Workshop

Complex physical, biological, geophysical and environmental systems display variability over a broad range of spatial and temporal scales. To make progress in understanding and modelling such systems, a combination of computational, analytical, and experimental techniques is required. There are issues that emerge prominently in each of these categories and in all these stochastic methods are playing a fundamental role. Firstly, at present and for the conceivable future, not all scales of variability can be explicitly modeled: representations of the effects of the unresolved scales of variability on the resolved scales are needed -- the so-called parameterization problem. Recently, stochastic parameterization schemes have been proposed and used in, for example, geophysical and climate simulations. Some of these stochastic parameterization methods may be verified under certain conditions. Secondly, critical insight can be gained by deriving and studying models that are simple enough to be amenable to analysis. Stochastic evolutionary equations are used to build reduced models to improve our understanding of non-equilibrium multiscale systems. Central topics for non-equilibrium systems investigations are, for example, time evolution of the statistics of dependent variables, such as passive tracers and their increments, slow-fast motion decomposition, coarse-graining and model reduction. Fast motion or microscopic processes may be represented by stochastic processes and this has brought in new insights for model reduction (such as slow manifold reduction) of multiscale systems. Thirdly, engineering and natural systems are often subject to uncertainty or random influence, such as stochastic forcing, uncertain parameters, and random boundary conditions. Taking stochastic effects into account is of central importance for developing mathematical models capable of interpreting experimental data. Stochastic partial or ordinary differential equations are appropriate models for randomly influenced multiscale systems. The last decade has seen important developments in both the areas of stochastic analysis and of dynamical systems. Random dynamical systems (e.g., basic background materials in Ludwig Arnold's 1998 book Random Dynamical Systems), as the combination of these two seemly remote mathematical areas, have attracted considerable attention in multiscale modeling community recently. Stochastic dynamical systems provide a unified framework and useful techniques for investigating multiscale systems under uncertainty. The research in the above-mentioned areas has reached a point where it is important and timely to summarize the past achievements, to synthesize various ideas, methods, and paradigms, and use these to stimulate new research by discussing directions and outlooks. Many challenging problems lie ahead, such as invariant manifold reduction under uncertainty, predictability of dynamical behavior, and systematic coarse-graining and model reduction, to name just a few. Solutions for these problems will greatly enhance our ability in understanding, quantifying, and managing uncertainty and predictability in engineering and science. Breakthroughs in solving these challenging problems are stimulated when we take concentrated and focused actions. With this in mind, we propose to organize a workshop at MSRI on mathematical issues in stochastic approaches for multiscale modeling. The major difference with other workshops is that we focus on mathematical issues about stochastic methods for multiscale non-equilibrium systems. The planned workshop will survey past achievements in stochastic approaches for multiscale modeling, and discuss future research directions. It is hoped that this workshop will provide a unique opportunity for interaction of established mathematicians and newcomers (Ph.D. students, post-docs, and junior researchers). Major goals of this workshop are: (i) to stimulate new research on stochastic modeling of multiscale systems, and (ii) to attract Ph.D. students, post-docs, and junior researchers into this exciting field. The engagement of newcomers into the field is an important feature of this workshop. Newcomers will have informal individual meetings with senior participants, discussing possible research topics. Round table meetings will be held, with senior participants outlining what areas they deem important and what opportunities exist for junior people to get involved in addressing open problems. This will offer newcomers a comprehensive view of the state of art of the field and pick up appropriate research projects. To facilitate the continued interaction among participants, informal electronic newsletters (via an E-mail list) will be edited by the organizers, discussing research and teaching of stochastic modeling of multiscale systems. A block of rooms has been reserved at the Hotel Durant. Please mention the workshop name and reference the following code when making reservations via phone, fax or e-mail: KD0000. The cut-off date for reservations is April 20, 2007. A block of rooms has been reserved at the Rose Garden Inn. Reservations may be made by calling 1-800-992-9005 OR directly on their website. Click on Corporate at the bottom of the screen and when prompted enter code MATH (this code is not case sensitive). By using this code a new calendar will appear and will show MSRI rate on all room types available. The following Group Code should be entered in the "Comment" field when booking through our Website. Group Code = CGMS39. Please note: The Original Queen room type does not have an in-room T1 connection, but we do have WIFI available in our main lobby area. The cut-off date for reservations is April 20, 2007.