| Summary: | In this thesis we develop a suite of three methods for multifidelity covariance estimation. We begin with a straightforward extension of scalar multifidelity Monte Carlo to matrices, obtaining what we refer to as the Euclidean or linear control variate mutifidelity covariance estimator. The mean squared error of this estimator is available in closed form, which enables analytic optimization of sample allocations and weights to minimize expected squared Frobenius error subject to computational budget constraints. Despite its nice analytical properties and familiar closed-form construction, however, the Euclidean estimator can be subject to loss of positive-definiteness. Given this liability, we subsequently develop two multifidelity covariance estimators which preserve positive definiteness by construction, utilizing, to varying degrees, the geometry of the manifold of symmetric positive definite (SPD) matrices.
Our first positive-definiteness-preserving estimator, referred to as the tangent space or log-linear control variate estimator, constructs a multifidelity covariance estimate by appplying linear control variates to sample covariance matrix logarithms, which are symmetric matrices residing in tangent spaces to the SPD manifold. Though the tangent space estimator preserves positive-definiteness and is straightforward to construct, obtaining its expected squared error, and thus choosing optimal sample allocations and control variate weights, are not tractable. When first-order approximations of the matrix logarithms involved are made, however, the optimal sample allocations and control variate weights for the tangent space estimator are the same as those of the Euclidean estimator, and in practice the tangent space estimator has been shown to yield variance reduction in example problems.
In a departure from the control variate formulations of the Euclidean and tangent-space estimators, our third multifidelity covariance estimator is defined as the solution to a regression problem on tangent spaces to product manifolds of SPD matrices. Given a set of high- and low-fidelity sample covariance matrices, which we view as a sample of a product-manifold-valued random variable, we estimate the underlying true covariance matrices by minimizing an intrinsic notion of squared Mahalanobis distance between the data and a model for its variation about its mean. The resulting estimates are guaranteeably positive definite and the Mahalanobis distance which they minimize has desirable properties, including tangent-space agnosticism and affine-invariance. Mahalanobis distance minimization can be carried out using unconstrained gradient-descent methods when a reparametrization in terms of SPD matrix square roots is employed, and we introduce a new Julia package, CovarianceRegression.jl, providing a convenient API for solving these multifidelity covariance regression problems. Using its machinery, we demonstrate that our estimator can provide significant reductions in MSE over single-fidelity covariance estimators in forward uncertainty quantification problems.
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