# Poster Session

Tuesday, January 15, 2013 - 4:05pm - 6:00pm

Lind 400

**Numerical Methods and Analysis via Random Field Based Malliavin Calculus for Backward Stochastic PDEs**

Wanyang Dai (Nanjing University)

We design numerical methods to compute the adapted solution of a unified backward stochastic partial differential equation (B-SPDE). In the equation, both drift and diffusion coefficients may involve nonlinear and high-order partial differential operators. Under certain generalized Lipschitz and linear growth conditions, the existence and uniqueness of adapted solution to the B-SPDE are guaranteed. The analysis concerning error estimation and convergence of the methods is conducted by developing theory for random field based Malliavin calculus and related Malliavin derivative based B-SPDEs under random environment.**The L^p Exact Limits of Global Strong Solutions of**

Some n-Dimensional Nonlinear Systems of Fluid Dynamics Equations

Linghai Zhang (Lehigh University)

We study the L^p exact limits of the global strong solutions of the Cauchy problems for

n-dimensional nonlinear systems of fluid dynamics equations, for all 1The main technical ingredients in the rigorous mathematical analysis include the Fourier transformation,

linear analysis and energy estimates.

The exact limits obtained in this work are stronger and broader than the results established in

many previous papers, though the ideas given in this paper are not very complicated.**Center manifolds for stochastic evolution equations**

Jinqiao Duan (Institute for Pure and Applied Mathematics (IPAM))

Stochastic invariant manifolds are crucial in modelling the dynamical behavior of dynamical systems under uncertainty.

Under the assumption of exponential trichotomy, existence and smoothness of center manifolds for a class of stochastic evolution equations with linearly multiplicative noise are proved.

The exponential attraction and approximation to center manifolds are also discussed.

This is a joint work with Xiaopeng Chen and Anthony J. Roberts. The manuscript for this poster is downloadable at: arXiv:1210.5924 [math.DS]**Stochastic wave equation: Analysis and Computation**

Mohammad Motamed (King Abdullah University of Science & Technology)

We propose a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions. The speed is assumed to depend on the physical space and a finite number of random variables. The numerical scheme consists of a finite difference or finite element method in the physical space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space.

We provide a rigorous convergence analysis and demonstrate different types of convergence of the probability error with respect to the number of collocation points under some regularity assumptions on the data. In particular, we show that, unlike in elliptic and parabolic problems, the solution to hyperbolic problems is not in general analytic with respect to the random variables. Therefore, the rate of convergence is only algebraic. A fast spectral rate of convergence is still possible for some quantities of interest and for the wave solution with particular types of data. We show that the collocation method is a valid alternative to the more traditional Monte Carlo method for this class of problems.**An Application of a Hybrid Monte Carlo Method in Path Space**

Frank Pinski (University of Cincinnati)

We are investigating collections of atoms as their positions evolve under

Brownian (over-damped Langevin) dynamics. In the cases where a collection

changes its conformation, an energy barrier often exists. These

transitions are rare events when the thermal energy is small compared to

the barrier height. The understanding of such rare events is the goal of

our studies.

We use a Hybrid Monte Carlo (HMC) Method in Path Space to sample

transition paths efficiently in a thermodynamic significant manner. The

relative probability of paths is computed using the continuum limit of the

Onsager-Machlup functional. In implementing the HMC, we introduce

auxiliary variables (velocities) and choose the masses such that all modes

evolve at the same rate. In addition, the method correctly handles the

fractal nature of the Brownian paths.

