To determine the optimal set of hyperparameters of a Gaussian process based on a large number of training data, both a linear system and a trace estimation problem must be solved. In this paper, we focus on establishing numerical methods for the case where the covariance matrix is given as the sum of possibly multiple Kronecker products, i.e., can be identified as a tensor. As such, we will represent this operator and the training data in the tensor train format. Based on the AMEn method and Krylov subspace methods, we derive an efficient scheme for computing the matrix functions required for evaluating the gradient and the objective function in hyperparameter optimization.

Balanced truncation is a well-established model order reduction method in system theory that has been applied to a variety of problems. Recently, a connection between linear Gaussian Bayesian inference problems and the system theoretic concept of balanced truncation was drawn for the first time. Although this connection is new, the application of balanced truncation to data assimilation is not a novel concept: It has already been used in four-dimensional variational data assimilation (4D-Var) in its discrete formulation. In this paper, the link between system theory  and data assimilation is further strengthened by discussing the application of balanced truncation to standard linear Gaussian Bayesian inference, and, in particular, the 4D-Var method.  similarities between both data assimilation problems allow a discussion of established methods as well as a generalisation of the state-of-the-art approach to arbitrary prior covariances as  reachability Gramians. Furthermore, we propose an enhanced approach using time-limited balanced truncation that allows to balance Bayesian inference for unstable systems and in addition mproves the numerical results for short observation periods.

Four-dimensional variational data assimilation (4D-Var) is a data assimilation method often used in weather forecasting. Based on a numerical model and observations of a system, it predicts the system state beyond the last time of measurement. This requires the minimisation of a functional. At each step of the optimisation algorithm, a full nonlinear model evaluation and its adjoint is required. This quickly becomes very costly, especially in high dimensions. For this reason, a surrogate model is needed that approximates the full model well, but requires significantly less computational effort. In this paper, we propose time-limited balanced truncation to build such a reduced-order model. Our approach is able to deal with unstable system matrices. We demonstrate its performance in experiments and compare it with α-bounded balanced truncation, which is an another reduction approach for unstable systems.