advances and challenges in the solution of stochastic partial differential … · 2006-10-02 ·...
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2006 SPDE Workshop on
Advances and Challenges in the Solution of Stochastic Partial
Differential Equation
October 20-22, 2006
Brown University, Rhode Island, USA
Sponsored by AFOSR
Advances and Challenges in the Solution of Stochastic Partial Differential Equations
October 20-22, 2006 Brown University Barus and Holley
Room 190
Program Friday Oct 20
8:30-9:00 Registration/Coffee/Juice 9:00-9:10 Welcome
9:10-10:00 A. Chorin, University of California, Berkeley
Problem Reduction and Memory
10:00-10:30 Coffee Break 10:30-11:20 B. Rozovski, Brown University
A Filtering Approach to Tracking Volatility from Prices Observed at Random Times
11:20-12:10 G. Papanicolau, Stanford University Self-averaging from lateral diversity in the Ito-Schrodinger equation
12:10- 1:30 Lunch break (on your own)
1:30 -2:20 P. Kotelenez, Case Western Reserve University Stochastic Partial Differential Equations in the Transition from Microscopic to Macroscopic Equations
2:20 -3:10 G. Karniadakis, Brown University Multi-Element Polynomial Chaos: Algorithms and Applications
3:10 -3:40 Coffee break 3:40 -4:30 E. Vanden-Ejinden, Courant Institute NYU
Rare Events in Spatially Extended Media 4:30 –5:20 J. Mattingly, Duke University
Degenerately Forced Stochastic PDEs: Ergodicity and the Spread of Randomness
6:30 - 8:30 Reception in Faculty Club
Saturday Oct 21
8:30 -9:00 Registration/Coffee/Juice 9:00- 9:50 B. Øksendal, University of Oslo
Stochastic Partial Differential Equations Driven by Multi-Parameter Levy White Noise
9:50-10:20 Coffee Break 10:20-11:10 H. Kushner, Brown University
Numerical Approximations for Nonlinear Stochastic Systems with Delay
11:10-12:00 A. Stuart, Warwick University, UK Sampling Conditioned Diffusions
12:00- 1:30 Lunch break (on your own) 1:30 - 2:20 J. Tribbia, National Center for Atmospheric Research Stochastic PDEs in Weather and Climate Prediction 2:20 - 3:10 R. Ghanem, University of South Carolina
Multiscale Analysis, Stochastic Analysis, and Model Validation: A Unifying Perspective through Polynomial Chaos Decompositions
3:10 - 3:40 Coffee break
3:40 - 4:30 T. Hou, CalTech Numerical Solutions of Fluid Dynamics Equations with Random Forcing
4:30 - 5:20 S.V. Lototsky, University of Southern California Stochastic Integration with Respect to Gaussian Processes and Fields
Sunday Oct 22 8:30 -9:00 Registration/Coffee/Juice 9:00- 9:50 D. Crisan, Imperial, UK
Particle Methods for a Class of Stochastic Partial Differential Equations
9:50-10:40 J. Glimm, Stony Brook University and Brookhaven National Lab Stochastic Phenomena in the Description of Turbulent Fluid Mixing 10:40-11:10 Coffee break 11:10-12:00 B. Birnir, University of California, Santa Barbara Turbulence of Uniform Flow 12:00-12:50 R. Temam, Indiana University Some Remarks on the Numerical Approximation of Stochastic Differential Equations 12:50- 1:00 Closing comments
Turbulence of Uniform Flow
Bjorn BirnirCenter for Complex and Nonlinear Science
University of California, Santa Barbara
Abstract
The existence of stochastic processes is discussed, describing turbulent solu-tions of the full Navier-Stokes equation, driven by unidirectional flow, in dimen-sions one, two and three. These solutions turn out to have a finite velocity andvelocity gradient but they are not smooth instead the velocity is Holder contin-uous with a Holder exponent depending on the dimension. They scale with theKolmogorov scaling in three dimensions and the Batchelor-Kraichnan scaling intwo dimension. In one dimensions they scale with the exponent 3/4, that is relatedto Hack’s law of river basins; stating that the lenght of the main river, in matureriver basins, scales with the area of the basinl ∼ Ah. h = 0.58 being Hack’s expo-nent. The existence of these turbulent solutions is then used to proof the existenceof an invariant measure in dimensions one, two and three. The invariant measurecharacterizes the statistically stationary state of turbulence and it can be used tocompute the statistically stationary quantities. These include all the deterministicproperties of turbulence and everything that can be computed and measured. Inparticular, the invariant measure determines the probability density of the turbu-lent solutions and this can be used to develop accurate sub-grid models in com-putations of turbulence, bypassing the problem that three-dimensional turbulencecannot be fully resolved with currently existing computer technology.
