Localization for high dimensional data assimilation and MCMC

High dimensionality often appears in data assimilation and Bayesian sampling problems.  It is prohibitive for most classical computational methods,  of which the necessary sample size grows linearly or even exponentially with the underlying dimension.  Yet the component correlation in many high dimension problems is local. The localization technique was designed to exploit this structural feature to improve sampling accuracy in the ensemble Kalman filter (EnKF). In this talk, we will first discuss the mathematical mechanism that guarantees a localized EnKF perform well with a small sample size, assuming the dynamical system is linear and preserves local structures. Then we discuss how to generalize the localization technique for non-Gaussian high dimensional sampling problems. Two Markov chain Monte Carlo (MCMC) algorithms will be devised for different settings, where the necessary sample size is independent of the state dimension.