[Paper Review] Multivariate extensions of the Multilevel Best Linear Unbiased Estimator for ensemble-variational data assimilation

Mayeul Destouches, Paul Mycek+1 more|arXiv (Cornell University)|Jun 12, 2023

Soil Geostatistics and Mapping14 references8 citations

TL;DR

This technical report extends MLBLUE to multidimensional cases, proposing several multilevel estimators for random vectors and covariance matrices, and discusses optimal weighting, localization, and MOSAP under variance minimization.

ABSTRACT

Multilevel estimators aim at reducing the variance of Monte Carlo statistical estimators, by combining samples generated with simulators of different costs and accuracies. In particular, the recent work of Schaden and Ullmann (2020) on the multilevel best linear unbiased estimator (MLBLUE) introduces a framework unifying several multilevel and multifidelity techniques. The MLBLUE is reintroduced here using a variance minimization approach rather than the regression approach of Schaden and Ullmann. We then discuss possible extensions of the scalar MLBLUE to a multidimensional setting, i.e. from the expectation of scalar random variables to the expectation of random vectors. Several estimators of increasing complexity are proposed: a) multilevel estimators with scalar weights, b) with element-wise weights, c) with spectral weights and d) with general matrix weights. The computational cost of each method is discussed. We finally extend the MLBLUE to the estimation of second-order moments in the multidimensional case, i.e. to the estimation of covariance matrices. The multilevel estimators proposed are d) a multilevel estimator with scalar weights and e) with element-wise weights. In large-dimension applications such as data assimilation for geosciences, the latter estimator is computationnally unaffordable. As a remedy, we also propose f) a multilevel covariance matrix estimator with optimal multilevel localization, inspired by the optimal localization theory of Ménétrier and Auligné (2015). Some practical details on weighted MLMC estimators of covariance matrices are given in appendix.

Motivation & Objective

• Motivate variance-minimizing construction of multilevel estimators for biased-free estimation at multiple fidelities.
• Extend MLBLUE from scalar expectations to random vectors and covariance matrices.
• Propose estimators with scalar, field, and matrix weights, including change of basis and localization approaches.
• Discuss computational cost, sample allocation, and model selection under budget constraints for large-scale geoscience applications.

Proposed method

• Reformulate MLBLUE via variance minimization under a no-bias constraint to derive optimal weights.
• Generalize the mean estimation to the expectation of random vectors and derive a multidimensional MLBLUE with field and matrix weights.
• Introduce scalar weights, field weights, and field weights with change of basis to handle high-dimensional outputs.
• Extend the framework to covariance estimation and covariance matrices with localization inspired by optimal localization theory.
• Present a MOSAP approach and semidefinite programming formulation for model selection under budget constraints.

Experimental results

Research questions

• RQ1 How can the MLBLUE framework be extended from scalar expectations to random vectors and their covariance structures?
• RQ2 What are the optimal weighting schemes (scalar, field, and matrix) for multidimensional MLBLUE under unbiasedness constraints?
• RQ3 How can covariance estimation be integrated within MLBLUE, including localization and computationally feasible strategies for large dimensions?
• RQ4 What are the cost-aware sample allocation and model selection methods that preserve unbiasedness and minimize estimator variance?
• RQ5 Can multilevel localization be optimally designed for covariance matrices in geoscience data assimilation?

Key findings

• A variance-minimizing derivation of MLBLUE yields explicit optimal weights for multidimensional estimators.
• Scalar, field, and matrix weighting schemes extend MLBLUE to random vectors and allow space-dependent and basis-transformed estimators.
• Covariance estimation and localization can be incorporated, with an approach that scales linearly with dimension for practical large-scale problems.
• A practical MOSAP formulation via semidefinite programming enables budget-constrained model selection and sample allocation.
• The framework supports estimation of scalar expectations, random vectors, and covariance matrices with provable convexity properties of the variance.