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Problem reduction and memory
Alexander Chorin
Department of Mathematics University of California, Berkeley
Abstract
I will present methods for the reduction of the complexity of computational problems, both time-dependent and stationary, and their connections to probability, renormalization, scaling, and statistical mechanics, together with examples. The main points, are: (i) in time dependent problems, it is no legitimate to average equations without taking into account memory effects and noise; (ii) mathematical tools developed in physics for carrying out renormalization group transformations yield effective block Monte-Carlo methods; (iii) the Mori-Zwanzig formalism, which in principle yields exact reduction methods but is often hard to use, can be tamed by approximation; and (iv) more generally, problem reduction is a search for hidden similarities. In the examples I will emphasize the "t-model" for problems where the memory (=autocorrelation of the noise) has a large support; this is important in applications to hydrodynamics. In particular, I will show results, due to Bernstein, Hald, and Stinis, of t-model estimates of the rate of decay of the solutions of the Burgers equation and the Euler equations in 2 and 3 dimensions.
Particle methods for a class of stochastic partialDifferential equations1
Dan Crisan2
Imperial, UK
Let (, F, P ) be a probability space on which we have defined an m-dimensional standard Brownian motion W . In turn, W drives the followingsemi-linear stochastic partial differential equation
(1) dt() = t(A)dt + mk = 1(t(k) − t(k)t() + t(Bk))(dWkt − t(k)dt).
Here is a measure valued process, A is a second order differential operator andBk, k = 1, ..., m are first order differential operators. Equation (1)is knownas the Fujisaki-Kallianpur-K unit a or the Kushner-Stratonovitch equation.It plays a central role in nonlinear filtering: The solution of(1)gives the con-ditional distribution of a stochastic process X (the signal)given an associatedobservation process W (the process W becomes a Brownian motion after asuitable change of measure). The aim of the talk is to present an overviewof existing particle methods for approximating the solution of (1),includinga hybrid method (see[1])that merges the weighted approximation approach,aspresented in Kurtzand Xiong [6] for a general class of nonlinear stochasticpartial differential equation to which (1) belongs, with the branching correc-tions approach introduced by Crisan and Lyons in [3]. This recent result is ageneralization of existing results (see for example [2,4,5]) where the differen-tial terms t(Bk), k = 1, ..., m are missing. The addition of these differentialterms in (1) is important. They appear in the case where the signal noiseand the observation noise are correlated (a common feature in financial ap-plications).
1Advances and Challenges in the Solution of Stochastic Partial Differential Equations,Brown University, 20-22 October 2006.
2Department of Mathematics,Imperial College London,180 Queens Gate, London SW7 2AZ,UK
1
References
[1] D. Crisan, Particle approximations for a class of stochastic partial differ-ential equations, to appear in the Applied Mathematics and OptimizationJournal, 2006.
[2] D. Crisan, T. Lyons, A particle approximation of the solution ofthe Kushner-Stratonovitch equation. Probab. Theory Related Fields115,no.4, 549578,1999.
[3] D. Crisan, T. Lyons, Minimal entropy approximations optimal algorithmsfor the Filtering Problem, Monte Carlo Methods and Applications, Vol8,No 4 pp343-356, 2002.
