Mathematical Sciences: Conference on Dynamical Numerical Analysis

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Dieci The investigator and his colleagues organize an international conference on numerical methods for dynamical systems. The interaction between dynamical systems and numerical analysis has grown in recent years as the limitations of classical quantitative numerical analysis have become increasingly apparent. In practice, a common goal of computation is to obtain information about solutions over moderate to long time intervals, yet classical error analysis is generally not meaningful past a short initial transient. It is thus fundamental to the numerical analysis of differential equations to address both the efficacy of existing computational methods and the design of special methods for computation over long time intervals. Specific topics to be covered include convergence of integration algorithms over long time intervals (including long-time error bounds and convergence of invariant sets), stability of integration algorithms over long time intervals (including preservation of dynamical structure and spurious solutions), new approaches to error analysis (including shadowing results and backward error analysis), and design of computational techniques for invariant sets such as algorithms to compute invariant torii, inertial manifolds, heteroclinic orbits, homoclinic orbits, and Lyapunov exponents. Differential equations describe how one position in space of a physical system at a particular time influences neighboring positions in the immediate future. It is this local nature of a differential equation that makes it possible to model complicated physical situations, because it means that the entire system does not have to be described simultaneously. But this also means that obtaining the correct differential equation model of a system is not the end of the story, because it is necessary to solve the differential equation to obtain information about the original system. Accordingly, the study of solutions of differential equations has grown int o one of the major areas of mathematics while becoming centrally important in science and engineering. The introduction of the computer made it possible to approximately solve differential equations on a wide scale for the first time. Over the last thirty years, numerical mathematicians have made great progress in devising methods to approximate a specific solution of a differential equation and in analyzing the accuracy of the approximate solution. The classical approach to study numerical methods for differential equations is very good at describing the behavior of the approximation of a particular solution in a small region of space over a short time interval. But understanding the nature of nonlinear equations requires knowledge about many solutions over relatively long time intervals. In response to this need, there is increasing interest in using numerical methods to study structures and patterns in solutions of differential equations, which is known as the dynamical behavior of the equation. This conference gathers leading experts from the United States and from around the world, representing nearly all aspects of the area, together with interested students and other researchers, to discuss the current state of the art and to stimulate new developments. The conference is expected to have a long-ranging impact on many applications in science and engineering.