Georgia Institute of Technology College of Sciences

Multiscale numerical methods seek to compute approximate solutions to physical problems at a reduced computational cost compared to direct numerical simulations. This talk will cover two methods which have a fine scale atomistic model that couples to a coarse scale continuum approximation. The quasicontinuum method directly couples a continuum approximation to an atomistic model to create a coherent model for computing deformed configurations of crystalline lattices at zero temperature. The details of the interface between these two models greatly affects the model properties, and we will discuss the interface consistency, material stability, and error for energy-based and force-based quasicontinuum variants along with the implications for algorithm selection. In the case of crystalline lattices at zero temperature, the constitutive law between stress and strain is computed using the Cauchy-Born rule (the lattice deformation is locally linear and equal to the gradient). For the case of complex fluids, computing the stress-strain relation using a molecular model is more challenging since imposing a strain requires forcing the fluid out of equilibrium, the subject of nonequilibrium molecular dynamics. I will describe the derivation of a stochastic model for the simulation of a molecular system at a given strain rate and temperature.

It is well known that typically equations do not have analytic (expressed by formulas) solutions. Therefore a classical approach to the analysis of dynamical systems (from abstract areas of Math, e.g. the Number theory to Applied Math.) is to study their asymptotic (when an independent variable, "time", tends to infinity) behavior. Recently, quite surprisingly, it was demonstrated a possibility to study rigorously (at least some) interesting finite time properties of dynamical systems. Most of already obtained results are surprising, although rigorously proven. Possible PhD topics range from understanding these (already proven!) surprises and finding (and proving) new ones to numerical investigation of some systems/models in various areas of Math and applications, notably for dynamical analysis of dynamical networks. I'll present some visual examples, formulate some results and explain them (when I know how).

We offer several perspectives on the behavior at infinity of solutions of discrete Schroedinger equations. First we study pairs of discrete Schroedinger equations whose potential functions differ by a quantity that can be considered small in a suitable sense as the index n \rightarrow \infty. With simple assumptions on the growth rate of the solutions of the original system, we show that the perturbed system has a fundamental set of solutions with the same behavior at infinity, employing a variation-of-constants scheme to produce a convergent iteration for the solutions of the second equation in terms of those of the original one. We use the relations between the solution sets to derive exponential dichotomy of solutions and elucidate the structure of transfer matrices. Later, we present a sharp discrete analogue of the Liouville-Green (WKB) transformation, making it possible to derive exponential behavior at infinity of a single difference equation, by explicitly constructing a comparison equation to which our perturbation results apply. In addition, we point out an exact relationship connecting the diagonal part of the Green matrix to the asymptotic behavior of solutions. With both of these tools it is possible to identify an Agmon metric, in terms of which, in some situations, any decreasing solution must decrease exponentially.This talk is based on joint work with Evans Harrell.

In this talk we will describe the different behaviors of solutions of the random Schrödinger with long-range correlations. While in the case of arandom potential with rapidly decaying correlations nontrivial phenomenaappear on the same scale, different phenomena appear on different scalesfor a random potential with slowly decaying correlations nontrivial .

I'll show how on metric measure spaces with Ricci curvature bounded from below in the sense of Lott-Sturm-Villani there is a well defined notion of Heat flow, and how the study of the properties of this flow leads to interesting geometric and analytic properties of the spaces themselves. A particular attention will be given to the class of spaces where the Heat flow is linear. (From a collaboration with Ambrosio and Savare')