[4] D. Crisan, P. DelMoral, T. Lyons, Interacting particle systems approx-imations of the Kushner-Stratonovitch equation, Adv. in Appl. Probab.31, no.3, 819838, 1999.
[5] P. DelMoral Feynman-Kac formulae. Genealogical and interacting parti-cle systems with applications. Probability and its Applications, Springer-Verlag, New York,2004.
[6] T.G Kurtz, J.Xiong, Particle representations for a class of nonlinearSPDEs. Stochastic Process. Appl.83 (1999), no.1, 103126.
2
Multiscale analysis, stochastic analysis, and model validation: A unifying perspective through polynomial
chaos decompositions
Roger Ghanem
University of Southern California Department of Aerospace and Mechanical Engineering
Los Angeles, CA [email protected]
Abstract
This talk will present some current challenges that stochastic analysis must address as it becomes a corner stone of prediction science. Hilbert space decompositions of stochastic processes will be used as the analysis vehicle of choice to delineate and characterize these challenges. Particular attention will be devoted to the definition of an error budget to be used in the adaptive refinement of prediction instruments. Through a suitable identification of these instruments with operators on product spaces, the adaptive refinement in question can be viewed as defining a constructive path for model validation. Evaluating the error budget entails defining novel statistics that permit the propagation of the effect of data limitations to the stochastic predictions. Novel procedures for estimating these statistics are described and challenges in their implementation are noted. Furthermore, the need for reduced-order analysis and models is reiterated for the accurate and proper application of stochastic analysis to physically realistic problems. Procedures for the construction of such models are described together with related theoretical and computational challenges. Since issues of reduced-order analysis pertain to stochastic multi-scale representations, challenges and on-going research in multiscale stochastic analysis will also be described.
Stochastic Phenomena in the Description of Turbulent Fluid Mixing
James Glimm Dept. of Applied Mathematics & Statistics
Stony Brook University and Brookhaven National Laboratory Stony Brook, NY
Abstract
Classical fluid instabilities give rise to chaotic flow regimes. The acceleration driven instabilities (Rayleigh-Taylor or steady acceleration and Richtmyer-Meshkov, or impulsive acceleration) are studied here. A scientific program for the study of chaotic flow has three components: 1. Simulations and theoretical calculations in agreement with experiments (validation of simulations). 2. Equations for averaged quantities and their fluctuations. 3. Quantification of uncertainty (error bars) for steps 1 and 2. We report progress for the surprisingly difficult step 1. We speculate on some of the numerical difficulties and point out open problems. Step 2, which is rather controversal, appears from our point of view to be less difficult than anticipated. Finally, we propose an approach to step 3, based on stochastic points of view.
Numerical Solutions of Fluid Dynamics Equations with Random Forcing
Thomas Hou Applied Mathematics
California Institute of Technology Pasadena, CA
Abstract
We study a numerical method based on the Wiener Chaos expansion for solving the stochastic Burgers and Navier-Stokes equations driven by Brownian motion. The main advantage of the Wiener Chaos approach is that it allows for separation of random and deterministic effects in a rigorous and effective manner. Based on our error analysis, we design a sparse truncation strategy for the Wiener Chaos expansion, which reduces the dimension of the resulting PDE system substantially while retaining the same asymptotic convergence rate. We demonstrate that the numerical method based on the Wiener Chaos expansion is more efficient and accurate than Monte Carlo simulations for short to moderate time computations. For long time solutions, we propose a new computational strategy where Monte-Carlo simulations are used to correct the unresolved small scales in the sparse Wiener Chaos expansion solutions. Using this improved hybrid method, we can simulate the long time front propagation for a reaction-diffusion equation in random shear flows and confirm that the front speed propagation obeys a quadratic enhancing law.
