Abstract: Recently, some works have suggested methods to combine variational probabilistic inference with Monte Carlo sampling. One promising approach is via local optimal transport. In this approach, a gradient steepest descent method based on local optimal transport principles is formulated to transform deterministically point samples from an intermediate density to a posterior density. The local mappings that transform the intermediate densities are embedded in a reproducing kernel Hilbert space (RKHS). This variational mapping method requires the evaluation of the log-posterior density gradient and therefore the adjoint of the observational operator. In this work, we evaluate nonlinear observational mappings in the variational mapping method using two approximations that avoid the adjoint, an ensemble based approximation in which the gradient is approximated by the particle covariances in the state and observational spaces the so-called ensemble space and an RKHS approximation in which the observational mapping is embedded in an RKHS and the gradient is derived there. The approximations are evaluated for highly nonlinear observational operators and in a low-dimensional chaotic dynamical system. The RKHS approximation is shown to be highly successful and superior to the ensemble approximation.

Abstract: Ensemble optimal interpolation (EnOI) have been introduced to drastically reduce the computational cost of the ensemble Kalman filter (EnKF). The idea is to use a static (pre-selected) ensemble to parameterize the background covariance matrix, which avoids the costly integration step of the ensemble members with the dynamical model. To better represent the strong variability of the Red Sea circulation, we propose new adaptive EnOI schemes in which the ensemble members are adaptively selected at every assimilation cycle from a large dictionary of ocean states describing the variability of the Red Sea system. Those members would account for the strong eddy and seasonal variability of the Red Sea circulation and enforce climatological smoothness in the filter update. We implement and test different schemes to adaptively choose the ensemble members based on (i) the similarity to the forecast, or (ii) an Orthogonal Matching Pursuit (OMP) algorithm. Results of numerical experiments assimilating remote sensing data into a high-resolution MIT general circulation model (MITgcm) of the Red Sea will be presented to demonstrate the efficiency of the proposed approach.

Abstract: This paper demonstrates the use of machine learning techniques to study the uncertainty in numerical weather prediction models due to the interaction of multiple physical processes. We aim to address the following problems: 1)estimation of systematic model errors in output quantities of interest at future times and 2)identification of specific physical processes that contribute most to the forecast uncertainty in the quantity of interest under specified meteorological conditions. To address these problems, we employ simple machine learning algorithms and perform numerical experiments with Weather Research and Forecasting (WRF) model. The results demonstrate the potential of machine learning approaches to aid the study of model errors.

Abstract: A Weak Constraint Gaussian Process (WCGP) model is presented to integrate noisy inputs into the classical Gaussian Process predictive distribution. This follows a Data Assimilation approach i.e. by considering information provided by observed values of a noisy input in a time window. Due to the increased number of states processed from real applications and the time complexity of GP algorithms, the problem mandates a solution in a high performance computing environment. In this paper, parallelism is explored by defining the parallel WCGP model based on domain decomposition. Both a mathematical formulation of the model and a parallel algorithm are provided. We prove that the parallel implementation preserves the accuracy of the sequential one. The algorithm’s scalability is further proved to be O(p^2) where p is the number of processors.