Data Assimilation, Noise Models, and Dimensional Reduction, with Applications

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There are many problems in science where one must draw conclusions and make predictions on the basis of imperfect models supplemented by noisy and/or incomplete data; examples are weather forecasting, climate prediction, economics, and materials science. The goal of the project is to make carrying out these task more efficient and accurate, and extend the methodology to new situations. In earlier work we showed how to optimize the task of finding the best conclusions. Under the new proposal we will look for new, more efficient ways to estimate margins of error, and also use the estimates of the error to improve the conclusions. The same technology can be used to improve the models themselves on the basis of data; we will try to expand this to problems where the data are qualitative, for example, devise methods that will eventually allow an engineer to translate observations about when an engine runs smoothly into precise quantitative information that can be used for design. Finally, we will extend these methods to problems where the mathematical models are unreliable, not because of incomplete knowledge or of experimental error, but because the computing power available to solve them is insufficient; in such cases, our methods would make it possible to use statistical methods to enhance the accuracy of the computations.The proposal has several thrusts. (i) One of the main obstacles to further improvement of data assimilation tools is the difficulty in deriving realistic noise models, in particular because the noise depends on the signal and is therefore hard to separate from it. It is proposed to perform this separation by using tools from statistical physics, in particular recent generalizations of the Mori-Zwanzig formalism. (ii) The problem of estimating the noise in noisy models and the problem of deriving reduced models of complex dynamics are very similar; this remark can be used to develop data assimilation methods for qualitative data. This requires creating maps between data sets and parameter sets in the models, which becomes feasible through the use of reduced dynamics, to be derived by the methods in the present proposal. (iii) One can interpret the missing components of the solution in underresolved numerical dynamics as an added noise. This remark suggests that these missing components can be estimated from numerical or experimental data, so as to enhance the accuracy of the numerical solutions. It is proposed to develop methods for doing so. (iv) It is proposed to develop reduced descriptions in a variety of areas of scientific computation. The applications of this work will include problems from geophysics, stochastic control, robotics, combustion, and fluid mechanics.