January 14 - 18, 2013
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.
Keywords of the presentation: wave propagation imaging random media
We analyze long range wave propagation in three-dimensional random
waveguides. The waves are trapped by top and bottom boundaries, but
the medium is unbounded in the two remaining directions. We consider
scalar waves, and motivated by applications in underwater acoustics,
we take a pressure release boundary condition at the top surface and
a rigid bottom boundary. The wave speed in the waveguide is known
and smooth, but the top boundary has small random fluctuations that
cause significant cumulative scattering of the waves over long
distances of propagation. To quantify the scattering effects, we
study the evolution of the random amplitudes of the waveguide modes.
We obtain that in the long range limit they satisfy a system of
paraxial equations driven by a Brownian field. We use this
system to estimate three important mode-dependent scales: the
scattering mean free path, the cross-range decoherence length and
the decoherence frequency. Understanding these scales is important
in imaging and communication problems, because they encode
the cumulative scattering effects in the wave field measured by
remote sensors. As an application of the theory, we analyze time
reversal and coherent interferometric imaging in strong cumulative
scattering regimes.
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.
Large Deviations and Variational Representations for Infinite Dimensional Systems: IIJanuary 16, 2013
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- Lecture Slides(pdf)
Keywords of the presentation: large deviations, SPDEs, infinite dimensional Brownian motion, Poisson random measures
In this talk we consider Stochastic dynamical systems with jumps. Large deviation results for finite dimensional stochastic differential equations with a Poisson
noise term have been studied by several authors, however for infinite dimensional models
with jumps, very little is available. The goal of this work is to develop a systematic approach for the study of large deviation properties of such infinite dimensional systems.
Our starting point is a variational representation for exponential functionals of general Poisson random measures and cylindrical Brownian motions. The representation is then used to give a general sufficient condition for a large deviation principle to hold for systems that have both Brownian and Poisson noise terms. Finally we give examples to illustrate the approach.
We prove path-wise uniqueness for an abstract stochastic reaction-diffusion
equation in Banach spaces. The drift contains a bounded H"{o}lder term; in
spite of this, due to the space-time white noise it is possible to prove
path-wise uniqueness. The proof is based on a detailed analysis of the
associated Kolmogorov equation. The model includes examples not covered by the
previous works based on Hilbert spaces or concrete SPDEs. Joint work with G. Da Prato and F. Flandoli.
We consider large scale behavior of the solution set of values u(t,x) for x in the d-dimensional integer lattice of the parabolic Anderson equation. We establish that the properly normalized sums of the u(t,x), over spatially growing boxes have an asymptotically normal distribution if the box grows sufficiently quickly with t and provided intermittency holds. The asymptotic distribution of properly normalized sums over spatially growing disjoint boxes is asymptotically independent. Thus, on sufficiently large scales the field of solutions averaged over disjoint large boxes looks like an iid Gaussian field. We identify the variance of this Gaussian distribution in terms of the eigenfunction of the positive eigenvalue of a certain operator.
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.
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]
Large Deviations and Variational Representations for Infinite Dimensional Systems: IJanuary 15, 2013
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- Lecture Slides(pdf)
Keywords of the presentation: large deviation, variational representation, SPDEs, infinite dimensional Brownian motion
In this talk and a sequel given by Amarjit Budhiraja we will discuss how variational representations can be used to develop an efficient methodology for large deviations analysis, especially in the infinite dimensional setting. This talk will start by reviewing the use of representations in a simple setting. We then discuss the proof of representations for infinite dimensional Brownian motion, and an application to an estimation problem in image matching. If time permits we will discuss some of the new features present when there is Poisson noise, a topic that will be completed in the sequel.
Keywords of the presentation: Stochastic perturbations, multiscale asymptotics, metastability, averaging
I will consider long time influence of small deterministic and stochastic perturbations of various dynamical systems
and stochastic processes. The long time evolution of the perturbed system can be described by a motion in the cone
of invariant measures of the non-perturbed system. The set of extreme points of the cone can be often parametrized
by a graph or by an "open book". The slow component of the perturbed system, is a process on this object.
I will demonstrate how this general approach works in the case of perturbations of systems with several asymptotically stable attractors, perturbations of an oscillator, of elastic systems, of the Landau-Lifshitz equation and its generalization. The same approach works also when PDEs with a small parameter are considered: the Neumann problem with a small parameter for second order elliptic PDEs will be considered.
We discuss recent results on the inviscid limit for invariant measures of the stochastic
Navier-Stokes equations and related systems.
