Applied and Computational Math Seminar

Scheduled Talks (Fall 2017): Room 423, 12:00 - 1:00pm

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.

Past Talks:

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.