples comparing the two MOR approaches.

September 18
Speaker: Christel Hohenegger, Department of Mathematics, University of Utah
Title: Diffusion in complex fluids
Abstract: Complex fluids are omnipresent in our everyday life, as they encompass material from polymer melts, to mucus, blood, cake batter, and sill putty. Depending on the applied stress, they can exhibit a liquid or solid like response. As a result, immersed and passive particles show subdiffusive behavior, which is to say the variance of the particle displacement grows sublinearly with time. Passive microrheology records displacement of such particles and extract mechanical properties of the bulk fluid. It is premised on the idea that statistics of particles trajectories can reveal fundamental information about the fluid environment. We probe this hypothesis on two fronts. First, we present a Landau-Lifshitz-Navier-Stokes model of a passive particle advected in a viscoelastic fluid and show how the mean square displacement and first step auto-correlation in the increment process are related to those of the fluid's modes. Second, we address the uncertainty in reconstructing loss and storage moduli which characterize the elastic and viscous properties of the fluid from simulated data as an inverse problem.

September 25
Speaker: Andrej Cherkaev, Department of Mathematics, University of Utah
Title: Modeling of damage spread in beam lattices and robust design of fault-tolerant lattices
Abstract: We discuss modeling of unstable process of damage spread in triangular and hexagonal lattices, introduce quantitative criteria for measurement of lattice damage, simulate the damage spread, and suggest a fault tolerant design of isotropic periodic lattice that is a hybrid between triangular and hexagonal structures.

Joint work with M. Ryvkin (Tel Aviv University)

October 16
Speaker: Chiu-Yen Kao, Department of Mathematics, Claremont Mckenna College
Title: Extremal Eigenvalues for Inhomogeneous Rods and Plates
Abstract: Optimizing eigenvalues of biharmonic equations appears in the frequency control based on density distribution of composite rods and thin plates with clamped or simply supported boundary conditions. We use a rearrangement algorithm to find the optimal density distribution which minimizes a specific eigenvalue. We answer the open question regarding optimal density configurations to minimize k-th eigenvalue for clamped rods and analytically show that the optimal configurations are distinct for clamped rods and simply supported rods. Many numerical simulations in both one and two dimensions demonstrate the robustness and efficiency of the proposed approach.

October 23
Speaker: Akil Narayan, Department of Mathematics and SCI, University of Utah
Title: Multivariate polynomial quadrature for parameterized problems
Abstract: Numerical quadrature rules that use point values are ubiquitous tools for approximating integrals. Some of the most popular rules achieve accuracy by enforcing exactness for integrands in a finite-dimensional polynomial space. When the integration domain is one-dimensional, classical rules are available and plentiful. In multidimensional domains with non-standard polynomial spaces and weights, the situation is far more complicated. We will present a general methodology for numerically generating approximate quadrature rules in the multidimensional case. The need for flexible multivariate quadrature rules will be motivated by solutions to parametric partial differential equations, and the efficacy of our approach will be shown on various test problems in scientific computing.

October 30
Speaker: Frank Stenger, School of Computing, University of Utah
Title: Stenger's proof of the Riemann hypothesis
Abstract: This talk will be about the speaker's proof of the Riemann hypothesis

November 27
Speaker: Varun Shankar, Department of Mathematics, University of Utah
Title: A high-order meshfree framework for solving PDEs on irregular domains
Abstract: We present meshfree methods based on Radial Basis Function (RBF) interpolation for solving partial differential equations (PDEs) on irregular domains in O(N) complexity. First, we present the Overlapped RBF-Finite Difference (RBF-FD) method, a recent generalization of RBF-FD methods that exploits certain properties of RBF interpolants to dramatically reduce the cost of high-order methods, both on serial and parallel architectures. To rectify the spectra of RBF-FD differentiation matrices on scattered node sets, we develop a quasi-analytic hyperviscosity-based stabilization technique based on a semi-discrete Von-Neumann analysis applied to differential operators that are continuous analogues of the discrete RBF-FD differentiation matrices. We also introduce a simple ghost node technique to ameliorate boundary-related instabilities seen in RBF-FD methods. We verify high-order convergence rates on some model problems on domains with curved boundaries, and demonstrate the solution of some PDEs inspired by biological applications on more irregular domains.

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