A Theory of Regularity StructuresJanuary 16, 2013
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- Lecture Slides(pdf)
The classical way of measuring the regularity of a function is by comparing it
in the neighbourhood of any point with a polynomial of sufficiently high degree.
Would it be possible to replace monomials by functions with less regular behaviour
or even by distributions? It turns out that the answer to this question has
surprisingly far-reaching consequences for building solution theories for semilinear
PDEs with very rough input signals, revisiting the age-old problem of multiplying
distributions of negative order, and understanding renormalisation theory.
As an application, we build the natural "Langevin equation" associated
with Phi^4 Euclidean quantum field theory in dimension 3.
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.
Keywords of the presentation: Nonlinear noise excitation, stochastic PDEs
We present a part of an ongoing effort that attempts to understand
why solutions to many stochastic PDEs are intermittent. In particular,
we show that there is a strong sense in which large families of SPDEs
with intermittent solutions are extremely "excitable." More significantly,
we show that this highly nonlinear level of noise excitation is, in a sense
dichotomous: "Semidiscrete" equations are nearly always far
less excitable than "continuous" equations. The reason for this dichotomy is also
identified, and somewhat surprisingly has to do with the structure theory
of certain topological groups. This is based on on-going work with Kunwoo Kim.
Keywords of the presentation: Bayesian inference, inverse problems, polynomial approximation, cross-entropy method, dimension reduction
The interplay of experimental observations with mathematical models often requires conditioning models on data---for example, inferring the coefficients or boundary conditions of partial differential equations from noisy functionals of the solution field. The Bayesian approach to these problems in principle requires posterior sampling in high or infinite-dimensional parameter spaces, where generating each sample requires the numerical solution of a deterministic PDE.
We present two developments designed to reduce the significant computational costs of the Bayesian approach. First, we consider local polynomial approximations or "surrogates" for the parameter-to observable map. While surrogates can substantially accelerate Bayesian inference in inverse problems, the construction of globally accurate surrogates for complex models can be prohibitive and in a sense unnecessary, as the posterior distribution may concentrate on a small fraction of the prior support. We present a new approach that uses stochastic optimization to construct polynomial approximations over a sequence of measures adaptively determined from the data, eventually concentrating on the posterior distribution. Second, while the posterior distribution may appear high-dimensional, the intrinsic dimensionality of the inference problem is affected by prior information, limited data, and the smoothing properties of the forward operator. Often only a few directions are needed to capture the change from prior to posterior. We describe a method for identifying these directions through the solution of a generalized eigenvalue problem, and extend it to nonlinear problems where the Hessian of the log-likelihood varies over parameter space. Identifying these directions leads to more efficient Rao-Blackwellized posterior sampling schemes.
Joint work with Jinglai Li, James Martin, Tiangang Cui, and Tarek Moselhy.
Keywords of the presentation: stochastic D Navier Stokes equations, strong speed of convergence, splitting method
We will present some convergence results for the stochastic 2D Navier Stokes equations. The talk will focus on the strong speed of convergence of a splitting method; this is a joint work with H. Bessaih and Z. Brzezniak.
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.
Keywords of the presentation: Brownian sheet, multiple points
An old but still unsolved problem was to show that in the critical dimension, the Brownian sheet does not have multiple points. Recent work of Dalang, Khoshnevisan, and Xiao established that for a broad class of multiple points, the assertion is true. This class included many, but not all possible multiple points. In particular, the time parameters had a particular order relation. In this work, joint with Robert Dalang, we deal with the general situation by showing that there are no multiple points for the Brownian sheet in the critical case.
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.
I will talk about solutions to an elliptic PDE with conductivity coefficient that varies randomly with respect to the spatial variable. It has been known for some time that homogenization may occur when the coefficients are scaled suitably. Less is known about fluctuations of the solution around its mean behavior. For example, if an electric potential is imposed at the boundary, some current will flow through the material. What is the net current? For a finite random sample of the material, this quantity is random. In the limit of large sample size it converges to a deterministic constant. I will describe a central limit theorem: the probability law of the energy dissipation rate is very close to that of a normal random variable having the same mean and variance, when the domain is large. I'll give an error estimate for this approximation in total variation; the estimate scales optimally with the domain size.
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.
Keywords of the presentation: Stochastic Partial Differential Equations, Queueing theory, asymptotic coupling
Finite-dimensional diffusions have been successfully used as tractable approximations to gain
insight into a large class of queueing systems. We show that certain classes of queueing
systems lead naturally to infinite-dimensional diffusion approximations.
For the particular model of many-server queues, we show that the limit state process
converges to the solution of a stochastic partial differential equation subject to an unusual boundary
condition that involves a coupled Ito equation. We also establish uniqueness of the stationary distribution of the pair
of processes, and describe its implications for the original queueing system. This talk is based on
joint works with Mohammadreza Aghajani and Haya Kaspi.
