CAMP/Nonlinear PDEs Seminar

Every wednesday at 4pm in Eckhart 202 (unless otherwise specified).

Fall 2010 Schedule.

September 20th. Special room and time: E206, 4pm
Markos Katsoulakis, University of Massachusetts, Amherst
Accelerated Kinetic Monte Carlo methods: hierarchical parallel algorithms and coarse-graining
In this talk we present two intimately related approaches in speeding-up molecular simulations via Monte Carlo simulations. First, we discuss coarse-graining algorithms for systems with complex, and often competing particle interactions, both in the equilibrium and non-equilibrium settings, which rely on multilevel sampling and communication. Second, we address mathematical, numerical and algorithmic issues arising in the parallelization of spatially distributed Kinetic Monte Carlo simulations, by developing a new hierarchical operator splitting of the underlying high-dimensional generator, as means of decomposing efficiently and systematically the computational load and communication between multiple processors. The common theme in both methods is the desire to identify and decompose the particle system in components that communicate minimally and thus local information can be either described by suitable coarse-variables (coarse-graining), or computed locally on a individual processors within a parallel architecture.
September 29th
Hitoshi Ishii, Waseda University, Japan.
Asymptotic solutions of Hamilton-Jacobi equations
I will discuss the long-time asymptotic behavior of solutions of Hamilton-Jacobi equations mostly in the case where the Hamiltonian of the equation is convex and coercive. I will focus mainly on a special form of solutions, called asymptotic solutions, which describe the long-time behavior of solutions.
October 1st. Eckhart 202. 4pm.
Neil Trudinger, Australian National University.
On regularity of weak solutions of Monge-Ampere type problems
We explain two notions of weak solution, arising from the notion of Monge Ampere measure, introduced by Aleksandrov many years ago. The first is applicable to fourth order PDEs that are the Euler -Lagrange equations of second order functionals, exemplified by the affine area of a graph, and the second is satisfied by potentials in optimal transportation. We will present some regularity results, resulting from joint work with Xu-jia Wang and others.
October 4th, 6th and 8th. There is no CAMP seminar because of the Amick Lectures
Claude Le Bris, CERMICS/ENPC.
Randomness in Multiscale Computational Mechanics and how to deal with it
October 4, 2010 at 4pm

The common denominator of all the contributions presented is random modelling for materials. Models for some complex fluids (typically solutions of flexible polymers) and models for some composite solid materials will be presented. In each category, emphasis will be laid upon the random component of the modelling, how this component is derived from the microscale and formalized, how it affects the overall mathematical nature of the problem giving birth to new questions of interest, how its coupling with the deterministic part of the system is dealt with computationally. The talk will overview works by, and joint works with various coworkers: PL. Lions (College de France), X. Blanc (CEA, Paris), A. Anantharaman, S. Boyaval , R. Costaouec, B. Jourdain, T. Lelievre, F. Legoll, G. Stoltz, F. Thomines (ENPC, Paris).

Molecular dynamics and computational statistical mechanics: some mathematical approaches
October 6, 2010 at 4pm

Molecular dynamics is a commonly used technique to compute ensemble averages, and beyond, thermodynamic quantities. Mathematically, molecular dynamics imply ordinary and stochastic differential equations, possibly multiscale in nature. A selection of various relevant mathematical questions will be addressed: long-time integration of, possibly highly oscillatory, Hamiltonian systems; simulation of constrained stochastic differential equations, long-time integration of Langevin type equations for metastable systems, large deviation theory for coarse-grained models, etc. The talk is based upon a series of works by, and joint works with X. Blanc (Paris 6 and CEA), M. Dobson, F. Legoll, T. Lelievre, G. Stoltz, (ENPC and INRIA).

Numerical approaches for non periodic and random composite materials: some recent progress
October 8, 2010 at 4:30pm

The talk will overview some recent contributions on several theoretical aspects and numerical approaches in stochastic homogenization. In particular, some variants of the theory of classical stochastic homogenization will be introduced. The relation between such homogenization problems and other multiscale problems in materials science will be emphasized. On the numerical front, some approaches will be presented, for acceleration of convergence in stochastic homogenization (variance reduction issues, etc) as well as for approximation of the stochastic problem when the random character is only a perturbation of a deterministic model. The talk is based upon a series of joint works with X. Blanc (CEA, Paris), PL. Lions (College de France, Paris), and A. Anantharaman, R. Costaouec, F. Legoll, F. Thomines (ENPC, Paris).