We illustrate this method by investigating one of the low energy modes in

the 13-atom and 14-atom Lennard-Jones clusters. The 14-atom cluster

consists of one atom sitting on the surface of a

close-packed structure of the others. The mode we investigated corresponds

to this extra atom penetrating the cluster and pushing an atom onto the

surface on the opposite side of the cluster. We also investigated a

similar mode present in the 13-atom cluster.**Optimal Exercise of Real Options: When Should You Sell Your Mansion?**

Anna Amirdjanova (University of California, Berkeley)

In this paper a class of mixed stochastic control/optimal stopping problems arising in the problem of finding the best time to sell an indivisible real asset, owned by a risk averse utility maximizing agent, is considered. The agent has power type utility based on the $ell_{alpha}$-type aggregator and has access to a frictionless financial market which can be used to partially hedge the risk associated with the real asset if correlations between the financial assets and the real asset are nonzero. The solution to the problem of finding the optimal time to sell the real asset is characterized in terms of solution to a certain free boundary problem. Comparisons with the case of exponential utility are also given.**Well-Posedness for Degenerate Parabolic SPDEs**

Martina Hofmanova (École Normale Supérieure de Cachan)

We present a well-posedness result for degenerate parabolic SPDEs,

equations that are widely used in fluid mechanics to model the phenomenon

of convection-diffusion of ideal fluid in porous media. In particular, we

adapt the notion of kinetic solution which is well suited for degenerate

parabolic problems and supplies a good technical framework to prove the

comparison principle. The proof of existence is based on the vanishing

viscosity method: the solution is obtained by a compactness argument as

the limit of solutions of nondegenerate approximations.**Stochastic Ferromagnetism**

Mikhail Neklyudov (Universitaet Tuebingen)

The dynamics of ferromagnetic systems at nanoscale is described by

the stochastic Landau-Lifshitz-Gilbert equation (SPDE). The noise describes thermal effects of the system. Recently the authors constructed

implementable nite element based numerical schemes which converge to

a solution of the corresponding SPDE and were able to estimate the rate

of convergence in the case of nite ferromagnetic ensembles (so-called

Arrhenius law). The poster provides an overview of the analytical and

numerical results of the authors. Further details will be provided in a

forthcoming book.**An Initial and Boundary-Value Problem for the Zakharov-Kuznestov Equation in a Bounded Domain**

Chuntian Wang (Indiana University)

Motivated by the study of boundary control problems for the Zakharov-Kuznetsov equation, we study in this article the initial and boundary value problem for the ZK equation posed in a limited domain (0,1)_{x} times(-pi /2, pi /2)^d, d=1,2. This article is related to the previous paper (An initial boundary-value problem for the

Zakharov-Kuznetsov equation, Saut, Jean-Claude and Temam, Roger) in which the authors studied the same problem in the band (0,1)_{x}timesmathbb R^d, d=1,2, but this article is not a straightforward adaptation indeed many new issues arise, in particular for the function spaces, due to the loss of the Fourier transform in the tangential directions (orthogonal to the x axis).

In this article, after studying a number of suitable function spaces, we show the existence and uniqueness of solutions for the linearized equation using the linear semigroup theory. We then continue with the nonlinear equation with the homogeneous boundary conditions. The case of the

full nonlinear equation with nonhomogeneous boundary conditions especially needed for the control problems will be studied elsewhere.**Stability Issues for Numerical Methods for SDEs**

Evelyn Buckwar (Johannes Kepler Universität Linz)

Stochastic Differential Equations (SDEs) have become a standard modelling tool in many areas of science, e.g., from finance to neuroscience. Many numerical methods have been developed in the last decades and analysed for their strong or weak convergence behaviour.This poster provides an overview on recent progress in the analysis of stability properties of numerical methods for SDEs, in particular for systems of equations. We are interested in developing classes of test equations that allow insight into the stability behaviour of the methods and in developing approaches to analyse the resulting systems of equations.**Inviscid Limits for the Stochastic Navier-Stokes Equations and Related Questions**

Nathan Glatt-Holtz (University of Minnesota, Twin Cities)Vladimir Sverak (University of Minnesota, Twin Cities)Vlad Vicol (Princeton University)

We discuss recent results on the inviscid limit for invariant measures of the stochastic

Navier-Stokes equations and related systems.