# Applied and Computational Math Seminar

## Scheduled Talks (Fall 2018): Room 425, 1:30 - 2:30pm

Date: April 26, 2019
Speaker: Harbir Antil, George Mason University
Title: Fractional PDEs: Optimal Control and Applications
Abstract: Fractional calculus and its application to anomalous transport has recently received a tremendous amount of attention. In these studies, the anomalous transport (of charge, tracers, fluid, etc.) is presumed attributable to long-range correlations of material properties within an inherently complex, and in some cases self-similar, conducting medium. Rather than considering an exquisitely discretized (and computationally explosive) representation of the medium, the complex and spatially correlated heterogeneity is represented through reformulation of the PDE governing the relevant transport physics such that its coefficients are, instead, smooth but paired with fractional-order space derivatives. This talk will give an introduction to fractional diffusion. We will describe how to incorporate nonhomogeneous boundary conditions in fractional PDEs. We will cover from linear to quasilinear fractional PDEs. New notions of exterior optimal control and optimization under uncertainty will be presented. We will conclude the talk with an approach that allows the fractional exponent to be spatially dependent. This has enabled us to define novel Sobolev spaces and their trace spaces. Several applications in: imaging science, quantum random walks, geophysics, and manifold learning (data analysis) will be discussed.

## Past Talks:

Date: April 10, 2019
Speaker: Michael Levitin, University of Reading, UK
Title: Sharp eigenvalue asymptotics for Steklov problem on curvilinear polygons
Abstract: I will discuss a work in progress (joint with Leonid Parnovski, Iosif Polterovich and David Sher) on Steklov (or Dirichlet-to-Neumann map) eigenvalue asymptotics for curvilinear polygonal domains in R^2. The results are quite unexpected, and the asymptotics depends non-trivially on the arithmetic properties of the angles of the polygon. There are also connections to classical problems of hydrodynamics (the sloping beach problem and the sloshing problem) and to the Laplacian on quantum graphs.

Date: February 15, 2019
Speaker: Ricardo Nochetto, University of Maryland, College Park
Title: Thermally Actuated Bilayer Plates
Abstract: We present a simple mathematical model of polymer bilayers that undergo large bending deformations when actuated by non-mechanical stimuli such as thermal effects. The model consists of a nonlinear fourth order problem with a pointwise isometry constraint, which we discretize with either Kirchhoff quadrilaterals or discontinuous Galerkin methods. We prove $\Gamma$-convergence of the discrete model and propose an iterative method that decreases its energy and leads to stationary configurations. We investigate performance, as well as reduced model capabilities, via several insightful numerical experiments involving large (geometrically nonlinear) deformations. They include the folding of several practically useful compliant structures comprising of thin elastic layers. This work is joint with S. Bartels, A. Bonito, and D. Ntogkas.

Date: February 1, 2019
Speaker: Heather Harrington, Oxford University
Title: Comparing models and data using computational algebraic geometry and topology.
Abstract: I will overview my research for a very general math audience. I will start with motivation of the biological problems we have explored, such as tumor-induced angiogenesis (the growth of blood vessels to nourish a tumor), as well as signaling pathways involved in the dysfunction of cancer (sets of molecules that interact that turn genes on/off and ultimately determine whether a cell lives or dies). Both of these biological problems can be modeled using differential equations. The challenge with analyzing these types of mathematical models is that the rate constants, often referred to as parameter values, are difficult to measure or estimate from available data.

I will present mathematical methods we have developed to enable us to compare mathematical models with experimental data. Depending on the type of data available, and the type of model constructed, we have combined techniques from computational algebraic geometry and topology, with statistics, networks and optimization to compare and classify models without necessarily estimating parameters. Specifically, I will introduce our methods that use computational algebraic geometry (e.g., Grobner bases) and computational algebraic topology (e.g., persistent homology). I will present applications of our methodology on datasets involving cancer. Time permitting, I will conclude with our current work for analyzing spatio-temporal datasets with multiple parameters using computational algebraic topology. Mathematically, this is studying a module over a multivariate polynomial ring, and finding discriminating and computable invariants.

