P. Kloeden | Huazhong University of Science and Technology (original) (raw)

Papers by P. Kloeden

Suppose that a mapping J : n r-t fl, on a compact metric space 0, generates a discrete dynamical ... more Suppose that a mapping J : n r-t fl, on a compact metric space 0, generates a discrete dynamical system {fn(x) : n = 0, 1, 2, ... } with chaotic behavior. For brevity we will refer to the system f. Standard computer models of this system are dynamical systems '1/J defined on a finite subset L of fl. The set of all trajectories of an individual system '1/J can (liffer dramatically from that of the original system even for a fine discretization. For instance, the mappingf( x) = 2x (mod 1) is chaotic with a unique absolutely continuous invariant measure and cycles of all periods. Yet every N-digital binary discretization '1/J N, defined as the restriction of f to the set { ij2N : i = 1, ... , 2N -l} is asymptotically trivial, '¢1, ::=::: 0 , k 2:: N. Such effects are an inevitable consequence of discretization in the sense that there always exists some discretization which collapses a given system f onto a given !-invariant set, in particular onto a fixed point or cycle [6]. Degenerate, collapsing behavior such as this can be eliminated by instead modelling with systems cp which can be regarded as either stochastic or multivalued perturbations of the original system f. However, the choice of an appropriate model system cp introduces a conundrum which frequently arises in the theory of ill-posed problems. If the perturbation that is introduced is too large, then the behaviour of the system cp, while not degenerate can differ markedly from f. On the other hand, if the perturbation is not strong enough, collapsing effects will not be avoided. Consequently, questions about the robustness of systems to various levels of stochastic or multivalued perturbation are very important. Fundamental theoretical results concerning stochastic perturbations can be found in [11, 1, 2]. Approaches using multivalued systems are much less investigated. However, this approach seems to be efficient in investigating systems with fast changing and discontinuous characteristics. Often the most adequate mathematical descriptions of

This volume is a collection of articles based on the plenary talks presented at the 2008 meeting ... more This volume is a collection of articles based on the plenary talks presented at the 2008 meeting in Hong Kong of the Society for the Foundations of Computational Mathematics. The talks were given by some of the foremost world authorities in computational mathematics. The topics covered reflect the breadth of research within the area as well as the richness and fertility of interactions between seemingly unrelated branches of pure and applied mathematics. As a result this volume will be of interest to researchers in the field of computational mathematics and also to non-experts who wish to gain some insight into the state of the art in this active and significant field.

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2005

It is shown that the synchronization of dissipative systems persists when they are disturbed by a... more It is shown that the synchronization of dissipative systems persists when they are disturbed by additive noise, no matter how large the intensity of the noise, provided asymptotically stable stationary-stochastic solutions are used instead of asymptotically stable equilibria.

Physical Review E, 2006

Wilkie ͓Phys. Rev. E 70, 017701 ͑2004͔͒ used a heuristic approach to derive Runge-Kutta-based num... more Wilkie ͓Phys. Rev. E 70, 017701 ͑2004͔͒ used a heuristic approach to derive Runge-Kutta-based numerical methods for stochastic differential equations based on methods used for solving ordinary differential equations. The aim was to follow solution paths with high order. We point out that this approach is invalid in the general case and does not lead to high order methods. We warn readers against the inappropriate use of deterministic calculus in a stochastic setting.

Journal of Dynamics and Differential Equations, 2006

We prove the existence of a stationary random solution to a delay random ordinary differential sy... more We prove the existence of a stationary random solution to a delay random ordinary differential system which attracts all other solutions in both pullback and forwards senses. The equation contains a one-sided dissipative Lipschitz term without delay, while the random delay appears in a globally Lipschitz one. The delay function only needs to be continuous in time. Moreover, we also prove that the split implicit Euler scheme associated to the random delay differential system generates a discrete time random dynamical system which also possesses a stochastic stationary solution with the same attracting property, and which converges to the stationary solution of the delay random differential equation pathwise as the stepsize goes to zero.

Bulletin of the Australian Mathematical Society, 2009

An existence and uniqueness theorem for mild solutions of stochastic evolution equations is prese... more An existence and uniqueness theorem for mild solutions of stochastic evolution equations is presented and proved. The diffusion coefficient is handled in a unified way which allows a unified theorem to be formulated for different cases, in particular, of multiplicative space–time white noise and trace-class noise.

The Annals of Probability, 2010

The solution of a parabolic stochastic partial differential equation (SPDE) driven by an infinite... more The solution of a parabolic stochastic partial differential equation (SPDE) driven by an infinite-dimensional Brownian motion is in general not a semi-martingale anymore and does in general not satisfy an Itô formula like the solution of a finite-dimensional stochastic ordinary differential equation (SODE). In particular, it is not possible to derive stochastic Taylor expansions as for the solution of a SODE using an iterated application of the Itô formula. Consequently, until recently, only low order numerical approximation results for such a SPDE have been available. Here, the fact that the solution of a SPDE driven by additive noise can be interpreted in the mild sense with integrals involving the exponential of the dominant linear operator in the SPDE provides an alternative approach for deriving stochastic Taylor expansions for the solution of such a SPDE. Essentially, the exponential factor has a mollifying effect and ensures that all integrals take values in the Hilbert space under consideration. The iteration of such integrals allows us to derive stochastic Taylor expansions of arbitrarily high order, which are robust in the sense that they also hold for other types of driving noise processes such as fractional Brownian motion. Combinatorial concepts of trees and woods provide a compact formulation of the Taylor expansions.