Keywords of the presentation: Fokker-Planck-Kolmogorov equations, continuity equations, measure-valued solutions, infinite dimensional state spaces, well-posedness
We present a new uniqueness result for solutions to Fokker–Planck–Kolmogorov (FPK) equations for probability measures on infinite-dimensional spaces. We consider infinite-dimensional drifts that admit certain finite dimensional approximations. In contrast to most of the previous work on FPK-equations in infinite dimensions, we include cases with non-constant coefficients in the second order part and also include degenerate cases where these can even be zero, i.e. we prove uniqueness of solutions to continuity equations. Also new existence results are proved. Applications to proving well-posedness of Fokker–Planck–Kolmogorov equations associated with SPDEs and of continuity equations associated with PDE are discussed.
Joint work with Vladimir Bogachev, Giuseppe Da Prato and Stanislav Shaposhnikov
In this lecture I will present a general overview as well as recent results
about stochastic viscosity solutions for nonlinear degenerate
parabolic stochastic partial differential equations
with multiplicative noise dependence. Special case of the theory are
stochastic Hamiton-Jacobi equations and
conservation laws.
Keywords of the presentation: Filtering, Navier-Stokes Equation, SPDE
The 3DVAR filter is prototypical of methods used to combine observed data with a dynamical system, online, in order to improve estimation of the state of the system. Such methods are used for high dimensional data assimilation problems, such as those arising in weather forecasting. To gain understanding of filters in applications such as these, it is hence of interest to study their behaviour when applied to infinite dimensional dynamical systems. This motivates study of the problem of accuracy and stability of 3DVAR filters for the Navier-Stokes equation.
We work in the limit of high frequency observations and derive continuous time filters. This leads to a stochastic partial differential equation (SPDE) for state estimation, in the form of a damped-driven Navier-Stokes equation, with mean-reversion to the signal, and spatially-correlated time-white noise. Both forward and pullback accuracy and stability results are proved for this SPDE, showing in particular that when enough low Fourier modes are observed, and when the model uncertainty is larger than the data uncertainty in these modes (variance inflation), then the filter can lock on to a small neighbourhood of the true signal, recovering from order one initial error, if the error in the observations modes is small. Numerical examples are given to illustrate the theory.
Joint work with D. Bloemker, K Law and K Zygalakis.
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Numerical Approximation of PDEs with Stochastic CoefficientsJanuary 15, 2013
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- Lecture Slides(pdf)
Partial Differential Equations with stochastic coefficients are a suitable tool to describe systems whose parameters are not completely determined, either because of measurement errors or intrinsic lack of knowledge on the system.
In the case of linear elliptic PDEs with random inputs, an effective strategy to approximate the state variables and their statistical moments is to use polynomial based approximations like Stochastic Galerkin or Stochastic Collocation method. These approximations exploit the high regularity of the state variables with respect to the input random parameters and for a moderate number of input parameters, are remarkably more effective than classical sampling methods. However, the performance of polynomial approximations deteriorates as the number of input random variables increases, an effect known as the curse of dimensionality. To address this issue, we propose strategies to construct optimal polynomial spaces and and related generalized sparse grids.
In the second part of this talk we propose and analyze a Stochastic Collocation method for solving the second order wave equation with a random wave speed. The speed is piecewise smooth in the physical space and depends on a finite number of random variables. We consider both full and sparse tensor product spaces of orthogonal polynomials, providing a convergence analysis. 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 input random variables. Therefore, the rate of convergence may only be algebraic. Faster convergence rates are still possible for some quantities of interest and for the wave solution with particular types of data.
We will finally discuss the use and analysis of discrete least-squares projection on a polynomial space starting from random, noise-free observations. In particular, we are interested in their application within Uncertainty Quantification for computational models.
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.
Random Matrices and Annihilating Brownian MotionsJanuary 17, 2013
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- Slides(application/pdf)
Keywords of the presentation: Ginibre ensemble, annihilating Brownian motions, Pfaffian point processes
Consider the system of annihilating Brownian motions (ABM's) on the real line
under the maximal entrance law. It turns out that the law of particles' positions at a given time is a Pfaffian point process equivalent to the law of real eigenvalues for the real Ginibre ensemble. Moreover, multi-time intensities for the system of ABM's are an extended Pfaffian point process. Is there a characterisation of the evolution of real eigenvalues in the real Ginibre ensemble in terms of a simple interacting particle system?
Joint work with Roger Tribe
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 1