Date: December 11, 2018
Speaker: Yasumasa Nishiura, Advanced Institute for Materials Research, Tohoku University
Title: What is a good mathematical descriptor for the toughness of heterogeneous materials
Abstract: One of the dreams of materials scientists is to make a novel materials through the design of atomistic scale, however most of the modern composite materials is very heterogeneous in order to make it strong via network structure like epoxy resin matrix with carbon fibers used in the aircraft. Those are far from crystal structure nor completely random so that it is not apriori clear what type of mathematical concepts is appropriate to describe it, especially medium-range structure. As for the static profile, recent statistical methods as well as topological approach TDA clarify some aspects of it. On the other hand, the performance of the materials, especially its dynamic robusness against mechanical stress remains open and still heavily depends on trials and errors in the laboratories. The difficulty lies in that, firstly the lack of good macroscopic mathematical model to describe dynamical processes, secondly it is not clear how to implement the microscopic heterogeneity into the macroscopic model, thirdly how to extract an appropriate mathematical descriptor that can measure and predict the strength of it and even allows us to design novel materials. I would like to present a case study in this direction in the context of cracking phenomena for brittle materials. The stage is still early phase, however it suggests many interesting questions and challenge not only for for materials scientists but for mathematicians.

Date: November 16, 2018
Title: Surveillance-Evasion games under uncertainty
Abstract: Adversarial path planning problems are important in robotics applications and in modeling the behavior of humans in dangerous environments. Surveillance-Evasion (SE) games form an important subset of such problems and require a blend of numerical techniques from multiobjective dynamic programming, game theory, numerics for Hamilton-Jacobi PDEs, and convex optimization. We model the basic SE problem as a semi-infinite zero-sum game between two players: an Observer (O) and an Evader (E) traveling through a domain with occluding obstacles. O chooses a pdf over a finite set of predefined surveillance plans, while E chooses a pdf over an infinite set of trajectories that bring it to a target location. The focus of this game is on "E's expected cumulative exposure to O", and we have recently developed an algorithm for finding the Nash Equilibrium open-loop policies for both players. I will use numerical experiments to illustrate algorithmic extensions to handle multiple Evaders, moving Observes, and anisotropic observation sensors. Time permitting, I will also show preliminary results for a very large number of selfish/independent Evaders modeled via Mean Field Games. Joint work with M.Gilles, E.Cartee, and REU-2018 participants.

Date: November 2, 2018
Speaker: Abner Salgado, University of Tennessee, Knoxville
Title: Regularity and rate of approximation for obstacle problems for a class of integro-differential operators
Abstract: We consider obstacle problems for three nonlocal operators:
A) The integral fractional Laplacian
B) The integral fractional Laplacian with drift
C) A second order elliptic operator plus the integral fractional Laplacian
For the solution of the problem in Case A, we derive regularity results in weighted Sobolev spaces, where the weight is a power of the distance to the boundary. For cases B and C we derive, via a Lewy-Stampacchia type argument, regularity results in standard Sobolev spaces. We use these regularity results to derive error estimates for finite element schemes. The error estimates turn out to be optimal in Case A, whereas there is a loss of optimality in cases B and C, depending on the order of the integral operator.

Date: October 26, 2018
Speaker: Bill Kalies, Florida Atlantic University
Title: Order Theory in Dynamics
Abstract: Recurrent versus gradient-like behavior in global dynamics can be characterized via a surjective lattice homomorphism between certain bounded, distributive lattices, that is, between attracting blocks (or neighborhoods) and attractors. In this lecture we explain the basic order and lattice theory for dynamical systems which lays a foundation for a computational theory for dynamical systems that focuses on Morse decompositions and index lattices. We build combinatorial order-theoretic models for global dynamics. We give computational examples that illustrate the theory for both maps and flows.

