SDEs and SPDEs : Numerical Methods and Applications
31 March to 4th April 2003, Edinburgh
Home | Scientific Programme & Participants | Workshop Arrangements | Application FormAbstracts
Blömker, DirkPattern formation below the threshold of stability
We consider stochastic PDEs (or systems of SPDEs) on a bounded domain near a deterministic bifurcation. We suppose a cubic-type (possibly non-autonomous) nonlinearity. If the trivial solution is near criticality, and if the stochastic forcing and the deterministic (in)stability are of a comparable margin, then the behaviour of solutions is already given by a stochastic ODE on some very long time-scale. This result reduces the question of pattern formation for the SPDE to exit-problems for
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Buckwar, Evelyn
Euler-Maruyama And Milstein Approximations For Stochastic Functional Differential Equations With Distributed Memory Term
We consider the problem of strong approximations of the solution of stochastic functional differential equations of It\^{o} form with a distributed delay term in the drift and diffusion coefficient. We give a general convergence result and discuss the Euler-Maruyama and the Milstein scheme. Numerical examples illustrate the theoretical results.
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Crauel, Hans
Random dynamical systems and SPDE
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Debussche, Arnaud
Ergodicity for the 3D stochastic Navier--Stokes equations
We consider the Kolmogorov equation associated with the stochastic Navier—Stokes equations in 3D, we prove existence of a solution in the strict or mild sense. The method consists in finding several estimates for the solutions $u_m$ of the Galerkin approximations of $u$ and their derivatives. These estimates are obtained with the help of an auxiliary Kolmogorov equation with a very irregular negative potential. Although uniqueness is not proved, we are able to construct a transition semigroup for the 3D Navier—Stokes equations. Furthermore, this transition semigroup has a unique invariant measure, which is ergodic and strongly mixing. This is a joint work with G. Da Prato.
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Doering, Charles
Duality and wavefront propagation in a stochastic Fisher-KPP equation
We motivate and discuss the problem of front propagation in the Fisher-Kolmogorov-Petrovsky-Piscunov partial differential equation with multiplicative noise. This stochastic PDE is dual to an interacting particle system, and duality is exploited to elucidate quantitative features of its solutions.
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Fouque, Jean-Pierre
Multi-scale stochastic volatility asymptotics
In the first part of the talk we show evidence of the presence of a fast time-scale in the volatility of the S&P500 index. We then argue that this time-scale shouldn't be ignore when performing a change of measure to describe the evolution of the underlying under a risk-neutral pricing measure. A combination of singular and regular perturbations enables us to calibrate the implied volatility surface with few robust and stable parameters. Finally we explain how to use this technique to price other exotic derivatives.
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Gyongy, Istvan
On the numerical approximation of Stochastic Partial Differential Equations
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Hairer, Martin
Non-equilibrium stationary states
We present some recent advances in the study of the problem of a finite Hamiltonian chain of oscillators coupled by its ends to two heat baths at different temperatures. In this talk, we will focus on spectral properties of the generator of the associated Markov semigroups. We show that under quite general condition it tends to be away of the imaginary axis.
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Hausenblas, Erika
Time discretization of the stochastic Navier-Stokes
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Higham, Desmond J
Stability Issues in Long Term Simulations of Stochastic Differential Equations
I will give an overview of a number of questions that take the general form: can a numerical method capture the relevant long term behavior of a stochastic differential equation (SDE)?
First, I will look at a linear stability. Here it is possible to develop a theory that closely matches the traditional numerical analysis approach. However, the answers that arise depend on how stability is measured. I will focus on mean-square and asymptotic versions of the theory.
Second, I will briefly indicate how some positive results can be derived for more general nonlinear SDEs.
Third, I will mention some recent analysis for a linear stochastic oscillator. Here the issue is to conserve appropriate properties.
The talk will touch on joint work with Xuerong Mao, Jon Mattingly, Aslaug Strommen and Andrew Stuart.
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Jansons, Kalvis
Exponential time stepping of SDEs
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Kloeden, Peter
Order barrier for numerical methods for monotone SDEs
It is shown that the only strong Taylor or strong derivative-free numerical methods which preserve the monotone structure of a monotone stochastic differential equation have at most strong order $1.0$ when the noise appears multiplicatively and is generally unrestricted order when the noise in linear stochastic differential equation appears additively. The situation is much better for deterministic DEs. Classes of Runge-Kutta methods that preserve monotonicity for ODEs will also be identified.
