Nonlinear Data Assimilation for high-dimensional systems

Abstract

In this chapter the state-of-the-art in data assimilation for high-dimensional highly nonlinear systems is reviewed, and recent developments are highlighted. This knowledge is available in terms of probability density functions, and the nonlinearity means that the shape of these functions is unknown a-priori. We focus on sampling methods because of they flexibility. Traditional Monte-Carlo methods like Metropolis-Hastings and its variants are discussed, including exciting new developments in this field. However, because of the serial nature of the sampling, and the possibility to reject samples these methods are not efficient in high-dimensional systems in which each sample is very expensive computationally. The emphasis of this chapter is on so-called particle filters as these are emerging as most efficient for these high-dimensional systems. Up to recently their profile has been low when the dimensions are high, or rather when the number of independent observations is high, because the area of state space when the observations are is decreasing very rapidly with system dimension. However, recent developments have beaten this curse of dimensionality, as will be demonstrated both theoretically and in high-dimensional examples. But it is also emphasized that much more needs to be done.