Date: October 24, 2018, 5:00-6:00pm
Room: Hill 005
Speaker: Prof. Wojciech Chacholski, Department of Mathematics, KTH
Title: What is persistence
Abstract: It is not surprising that different units and scales are used to measure different phenomena. So why the Gromov-Hausdorff and bottleneck distances are the only one used to measure inputs and outcomes of topological data analysis applied to a variety of different data sets? My aim is to explain and illustrate a new approach to persistence. I will present both mathematical and real life data examples illustrating effectiveness of our approach to improve various classification tasks.

Date: October 19, 2018
Speaker: Francisco Sayas, University of Delaware
Title: Waves in viscoelastic solids
Abstract: I will first explain a transfer function based framework collecting well-known models of wave propagation in viscoelastic solids, fractional derivative extensions, and their couplings. I will briefly explain the associated Laplace domain stability results and their time domain counterparts, as well as how they are affected by finite element discretization in space. Finally, I will discuss a semigroup approach to a non-strictly diffusive Zener model. This is joint work with Tom Brown, Shukai Du, and Hasan Eruslu.

Date: October 12, 2018
Speaker: Vladimir Itskov, The Pennsylvania State University
Title: Directed complexes, sequence dimension and inverting a neural network.
Abstract: What is the embedding dimension, and more generally, the geometry of a set of sequences? This problem arises in the context of neural coding and neural networks. Here one would like to infer the geometry of a space that is measured by unknown quasiconvex functions. A natural object that captures all the inferable geometric information is the directed complexes (a.k.a. semi-simplicial sets). It turns out that the embedding dimension as well as some other geometric properties of data can be estimated from the homology of an associated directed complex. Moreover each such directed complex gives rise to a multi-parameter filtration that provides a dual topological description of the underlying space. I will also illustrate these methods in the neuroscience context of understanding the "olfactory space".

Date: September 28, 2018
Speaker: Carina Curto, The Pennsylvania State University
Title: Graph rules for inhibitory network dynamics
Abstract: Many networks in the nervous system possess an abundance of inhibition, which serves to shape and stabilize neural dynamics. The neurons in such networks exhibit intricate patterns of connectivity, whose structure controls the allowed patterns of neural activity. In this work, we examine inhibitory threshold-linear networks whose dynamics are dictated by an underlying directed graph. We develop a set of parameter-independent graph rules that enable us to predict features of the dynamics from properties of the graph. These rules provide a direct link between the structure and function of these networks, and provides new insights into how connectivity may shape dynamics in real neural circuits.

Date: December 1, 2017
Speaker: Harbir Antil, George Mason University
Title: Fractional Operators with Inhomogeneous Boundary Conditions: Analysis, Control, and Discretization
Abstract: In this talk we introduce new characterizations of spectral fractional Laplacian to incorporate non homogeneous Dirichlet and Neumann boundary conditions. The classical cases with homogeneous boundary conditions arise as a special case. We apply our definition to fractional elliptic equations of order $s \in (0,1)$ with nonzero Dirichlet and Neumann boundary conditions. Here the domain $\Omega$ is assumed to be a bounded, quasi-convex.

To impose the nonzero boundary conditions, we construct fractional harmonic extensions of the boundary data. It is shown that solving for the fractional harmonic extension is equivalent to solving for the standard harmonic extension in the very-weak form. The latter result is of independent interest as well. The remaining fractional elliptic problem (with homogeneous boundary data) can be realized using the existing techniques. We introduce finite element discretizations and derive discretization error estimates in natural norms, which are confirmed by the numerical experiments. We also apply our characterizations to Dirichlet and Neumann boundary optimal control problems with fractional elliptic equation as constraints.