Multi-Element Polynomial Chaos: Algorithms and Applications
George Em Karniadakis Division of Applied Mathematics
Brown University Providence, RI
Abstract
We will present an overview of generalized polynomial chaos in conjunction with Galerkin and collocation projections. Open issues related to long-time integration, parametric discontinuities, and high-dimensional stochastic inputs will be addressed. Examples from fluid mechanics will be given including a theoretical and numerical study of scattering of shock waves by randomly rough surfaces.
Stochastic Partial Differential Equations in theTransition from Microscopic to Macroscopic
Equations
Peter M. KotelenezDepartment of Mathematics
Case Western Reserve UniversityCleveland, Ohio 44106
email: [email protected]
Abstract
Stochastic partial differential equations (SPDE’s) are treated as a mesoscopic model for the distri-bution of particles and their generalizations. The positions of the particles are solutions of stochasticordinary differential equations (SODE’s) driven by correlated and uncorrelated Brownian motions.SODE’s are a more detailed mesoscopic model for the distributions of solute particles in fluids (consist-ing of solvent particles). The SODE’s can be derived from an infinite system of coupled deterministicoscillators which are, by definition, a microscopic model for both solute and solvent particles. Corre-lated Brownian motions are shown to be consistent with the depletion phenomenon which has beenan active area of research in the empirical sciences for a number of decades.
Empirical measures for large systems of SODE’s are extended by continuity on spaces of Borelmeasures. These extensions are shown to be weak solutions of the associated SPDE’s. Existence,uniqueness and smoothness hold for a large class of quasilinear coercive and non-coercive SPDE’swith mass conservation. For a certain class, the solutions are space-time homogeneous and isotropicrandom fields. A wider class of solvable SPDE’s can be obtained by combining the particle approachwith other methods, employing fractional steps. Numerical schemes for systems of SODE’s can beused to solve the associated SPDE’s numerically. This is a direct generalization of point vortex meth-ods in 2D fluid mechanics to arbitrary space dimensions. If necessary, fractional steps are employedas well.
At the end, we sketch the derivation of macroscopic (quasilinear) PDE’s for space dimension ≥ 2as the limit of SPDE’s if the spatial correlations tend to 0.
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Numerical Approximations for Nonlinear Stochastic Systems With Delays
Harold J. Kushner Brown University
Division of Applied Mathematics Providence, RI
Abstract
We will discuss the extension of the Markov chain approximation numerical methods to controlled nonlinear diffusion models, where the dynamical terms have delays As for the no-delay problem (where the methods are the current standards) the basic idea is the approximation of the control problem by a control problem with a suitable approximating Markov chain model, solve the Bellman equation for the approximation, and then prove the convergence of the costs or optimal costs to that for the original problem. The method is robust, the approximations have physical interpretations as control problems closely related to the original one, and, for the non-delay problem, there are many effective methods for getting the approximations, and for solving the Bellman equation if the dimension is not too large. One particular motivation arises in modern telecommunications, owing to the often significant delays between the controller and distant routers, due to the finite speed of electromagnetic transmission. While the basic idea and proofs are extensions of those for the no-delay case, they are not obvious. We quickly review the no-delay case to set the stage and refresh familiarity with the basic concepts. The fundamental assumption required for the convergence of the numerical procedure is the so-called “local consistency condition,” and this is also true for the delay problem. There are two “extreme” types of construction that are of interest, called the “explicit” and “implicit” methods, owing to the similarity of one particular way of constructing them to methods of the same names for solving parabolic PDE’s. [The proofs are purely probabilistic.] Each, together with the intermediate forms, plays an important role for the delay model. The dynamics depend on a segment of the recent path, and possibly of the control as well. The state of the problem, as needed for the numerics, consists of a segment of the path (over the delay interval) and of the control path as well (if the control is also delayed). This can consume a lot of memory, so we will be particularly concerned with efficient representations.
Stochastic integration with respect to Gaussian processes and fields.Sergey Lototsky, USC
(based on joint work with Karsten Stemmann)
While stochastic integral with respect to a standard Brownian motion is a well-studiedobject, integration with respect to other Gaussian processes is currently an area of activeresearch, and the fractional Brownian motion is receiving most of the attention. The ob-jective of this talk is to define and investigate stochastic integrals with respect to arbitraryGaussian processes and fields using chaos expansion.