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Kuske, Rachel
Multi-scale analysis for stochastic bifurcations
Many systems which are sensitive to noise exhibit dynamical features from both the underlying deterministic behavior and the stochastic elements. This is particularly true for stochastic dynamics near a bifurcation. We illustrate multi-scale methods which provide a viewpoint that separates the stochastic and deterministic effects. The resulting equations give explicit behavior and fast computational methods over long time scales, both above and below the deterministic transition point. The methods are illustrated via canonical examples, including stochastic delay equations and the stochastic Duffing-van der Pohl. Connections to two-point motion Lyapunov exponents, resonance, Fourier representations, and other applications are discussed.
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Langa, Jose A
Asymptotic behaviour of nonautonomous Lotka-Volterra eqautions
By means of the theory of non-autonomous attractors, we are able to entirely describe the 'pullback' asymptotic behaviour of a non-autonomous competitive two-species Lotka-Volterra system in which considering the forwards asymptotic behaviour is not helpful. In particular, we are still able to prove that there is an attracting time-dependent coexistent state.
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Lelievre, Tony
Micro-macro simulations of polymeric fluid
We will present the numerical and mathematical analysis of a micro-macro model to simulate polymeric fluid flows. These kinds of models couple the evolution of the configurations of polymers at the microscopic scale to the velocity field at the macroscopic scale, through the stress tensor. The microscopic evolution of the polymers is modeled through a SDE and the evolution of the macroscopic quantities (velocity, pressure) through the Navier-Stokes equations. We will give some existence results we have obtained on these kinds of systems and also an analysis of
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Lindenberg, Katja
Subdiffusion-limited reactions
We consider the coagulation dynamics A+A->A and annihilation dynamics A+A->0 and A+B->0 for particles moving subdiffusively in one dimension. The analysis combines the ``anomalous kinetics" and ``anomalous diffusion" problems, each of which leads to interesting dynamics separately and to even more interesting dynamics in combination. Our analysis is based on the fractional diffusion equation and its discrete analog.
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Maier-Paape, Stanislaus
Spinodal decomposition for the Cahn-Hilliard-Cook equation
The Cahn-Hilliard-Cook equation, is the Cahn-Hilliard equation perturbed by additive stochastic noise. We prove that most realizations of the solution which starts at a homogeneous state in the spinodal interval exhibit phase separation, leading to the formation of complex patterns of a characteristic size. In more detail, our results can be summarized as follows. The Cahn-Hilliard-Cook equation depends on a small positive parameter $\epsilon$ which models atomic scale interaction length. We quantify the equation behavior of solutions as $\epsilon \rightarrow 0$. Specifically, we show that for the solution starting at a homogeneous state the probability of staying near a finite-dimensional subspace $Y_\epsilon$ is high. The subspace $Y_\epsilon$ is an affine space corresponding to the highly unstable directions for the for the linearized deterministic equation. Joint work with D. Blomker and T. Wanner.
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Mannella, Riccardo
Exponentially fast MC simulations
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Mao, Xuerong
Stability of numerical solutions to hybrid SDEs
Stability of hybrid SDEs (i.e. SDEs with Markovian switching) has received a great deal of attention recently. However there is little known about the stability of numerical solutions to such hybrid SDEs. The talk will address some problems in this area.
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Mattingly, Jonathan
Erogdicity and scales in 2D Navier Stokes equation
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McKane, Alan J
Using optimal paths to calculate sotchastic outcomes
We give an overview of the use of optimal paths in the path-integral formulaton of stochastic differential equations, focusing on the development of the approach described in the theoretical physics literature. The method, used originally to describe noise-activated escape over potential barriers, has been extended and we describe some recent work in the area.
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Menozzi, Stephane
New results on the approximation of killed processes.
We are interested in approximating a multidimensional Ito process killed when it leaves a smooth domain $D$, by a discrete version for which the exit time is computed at discretization times with time step $h$. We first prove that the weak approximation error is bounded by $C \sqrt h$.
In the Markovian case of diffusion and when the approximation is computed with the Euler scheme, we also show that a lower bound with the same order holds true. In some cases, an expansion is even available.
This provides a theoretical explanation of the well-known bias that we can numerically observe in that kind of procedure.