Date: November 3, 2017
Speaker: Marcio Gameiro, University of Sao Paulo at Sao Carlos, Brazil
Title: Rigorous Multi-parameter Continuation of Solutions of Differential Equations
Abstract: We present a rigorous multi-parameter continuation method to compute solutions of differential equations depending on parameters. The method combines classical numerical methods, analytic estimates and the uniform contraction principle to prove the existence of solutions of nonlinear differential equations. The method is applied to the computation of equilibria for the Cahn-Hilliard equation and periodic solutions of the Kuramoto-Sivashinsky equation.

Date: September 22, 2017
Speaker: Qi Wang, University of South Carolina
Title: Energy quadratization strategy for numerical approximations of nonequilibrium models
Abstract: There are three fundamental laws in equilibrium thermodynamics. But, what are the laws in nonequilibrium thermodynamics that guides the development of theories/models to describe nonequilibrium phenomena? Continued efforts have been invested in the past on developing a general framework for nonequilibrium thermodynamic models, which include Onsager's maximum entropy theory, Prigogine's minimum entropy production rate theory, Poisson bracket formulation of Beris and Edwards, as well as the GENERIC formalism promoted by Ottinger and Grmela. To some extent, they are equivalent and all give practical means to develop nonequilibrium dynamic models. In this talk, I will focus on the Onsager approach, termed the Generalized Onsager Principle (GOP). I will review how one can derive thermodynamic and generalized hydrodynamic models using the generalized Onsager principle coupled with the variational principle. Then, I will discuss how we can exploit the mathematical structure of the models derived using GOP to design structure and property preserving numerical approximations to the governing system of partial differential equations. Since the approach is valid near equilibrium as pointed it out by Onsager, an energy quadratization strategy is proposed to arrive linear numerical schemes. This approach is so general that in principle we can use it to any nonequilibrium model so long as it has the desired variational and dissipative structure. Some numerical examples will be given to illustrate the usefulness of this approach.

Date: April 20, 2017
Speaker: Michael Neilan, University of Pittsburgh
Title: Discrete theories for elliptic problems in non--divergence form
Abstract: In this talk, two discrete theories for elliptic problems in non-divergence form are presented. The first, which is applicable to problems with continuous coefficients and is motivated by the strong solution concept, is based on discrete Calderon-Zygmund-type estimates. The second theory relies on discrete Miranda-Talenti estimates for elliptic problems with discontinuous coefficients satisfying the Cordes condition. Both theories lead to simple, efficient, and convergent finite element methods. We provide numerical experiments which confirm the theoretical results, and we discuss possible extensions to fully nonlinear second order PDEs.

Date: March 3, 2017
Speaker: Ridgway Scott, University of Chicago
Title: Electron correlation in van der Waals interactions
Abstract: We examine a technique of Slater and Kirkwood which provides an exact resolution of the asymptotic behavior of the van der Waals attraction between two hydrogens atoms. We modify their technique to make the problem more tractable analytically and more easily solvable by numerical methods. Moreover, we prove rigorously that this approach provides an exact solution for the asymptotic electron correlation. The proof makes use of recent results that utilize the Feshbach-Schur perturbation technique. We provide visual representations of the asymptotic electron correlation (entanglement) based on the use of Laguerre approximations.We also describe an a computational approach using the Feshbach-Schur perturbation and tensor-contraction techniques that make a standard finite difference approach tractable.

Date: April 22, 2016
Speaker: Guillaume Bal, Columbia University
Title: Boundary control in transport and diffusion equations
Abstract: Consider a prescribed solution to a diffusion equation in a small domain embedded in a larger one. Can one (approximately) control such a solution from the boundary of the larger domain? The answer is positive and this form of Runge approximation is a corollary of the unique continuation property (UCP) that holds for such equations. Now consider a (phase space, kinetic) transport equation, which models a large class of scattering phenomena, and whose vanishing mean free path limit is the above diffusion model. This talk will present positive as well as negative results on the control of transport solutions from the boundary. In particular, we will show that internal transport solutions can indeed be controlled from the boundary of a larger domain under sufficient convexity conditions. Such results are not based on a UCP. In fact, UCP does not hold for any positive mean free path even though it does apply in the (diffusion) limit of vanishing mean free path. These controls find applications in inverse problems that model a large class of coupled-physics medical imaging modalities. The stability of the reconstructions is enhanced when the answer to the control problem is positive.

