We introduce a multifidelity estimator of covariance matrices formulated as the solution to a regression problem on the manifold of symmetric positive definite matrices. The estimator is positive definite by construction, and the Mahalanobis distance minimized to obtain it possesses properties which enable practical computation. We show that our manifold regression multifidelity (MRMF) covariance estimator is a maximum likelihood estimator under a certain error model on manifold tangent space. More broadly, we show that our Riemannian regression framework encompasses existing multifidelity covariance estimators constructed from control variates. We demonstrate via numerical examples that our estimator can provide significant decreases, up to one order of magnitude, in squared estimation error relative to both single-fidelity and other multifidelity covariance estimators. Furthermore, preservation of positive definiteness ensures that our estimator is compatible with downstream tasks, such as data assimilation and metric learning, in which this property is essential.

We introduce a multi-fidelity estimator of covariance matrices that employs the log-Euclidean geometry of the symmetric positive-definite manifold. The estimator fuses samples from a hierarchy of data sources of differing fidelities and costs for variance reduction while guaranteeing definiteness, in contrast with previous approaches. The new estimator makes covariance estimation tractable in applications where simulation or data collection is expensive; to that end, we develop an optimal sample allocation scheme that minimizes the mean-squared error of the estimator given a fixed budget. Guaranteed definiteness is crucial to metric learning, data assimilation, and other downstream tasks. Evaluations of our approach using data from physical applications (heat conduction, fluid dynamics) demonstrate more accurate metric learning and speedups of more than one order of magnitude compared to benchmarks.

Bayesian inference is a ubiquitous and flexible tool for updating a belief (i.e., learning) about a quantity of interest when data are observed, which ultimately can be used to inform downstream decision-making. In particular, Bayesian inverse problems allow one to derive knowledge from data through the lens of physicsbased models. These problems can be formulated as follows: given observational data, a physics-based model, and prior information about the model inputs, find a posterior probability distribution for the inputs that reflects the knowledge about the inputs in terms of the observed data and prior. Typically, the physicsbased models are given in the form of an input-to-observation map that is based on a system of partial differential equations (PDEs). The computational task underlying Bayesian inference is approximating posterior probability distributions to compute expectations and to quantify uncertainties. There are multiple ways of computationally exploring posterior distributions to gain insights, reaching from Markov chain Monte Carlo to variational methods [24, 42, 28]. In this work, we make use of Stein variational gradient descent (SVGD) [32], which is a method for particle-based variational inference, to approximate posterior distributions. It builds on Stein's identity to formulate an update step for the particles that can be realized numerically in an efficient manner via

Multi-fidelity methods leverage low-cost surrogate models to speed up computations and make occasional recourse to expensive high-fidelity models to establish accuracy guarantees. Because surrogate and high-fidelity models are used together, poor predictions by the surrogate models can be compensated with frequent recourse to high-fidelity models. Thus, there is a trade-off between investing computational resources to improve surrogate models and the frequency of making recourse to expensive high-fidelity models; however, this trade-off is ignored by traditional modeling methods that construct surrogate models that are meant to replace high-fidelity models rather than being used together with high-fidelity models. This work considers multi-fidelity importance sampling and theoretically and computationally derives the optimal trade-off between improving the fidelity of surrogate models for constructing more accurate biasing densities and the number of samples that is required from the high-fidelity model to compensate poor biasing densities. Numerical examples demonstrate that such optimal---context-aware---surrogate models for multi-fidelity importance sampling have lower fidelity than what typically is set as tolerance in traditional model reduction, leading to runtime speedups of up to one order of magnitude in the presented examples.