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Moon, Kyoung-Sook
Adaptive Monte carlo Algorithm for Killed Diffusion
We will present an adaptive algorithm for weak approximations of stochastic differential equations by the Monte Carlo Euler method. The goal is to compute an expected value, $E[g(X(\tau),\tau)]$, of a given function, $g$, of the solution depending on the first exit time, $\tau$, from a given domain. In particular, we will explain why killed diffusions are good examples where adaptive methods are very useful. The algorithm is based on the error expansion, for the approximation of $E[g(X(T))]$ for a fixed final time $T>0$, with a posteriori leading order term introduced in [A. Szepessy, R. Tempone and G. Zouraris, Comm. Pure and Appl. Math., 54, 1169-1214, 2001] and on the almost optimal convergence rate proven in [K-S. Moon, A. Szepessy,
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Moro, Esteban
On the mesoscopic description of microscopic particle models using stochastic differential equations.
We address the question of which stochastic differential equation (SDE) is relevant to described the mesoscopic behavior of microscopic particle models and in particular the type of noise. Specifically we comment on the relevance of SDE for the description of the so called "pulled fronts" and under which conditions such mesoscopic SDE description is valid.
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Robinson, James
The structure of random attractors
Examples for which we have a good understanding of the structure of attractors are rare, even in the deterministic case. For gradient-like deterministic systems the attractor consists of the closure of the unstable manifolds of the fixed points. Here we investigate the random attractors for stochastic versions of such systems, and find dramatically different behaviours for additive and multiplicative noise.
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Sánchez, Angel
Reconciling analytics and numerics on the one-dimensional, stochastic sine-Gordon equation
Recently it has been proven that the one-dimensional sine-Gordon model can not have any phase transitions. However, simulations of the stochastic sine-Gordon eqaution suggest a flat phase at low noise and a rough phase at high noise. In this talk I will show how this contradiction can be understood.
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Sancho, Jose M
Multiplicative noise-induce phase transitions: Ito versus Stratonovich.
We present a model to study the intrinsic noise effects in stochastic multiplicative-noise partial differential equations, which are independent of the noise interpretation (It\^o vs. Stratonovich), in the context of noise-induced ordering phase transitions. The main feature of this model is the absence of a linear instability at the transition point. The dynamical properties of the resulting noise-induced growth processes are studied and compared in the two interpretations and with a reference Ginzburg-Landau type model.
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Schurz, Henri
Stability of Some Numerical Methods Along Lyapunov Functionals for SDEs
Numerical stability is one of the key concepts in proving the convergence of numerical methods for SDEs using the main principles of stochastic numerics, see Schurz (1997, 2002). We are going to discuss some general results concerning the numerical stability along certain Lyapunov functionals of related stochastic dynamics. These results can be used to prove the convergence rates of some numerical methods for SDEs on infinite time-intervals $[0,+\infty]$. Nonequidistant discretizations and simple test examples are also treated, whereas the general construction of discrete Lyapunov functionals seems to be very delicate.
References:
[1.] H. Schurz: Stability, stationarity, and boundedness of some implicit numerical methods for stochastic differential equations and applications, Logos-Verlag, Berlin, 1997.
[2.] H. Schurz: Numerical analysis of SDEs without tears, in: Handbook of Stochastic Analysis and Applications, D. Kannan and V. Lakshmikantham, Eds. (Marcel Dekker, Basel, 2002) 237-359.
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Schwab, Christoph
Numerical Solution of elliptic PDEs with stochastic data
We discuss the formulation of elliptic stochastic PDEs with stochastic input data in a bounded polyhedral domain D of dimension d.
For stochastic loadings with finite M-th moments, the solution's moments or k-point correlations, k<=M, belong to certain Sobolev spaces with control on the mixed derivatives of order M.
We present a deterministic Finite Element Method for the approximation of the M-th moment of the solution with log-linear complexity in the deterministic problem.
Numerical examples show the sharpness of the error bounds. We also address the problem of stochastic coeffcients.
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Shevchenko, Georgiy
Rate of convergence of discrete-time approximations for solutions of SDEs in Hilbert spaces
The approximations by time discretization of SDEs in Hilbert spaces are considered. New results concerning rate of convergence of approximations for stochastic semilinear equations and Ito-Volterra type equations are established. Several approximation schemes are studied. Also we consider approximations by finite-dimensional processes.
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Sircar, Ronnie
Fractional Brownian Motion Approximation of Financial Markets with Inert Investors
In a financial market where investor inertia is modelled by a semi-Markov process fluctuating on a short time-scale, we show that resulting stock prices are approximated by stochastic processes driven by fractional Brownian motion. This provides a link between the tendency of small traders to be inactive most of the time, and long-range dependence in stock price returns observed by Mandelbrot and others. We find the associated Hurst exponent from financial data using a wavelet-based estimator and discuss its variation over time.
Joint work with Erhan Bayraktar & Ulrich Horst.