Contemporary Mathematics, 1994

SIAM Journal on Numerical Analysis, 2016

Recent Trends in Optimization Theory and Applications, 1995

SIAM Journal on Mathematical Analysis, 2015

The appearance of delay terms in a chemostat model can be fully justified since the future behavi... more The appearance of delay terms in a chemostat model can be fully justified since the future behavior of a dynamical system does not in general depend only on the present but also on its history. Sometimes only a short piece of history provides the relevant influence (bounded or finite delay), while in other cases it is the whole history that has to be taken into account (unbounded or infinite delay). In this paper a chemostat model with time variable delays and wall growth, hence a nonautonomous problem, is investigated. The analysis provides sufficient conditions for the asymptotic stability of nontrivial equilibria of the chemostat with variable delays, as well as for the existence of nonautonomous pullback attractors.

This book presents a systematic theory of Taylor expansions of evolutionary-type stochastic parti... more This book presents a systematic theory of Taylor expansions of evolutionary-type stochastic partial differential equations (SPDEs). The authors show how Taylor expansions can be used to derive higher order numerical methods for SPDEs, with a focus on pathwise and strong convergence. In the case of multiplicative noise, the driving noise process is assumed to be a cylindrical Wiener process, while in the case of additive noise the SPDE is assumed to be driven by an arbitrary stochastic process with Hlder continuous sample paths. Recent developments on numerical methods for random and stochastic ordinary differential equations are also included since these are relevant for solving spatially discretised SPDEs as well as of interest in their own right. The authors include the proof of an existence and uniqueness theorem under general assumptions on the coefficients as well as regularity estimates in an appendix. Audience: Applied and pure mathematicians interested in using and further developing numerical methods for SPDEs will find this book helpful. It may also be used as a source of material for a graduate course. Contents: Preface; List of Figures; Chapter 1: Introduction; Part I: Random and Stochastic Ordinary Partial Differential Equations; Chapter 2: RODEs; Chapter 3: SODEs; Chapter 4: SODEs with Nonstandard Assumptions; Part II: Stochastic Partial Differential Equations; Chapter 5: SPDEs; Chapter 6: Numerical Methods for SPDEs; Chapter 7: Taylor Approximations for SPDEs with Additive Noise; Chapter 8: Taylor Approximations for SPDEs with Multiplicative Noise; Appendix: Regularity Estimates for SPDEs; Bibliography; Index.

Stochastics and Dynamics, 2008

The synchronization of Stratonovich stochastic differential equations (SDE) with a one-sided diss... more The synchronization of Stratonovich stochastic differential equations (SDE) with a one-sided dissipative Lipschitz drift and linear multiplicative noise is investigated by transforming the SDE to random ordinary differential equations (RODE) and synchronizing their dynamics. In terms of the original SDE, this gives synchronization only when the driving noises are the same. Otherwise, the synchronization is modulo exponential factors involving Ornstein–Uhlenbeck processes corresponding to the driving noises. Moreover, this occurs no matter how large the intensity coefficients of the noise.

SIAM Journal on Numerical Analysis, 1986

We consider a dynamical system described by a system of ordinary differential equations which pos... more We consider a dynamical system described by a system of ordinary differential equations which possesses a compact attracting set A of arbitrary shape. Under the assumption of uniform asymptotic stability of A in the sense of Lyapunov, we show that discretized versions of the dynamical system involving one-step numerical methods have nearby attracting sets A(h), which are also uniformly asymptotically stable. Our proof uses the properties of a Lyapunov function which characterizes the stability of A.

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2009

We consider the numerical approximation of parabolic stochastic partial differential equations dr... more We consider the numerical approximation of parabolic stochastic partial differential equations driven by additive space–time white noise. We introduce a new numerical scheme for the time discretization of the finite-dimensional Galerkin stochastic differential equations, which we call the exponential Euler scheme, and show that it converges (in the strong sense) faster than the classical numerical schemes, such as the linear-implicit Euler scheme or the Crank–Nicholson scheme, for this equation with the general noise. In particular, we prove that our scheme applied to a semilinear stochastic heat equation converges with an overall computational order 1/3 which exceeds the barrier order 1/6 for numerical schemes using only basic increments of the noise process reported previously. By contrast, our scheme takes advantage of the smoothening effect of the Laplace operator and of a linear functional of the noise and, therefore overcomes this order barrier.

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2011

The stochastic Euler scheme is known to converge to the exact solution of a stochastic differenti... more The stochastic Euler scheme is known to converge to the exact solution of a stochastic differential equation (SDE) with globally Lipschitz continuous drift and diffusion coefficients. Recent results extend this convergence to coefficients that grow, at most, linearly. For superlinearly growing coefficients, finite-time convergence in the strong mean-square sense remains. In this article, we answer this question to the negative and prove, for a large class of SDEs with non-globally Lipschitz continuous coefficients, that Euler’s approximation converges neither in the strong mean-square sense nor in the numerically weak sense to the exact solution at a finite time point. Even worse, the difference of the exact solution and of the numerical approximation at a finite time point diverges to infinity in the strong mean-square sense and in the numerically weak sense.

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2007

Random ordinary differential equations (RODEs) are ordinary differential equations (ODEs) with a ... more Random ordinary differential equations (RODEs) are ordinary differential equations (ODEs) with a stochastic process in their vector field. They can be analysed pathwise using deterministic calculus, but since the driving stochastic process is usually only Hölder continuous in time, the vector field is not differentiable in the time variable, so traditional numerical schemes for ODEs do not achieve their usual order of convergence when applied to RODEs. Nevertheless deterministic calculus can still be used to derive higher order numerical schemes for RODEs via integral versions of implicit Taylor-like expansions. The theory is developed systematically here and applied to illustrative examples involving Brownian motion and fractional Brownian motion as the driving processes.