**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:** 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.