It is more convenient to work with a field rather than a process: by definition, a field is aa collection of stochastic integrals for a class of deterministic integrands. More precisely,A generalized Gaussian field X over a Hilbert space H is a continuous linear mappingf 7→ X(f) from H to the space of Gaussian random variables. The corresponding chaosspace HX is the Hilbert space of square integrable random variables that are measurablewith respect to the sigma-algebra generated by X(f), f ∈ H, and the chaos expansionis an orthogonal decomposition of HX. Given an orthonormal basis {ξm, m ≥ 1} in HX,a square integrable H-valued random variable η has a chaos expansion η =
∑m≥1 ηmξm,
with ηm = E(ηξm) ∈ H. Thus, the definition of the stochastic integral X(η) requiresan extension of the linearity property of X to linear combinations with random coefficients.Two “natural” extensions of this property, using the Wick and ordinary products of randomvariables, lead to the Ito-Skorokhod and the Stratonovich stochastic integrals.
Many computations are simplified if X is a white noise over H, that is, a zero-mean generalizeGaussian field such that E
(X(f)X(g)
)= (f, g)H for all f, g ∈ H. It turns out that, for every
zero-mean Gaussian field X over H, there exists a different (usually larger) Hilbert spaceH′ such that X is a white noise over H′. Moreover, the space H′ is uniquely determined byX. On the other hand, every zero-mean Gaussian field X over H can be written in the formX(f) = B(K∗f), f ∈ H, where K∗ is a bounded linear operator on H and B is a white noiseover H, although this white noise representation of X is not necessarily unique. Differentwhite noise representations of X lead to different but equivalent formulas for computingX(f), and chaos expansion is an efficient way for deriving those formulas. In particular, forboth deterministic and random f , chaos expansion provides an explicit formula for X(f) interms of the Fourier coefficients of the integrand f .
To define stochastic integral with respect to a Gaussian process X = X(t), t ∈ [0, T ], weconstruct a Hilbert space HX and a white noise B over HX such that X(t) = B(χt), whereχt is the characteristic function of the interval [0, t]. The space HX is uniquely determinedby X; for example, the Wiener process on (0, T ) has HX = L2((0, T )). Then the equality∫ T0 f(s)dX(s) = B(f), f ∈ HB, is a canonical definition of the stochastic integral with
respect to X.
Here are the main questions that will be discussed:
(1) Possible white noise representations of a zero-mean generalized Gaussian field.(2) A connection between generalized Gaussian fields over L2((0, T )) and processes that
are representable in the form∫ t0 K(t, s)dW (s).
(3) Chaos expansions of the Ito-Skorokhod and Stratonovich integrals.(4) Examples of the corresponding stochastic equations.(5) Open problems.
Degenerately forced stochastic PDEs:
Ergodicity and the spread of randomness
Jonathan Mattingly Department of Mathematics
Duke University Durham, NC
Abstract
If one forces a stochastic PDE with a finite number of Brownian motions how does the injected randomness move to other degrees of freedom. I will mainly discuss the two dimensional Navier-Stokes equation but much of the results apply to other dissipative SPDEs such as reaction diffusion equations. In the setting of the 2D Navier-Stokes equation, such understating is needed to analysis the setting when the system is forced at one scale and the randomness is allowed to cascade to other scales. To answer this question, I will describe a version of Hörmander's "sum of squares theorem" which is applicable in this infinite dimensional setting. In particular, it will allow us to conclude that any finite dimensional projection of the transition kernel has a smooth density with respect to Lebesgue measure. I will then turn to the question of ergodicity and used the above understanding to prove that the 2D Navier-Stokes with sufficient forcing has a unique invariant measure. We will infact prove that the system has a spectral gap in the Wassertein distance. To prove ergodicity, we introduce a modification of the classical Strong Feller property of a Markov Chain, called the asymptotic strong Feller property.