October 13th
Andreas Hildebrandt, Center for Bioinformatics, Saarland University.
Biomolecules in a structured solvent - a nonlocal electrostatics treatment

The successful development of new drugs is one of science's most difficult tasks today. Even under optimal circumstances, it takes years to finish, has a very low probability of success, and is immensely expensive. In this talk, I want to show how molecular modeling techniques can help to make drug design cheaper, faster, and more reliable. In particular, I will focus on the accurate computation of electrostatic effects which play an important role in the energetics of biomolecules. Many of those effects are dominated by the shielding effect of the water that is always present in biochemical reactions. Therefore, a highly accurate computation of electrostatic potentials of biomolecules in water is an important precursor for many applications in bioinformatics, like the mentioned computer aided development of inhibitors for disease related enzymes.

In the literature, nonlocal extensions of classical macroscopic electrostatics have been proposed to capture the effects of the water on the electric potential. We propose a reformulation of the resulting equations, which we can be addressed numerically.

The method has been shown to yield very accurate results on small systems like mono- or polyatomic ions and initial results on selected proteins are highly promising.

October 14th. Eck 206. 3pm
Pierre Louis Lions, College de France.
Uniqueness and Mean Field Games
October 20th
Pierre Louis Lions, College de France.
Recent results on Mean Field Games
October 27th
Fabio Ramos, Universidade Federal do Rio de Janeiro.
Scaling in Turbulent Flows: Heuristics and Rigorous Results
In this talk I will present some classical heuristics and related recent rigorous results for the Kolmogorov-Obukhov spectral scaling for fluid flows at high Reynolds number. In particular, I will exhibit some universal bounds for the Littlewood-Paley first-order moments for weak solutions of the 3D Navier-Stokes equations, obtained in a recent work with Felix Otto.
November 3rd
Hiroshi Matano, The University of Tokyo.
Homogenization of recurrent traveling waves in a 2D cylinder with saw-toothed boundary

In this talk I will present a result on the homogenization limit of a certain traveling wave in a spatially non-periodic environment. Quite intriguingly, the rate of convergence to the homogenization limit depends on the number of independent frequencies.

To be more precise, we consdier a curvature-dependent motion of plane curves in a two-dimensional cylinder with spatially undulating boundary. In other words, the boundary has many bumps and we assume that the bumps are aligned in a spatially recurrent manner.

We study how the average speed of the traveling wave depends on the geometry of the domain boundary. More specifically, we consider the homogenization problem as the boundary undulation becomes finer and finer, and determine the homogenization limit of the average speed and the limit profile of the traveling wave. Quite surprisingly, this homogenized speed depends only on the maximal opening angles of the domain boundary and no other geometrical features are relevant. Next we consider the special case where the boundary undulation is quasi-periodic with m independent frequencies. We show that the rate of convergence to the homogenization limit depends on this number m.

This is joint work with Bendong Lou and Ken-Ichi Nakamura.

November 10th
Yifeng Yu, University of California. Irvine.
Asymptotics for turbulent flame speeds of the viscous G-equation enhanced by cellular and shear flows.
(Inviscid) G-equations are well-known front propagation models in turbulent combustion. The effective Hamiltonian from its cell problem gives a way to compute the turbulent flame speed sT. To determine sT is a fundamental problem in turbulent combustion theory. According to this model, for the cellular flow, sT increases almost linearly with respect to the turbulent intensity (weak "bending effect"). The viscous G-equation arises from either numerical approximations or regularizations by adding a small diffusion term d∆G. We proved that the corresponding effective Hamiltonian (sT,d) is uniformly bounded as the turbulent intensity increases (strong "bending effect"), i.e, sT,d ≤ C(d) as the turbulent intensity → +∞. So the presence of diffusion dramatically slows down front propagation. If time permits, we will also talk about diffusion effects for shear flows.
November 11th Special room and time: Eck 206, 3:00pm
Charles Fefferman, Princeton University.
The breakdown problem for the 2D Muskat equation
November 17th.
Vlad Vicol, University of Chicago.
On an active scalar equation arising in magnetodynamics.
We present some recent results on the well-posedness theory for an active scalar equation that has been proposed by Keith Mo ffatt as a model for magnetogeostrophic turbulence in the Earth's fluid core. We show that in the presence of critical diffusion are the equations are globally well-posed, while the non-diffusive case the system is ill-posed in Sobolev spaces.
November 24th
No seminar
December 1st
Michael Dabkowski, University of Wisconsin. Madison.
Eventual Regularity of Solutions to the Supercritical Dissipative Suraface Quasi-geostrophic Equation
We will prove that solutions to the supercritical dissipative surface quasi-geotrophic equation eventually become smooth by examining the evolution of the equation on a test class of functions that is dual to Holder $C^{\beta}$ functions. Using induction on scales, we will prove that there is a finite time after which the solution gains a certain degree of Holder continuity, which turns out to be a sufficient condition for smoothness.

For questions, contact Luis Silvestre at email at uchicago

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