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Stuart, Andrew
Fitting SDEs to partially observed dynamical systems
In many applications, such as molecular conformation, it is of interest to find coarse-grained stochastic models for the dynamics of large Hamiltonian sysems. The work of Kac and Zwanzig showed how Hamiltonian systems, in the form of a particle interacting with a heat bath, could be approximated by SDEs. This framework applies when the particle-bath coupling is linear. When the coupling is more complicated, it is of interest to find similar approximating SDEs as this is the situation which arises in practical applications. Linear response theory provides a rationale for the form of SDE to be used. Time series analysis, and the Kalman filter in particular, is then employed to fit SDEs to partially observed Hamiltonian systems.
Joint work with Raz Kupferman (Jerusalem) and Petter Wiberg (Warwick).
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Timofeeva, Yulia
Sparks and waves in a stochastic fire-diffuse-fire model of Ca2+ release
Calcium ions are an important second messenger in living cells. Indeed calcium signals in the form of waves have been the subject of much recent experimental interest. It is now well established that these waves are composed of elementary stochastic release events (calcium puffs) from spatially localized calcium stores.
We develop a computationally inexpensive model of calcium release based upon a stochastic generalization of the Fire-Diffuse-Fire (FDF) threshold model. Our model retains the discrete nature of calcium stores, but also incorporates a notion of release probability via the introduction of threshold noise. Numerical simulations of the model illustrate that stochastic calcium release leads to the spontaneous production of calcium sparks that may merge to form saltatory waves.
A statistical analysis of the model shows the interesting possibility of a non-equilibrium phase-transition between propagating and non-propagating waves suggesting that the model belongs to the directed percolation universality class. Moreover, threshold noise is shown to generate a form of array enhanced coherence resonance whereby all calcium stores release periodically and simultaneously.
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Tretyakov, Michael
Quasi-symplectic methods for Langevin equations
Many problems of physics can be formulated in the form of Langevin equations (second-order differential equations with noise). The proposed mean-square and weak quasi-symplectic methods preserve some structural properties of the Langevin equations. Their superiority in comparison with standard schemes of numerical integration of SDEs is demonstrated. The talk is based on a joint paper with Professor G.N. Milstein.
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Van den Broeck, Christian
Brownian ratchets versus Langevin theory
We present molecular dynamics simulations for Brownian ratchets and confront these results with the predictions of Langevin theories.
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Vanden-Eijnden, Eric
Effective Dynamics for some multi-scale SPDEs
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Winkler, Renate
Numerical Integration of Stochastic DAEs in Circuit Analysis
We describe how stochastic differential-algebraic equations (SDAEs) arise as a mathemaical model for electrical networks which are influenced by additional sources of Gaussian white noise. In industrial applications one has to deal with a large number of equations as well as a large number of small noise sources. We show how drift-implicit schemes for SDEs can be adapted to become directly applicable to stochastic DAEs, and prove that the convergence properties of these methods known for SDEs are preserved. For the drift-implicit Euler scheme we present a stepsize-control that is based on the mean square of local error estimates.
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Yan, Yubin
Finite element method for stochastic parabolic partial differential equations I will present our first attempts to prove error estimates for finite element approximations of a parabolic stochastic partial differential equation: du + Au dt = dW where A is an elliptic operator and W is space-time white noise. Based on appropriate nonsmooth data error estimates for deterministic finite element problems, we obtain the error estimates in semidiscrete and fully discrete cases in both strong and weak norms. Our results are general and can be applied to several spatial variables.
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Yuan, Chenggui
Invariance measure of numerical solutions to hybrid SDEs
Invariance measure for the solutions of hybrid SDEs (i.e. SDEs with Markovian switching) is one of the most important properties. There are some theoretical results aleady but few on the numerical solutions. This talk will discuss under what conditions the numerical solutions have the invariance measure.
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Zouraris, Georgios
Galerkin finite element approximations of Elliptic SPDEs
We describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous Dirichlet boundary conditions. The aim of the computations is to approximate the expected value of the solution. The first method generates iid approximations of the solution by sampling the coefficients of the equation and using a standard Galerkin finite element method. The Monte-Carlo methods uses these approximations to compute corresponding sample averages. The second method is based on the assumption that the stochasticity of the coefficients is specified by a known random vector. This assumption turns out the original stochastic problem into a deterministic parametric elliptic problem. A Galerkin finite element method of either $h$ or $p$ version, then approximates the solution of the corresponding deterministic problem yielding approximations of the expected value. We derive a priori error estimates for the approximation of the expected value and compare the asymptotical work required by each method to achieve a given accuracy.
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