Date: April 8, 2016
Speaker: John Sylvester, University of Washington
Title: Evanescence, Translation, and Uncertainty Principles in the Inverse Source Problem
Abstract: The inverse source problem for the Helmholtz equation (time harmonic wave equation) seeks to recover information about a radiating source from remote observations of a monochromatic (single frequency) radiated wave measured far from the source (the far field). The two properties of far fields that we use to deduce information about shape and location of sources depend on the physical phenomenon of evanescence, which limits imaging resolution to the size of a wavelength, and the formula for calculating how a far field changes when the source is translated. We will show how adaptations of "uncertainty principles", as described by Donoho and Stark [1] provide a very useful and simple tool for this kind of analysis.

Date:March 24, 2016
Speaker: Qi Wang , Interdisciplinary Mathematics Institute and NanoCenter at University of South Carolina
Title: Onsager principle, generalized hydrodynamic theories and energy stable numerical schemes
Abstract: In this talk, I will discuss the Onsager principle for nonequilibrium thermodynamics and present the generalized Onsager principle for deriving generalized hydrodynamic theories for complex fluids and active matter. For closed matter systems, the generalized Onsager principle combines variational principle with the dissipative property of the system to give a hydrodynamic system that dissipates the total energy. I will illustrate the idea using a few examples in complex fluids. For the hydrodynamic system of equations derived from the generalized Onsager principle, dissipation property preserving numerical schemes can be devised , known as energy stable schemes. These schemes are unconditional stable in time. Several applications of generalized hydrodynamic theories to active matter systems, like cell migration on solid substrates and cytokinesis of animal cells will be presented.

Date: February 26, 2016
Speaker: Andrea Bonito, Texas A&M University
Title: Bilayer Plates: From Model Reduction to Gamma-Convergent Finite Element Approximation
Abstract: The bending of bilayer plates is a mechanism which allows for large deformations via small externally induced lattice mismatches of the underlying materials. Its mathematical modeling consists of a geometric nonlinear fourth order problem with a nonlinear pointwise isometry constraint and where the lattice mismatches act as a spontaneous curvature. A gradient flow is proposed to decrease the system energy and is coupled with finite element approximations of the plate deformations based on Kirchhoff quadrilaterals. In this talk, we give a general overview on the model reduction procedure, discuss to the convergence of the iterative algorithm towards stationary configurations and the Gamma-convergence of their finite element approximations. We also explore the performances of the numerical algorithm as well as the reduced model capabilities via several insightful numerical experiments involving large (geometrically nonlinear) deformations. Finally, we briefly discuss applications to drug delivery, which requires replacing the gradient flow relaxation by a physical flow.

Date: February 26, 2016
Speaker: Lou Kondic, New Jersey Institute of Technology
Title: Force networks in particulate-based systems: persistence, percolation, and universality
Abstract: Force networks are mesoscale structures that form spontaneously as particulate-based systems (such as granulars, emulsions, colloids, foams) are exposed to shear, compression, or impact. The presentation will focus on few different but closely related questions involving properties of these networks:
(i) Are the networks universal, with their properties independent of those of the underlying particles?
(ii) What are percolation properties of these networks, and can we use the tools of percolation theory to explain their features?
(iii) How to use topological tools, and in particular persistence approach to quantify the properties of these networks?
The presentation will focus on the results of molecular dynamics/discrete element simulations to discuss these questions and (currently known) answers, but I will also comment and discuss how to relate and apply these results to physical experiments.