Stochastic partial differential equations driven by multi-parameter Lévy white noise
Bernt Øksendal Department of Mathematics
University of Oslo Oslo, Norway
Abstract
We construct a white noise theory for multi-parameter Lévy processes and for compensated Poisson random measures. The white noise of these processes and measures can be shown to exist in the space (S)* of stochastic distributions. This is used to solve linear (and certain non-linear) SPDE´s driven by multi-parameter Lévy processes. The procedure is as follows: STEP 1: SPDE´s involving such (additive or multiplicative) Lévy process noise terms can be interpreted as (S)* - valued PDE´s. If the equation has a multiplicative noise term, this multiplication corresponds to the (Lévy-) Wick product in (S)*. STEP 2: By (a Lévy version of) the Hermite transform, these (S)*-valued equations can in turn be transformed into complex-valued PDE´s involving ordinary complex multiplication and complex number parameters z. STEP 3: For each value of z we solve this PDE using classical solution methods. STEP 4: Then we use the inverse Hermite transform to transfer the solution of this PDE back to the solution of the original SPDE. We illustrate this method by solving the following SPDE´s: (i) The Dirichlet problem (Poisson equation) with a multi-parameter Lévy noise force term (ii) The heat equation with a multi-parameter Lévy noise heat source (iii) The Cauchy problem for the wave equation with Lévy noise initial data. In general the solutions of these equations will be situated in the space (S)* of stochastic distributions, but in some cases (depending on the dimension) the solution can be shown to be a classical multiparameter process (also called a random field) in L^2.
Self-averaging from lateral diversity
in the Ito-Schrodinger equation
George Papanicolaou
Mathematics Department Stanford University
Stanford, CA [email protected]
Abstract
We consider the random Schrödinger equation as it arises in the paraxial regime for wave propagation in random media. In the white noise limit it becomes the Itô -Schrödinger stochastic partial differential equation which we analyze here in the high frequency regime. We also consider the large lateral diversity limit where the typical width of the propagating beam is large compared to the correlation length of the random medium. We use the Wigner transform of the wave field and show that it becomes deterministic in the large diversity limit when integrated against test functions. This is the self-averaging property of the Wigner transform. We also show how these asymptotic results can be used in broadband array imaging in random media.
A filtering approach to tracking volatility from prices observed at random times
Boris Rozovski
Division of Applied Mathematics Brown University
Providence, RI [email protected]
Abstract
This presentation is concerned with nonlinear filtering of the coefficients in asset price models with stochastic volatility. More specifically, it is assumed that the asset price process S = (St)t≥0 is given by
dSt = r(θt)Stdt +_v(θt)StdBt,
where B = (Bt)t≥0 is a Brownian motion, v is a positive function, and the volatility θ = (θt)t≥0 is a a Markov process. The random process θ is unobservable. It is also assumed that the asset price St is observed only discretely, at random times 0 < τ1 < τ2 < . . .. This is an appropriate assumption when modeling high frequency financial data (e.g., tick-by-tick stock prices). In the above setting the problem of estimation of θ can be approached as a special nonlinear filtering problem. While quite natural, this problem does not fit into the ”standard” filtering framework and requires new technical tools. An optimal recursive Bayesian filter for θt based on the observations of Sτ1, Sτ2, . . . for all τk ≤ t is derived. It turns out that the filter is given by a recursive system of deterministic Kolmogorov-type equations, which makes the numerical implementation relatively easy.
