Explicit off-line criteria for stable accurate time filtering of strongly unstable spatially extended systems

Abstract

Many contemporary problems in science involve making predictions based on partial observation of extremely complicated spatially extended systems with many degrees of freedom and physical instabilities on both large and small scales. Various new ensemble filtering strategies have been developed recently for these applications, and new mathematical issues arise. Here, explicit off-line test criteria for stable accurate discrete filtering are developed for use in the above context and mimic the classical stability analysis for finite difference schemes. First, constant coefficient partial differential equations, which are randomly forced and damped to mimic mesh scale energy spectra in the above problems are developed as off-line filtering test problems. Then mathematical analysis is used to show that under natural suitable hypothesis the time filtering algorithms for general finite difference discrete approximations to an s ؋ s partial differential equation system with suitable observations decompose into much simpler independent s-dimensional filtering problems for each spatial wave number separately; in other test problems, such block diagonal models rigorously provide upper and lower bounds on the filtering algorithm. In this fashion, elementary off-line filtering criteria can be developed for complex spatially extended systems. The theory is illustrated for time filters by using both unstable and implicit difference scheme approximations to the stochastically forced heat equation where the combined effects of filter stability and model error are analyzed through the simpler off-line criteria.M any contemporary problems in science ranging from protein folding in molecular dynamics to scale up of smallscale effects in nanotechnology to making accurate predictions of the coupled atmosphere-ocean system involve partial observations of extremely complicated systems with many degrees of freedom. Novel mathematical issues arise in the attempt to quantify the behavior of such complex multiscale systems (1, 2). For example, in the coupled atmosphere-ocean system, the current practical models for prediction of both weather and climate involve general circulation models where the physical equations for these extremely complex flows are discretized in space and time and the effects of unresolved processes are parametrized according to various recipes; the result of this process involves a model for the prediction of weather and climate from partial observations of an extremely unstable, chaotic dynamical system with several billion degrees of freedom. Bayesian hierarchical modeling (3) and reduced order filtering strategies (4-12) have been developed with some success in these extremely complex systems. The basis for such dynamic prediction strategies for the complex spatially extended systems is the classical Kalman filtering algorithm (13-16). Mathematical issues arise in the practical application of these filtering strategies to complex spatially extended systems and are the focus of this article.One mathematical issu...

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