Collaborative Research: Determining Forms and Data Assimilation with Stochastic Data

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Among the numerous factors that make accurate weather forecasting a challenge is initialization. The finite number of data collecting instruments distributed about the globe and atmosphere give only an incomplete description of the state of the weather at any instant. Yet there is a wealth of such data over an extended time period in the past, which when combined with certain mathematical models can provide a more complete description. The general procedure for this is called data assimilation. It provides more accurate starting values for computer simulations of the weather going into the future. This award funds research into a new method of data assimilation that is flexible enough to be combined with a variety of simulation techniques. The work will concern both the implementation of this method, and the effect of measurement errors in the data, which inevitably occur. The assimilation process leads to another mathematical model that can be used to cleanse the past data of such noise. In this second direction, the goal is not to obtain a higher resolution condition for starting a simulation for the future, but to reconstruct a highly resolved version over the time period of the original data. This has natural applications in voice and pattern recognition. The research team will implement a recent data assimilation algorithm based on feedback control for several important physical systems. The approach can be applied with a variety of determining parameters, such as nodal values, and finite volume elements. The team will also carry out computations with a new determining form based on feedback control. The steady states of this ordinary differential equation are precisely the finite-dimensional projections of trajectories in the global attractor. The team will study both computationally and analytically the effects of stochastic perturbations in the data. In the case of the data assimilation algorithm, the plan is to quantify and control the effect of the noise on the highly resolved state to be used in a subsequent direct numerical simulation. In the case of the determining form, the idea is to use its evolutionary process to remove the noise from the data itself.