Sampling Conditioned Diffusions
Andrew Stuart Warwick Mathematics Institute
Warwick University, UK [email protected]
Abstract There are a wide variety of applications which can be cast as sampling problems for conditioned SDEs (diffusion processes). Examples include nonlinear filtering in signal processing, data assimilation in the ocean/atmosphere sciences, data interpolation in econometrics, and finding transition pathways in molecular systems. In all these examples the object to sample is a path in time, and is hence infinite dimensional. In many applications of interest the noise is additive and the target measure is absolutely continuous with respect to a Gaussian measure. We describe abstract MCMC methods for sampling such problems, exploiting the structure of the target measure. A central role is played by the Langevin equation for which this measure is invariant. This equation is a semilinear stochastic PDE of reaction-diffusion type, subject to additive space-time white noise. We give an overview of the subject area, describing the analytical, statistical and computational challenges, and illustrating applicability of the techniques being developed. Collaboration with: Alex Beskos, Martin Hairer, Jochen Voss, David White (Warwick) and Gareth Roberts (Lancaster).
Some remarks on the numerical approximation of
stochastic differential equations.
Roger Temam Department of Mathematics
Indiana University Bloomington, Indiana [email protected]
Abstract
We present of few remarks on the numerical approximation of stochastic partial differential equations, including consistency and order of approximation.
Stochastic PDEs in Weather and Climate Prediction
Joseph Tribbia
National Center for Atmospheric Research Boulder, CO
Abstract
Weather and climate prediction have, for the past century, developed from the vision of Vilhelm Bjerknes who first noted that the forecasting problem could be solved using a a closed set of deterministic PDEs obtained using the physical laws of classical hydrodynamics and thermodynamics. Bjerknes' realization set into motion the research which culminated in the development of the sophisticated computational models used today to predict weather variations out to two weeks in advance and a range of climate predictions out as far as the projection of anthropogenic climate change into the next century. The seeds of the stochastic extension of these prediction problems were initiated in the middle 1960's, when the limited nature of the predictability of the weather and climate systems were recognized by Lorenz while concurrently the formalism of stochastic-dynamic prediction was developing through the research of Epstein. This work was heavily influenced by methods used in the turbulence closure problem by Kraichnan and Herring. In the 1970's, the application of Monte Carlo techniques for stochastic prediction was proposed and studied by Leith. The current extension of these ideas has led the geophysical prediction community into the realm of predictions of forecast reliability and the desire for accurate uncertainty quantification. These needs have forced the weather and climate communities to explicitly account for the stochastic aspects of unresolved scales and physical processes and their impact on the uncertainty of of weather and climate predictions. My talk will review the current state of the art in stochastic modeling of climate and weather and predict (somewhat uncertainly) the developments in the near future.
Rare events in spatially extended media
Eric Vanden-Ejinden Courant Institute of Mathematical Sciences
New York University New York, NY
Abstract
The dynamical behavior of many systems arising in physics, chemistry, biology, etc. is dominated by rare but important transition events between long lived states. Important examples include nucleation events during phase transition, conformational changes of macromolecules, or chemical reactions. Understanding the mechanism and computing the rate of these transitions is a topic that has attracted a lot of attention for many years. In this talk, I will discuss some recent theoretical developments for the description of rare events, as well as several computational techniques which allow to determine their pathways and rate. I will illustrate this concepts on the specific example of some reaction-diffusion equations driven by white-noise arising e.g. in the context of population dynamics and in the description of the kinetics of phase transitions.
Chorin"Problem reduction and memory"
I will presentmethods for the reduction of the complexity of computationalproblems, both time-dependent and stationary,and their connections to probability, renormalization, scaling, andstatistical mechanics, together with examples.The main points,are: (i) in time dependent problems, it is notlegitimateto average equations without taking into account memory effects and noise;(ii)mathematical tools developed in physics for carrying out renormalization grouptransformations yield effective block Monte-Carlo methods; (iii) theMori-Zwanzig formalism,which in principle yields exact reduction methods but is often hardto use, can betamed by approximation; and (iv) more generally, problem reductionis a search forhidden similarities.In the examples I will emphasize the "t-model" for problems where thememory (=autocorrelation of the noise) has a large support; thisis important in applications to hydrodynamics. In particular, I will showresults, due to Bernstein, Hald, and Stinis, of t-model estimates of therate of decay of the solutions ofthe Burgers equation and the Euler equations in 2 and 3 dimensions.
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