arXiv subjects

Anthony T. Weaver

Publications and source records attributed to Anthony T. Weaver.

4 recordsLinked to original sources

arXiv2026

Block Alpha-Circulant Preconditioners for All-at-Once Diffusion-Based Covariance Operators

Covariance matrices are central to data assimilation and inverse methods derived from statistical estimation theory. Previous work has considered the application of an all-at-once diffusion-based representation of a covariance matrix operator in order to exploit inherent parallelism in the underlying problem. In this paper, we provide practical methods to apply block $α$-circulant preconditioners to the all-at-once system for the case where the main diffusion operation matrix cannot be readily diagonalized using a discrete Fourier transform. Our new framework applies the block $α$-circulant preconditioner approximately by solving an inner block diagonal problem via a choice of inner iterative approaches. Our first method applies Chebyshev semi-iteration to a symmetric positive definite matrix, shifted by a complex scaling of the identity. We extend theoretical results for Chebyshev semi-iteration in the symmetric positive definite setting, to obtain computable bounds on the asymptotic convergence factor for each of the complex sub-problems. The second approach transforms the complex sub-problem into a (generalized) saddle point system with real coefficients. Numerical experiments reveal that in the case of unlimited computational resources, both methods can match the iteration counts of the `best-case' block $α$-circulant preconditioner. We also provide a practical adaptation to the nested Chebyshev approach, which improves performance in the case of a limited computational budget. Using an appropriate choice of $α$ our new approaches are robust and efficient in terms of outer iterations and matrix--vector products.

math.NA↗

arXiv2025

A filtered multilevel Monte Carlo method for estimating the expectation of cell-centered discretized random fields

In this paper, we investigate the use of multilevel Monte Carlo (MLMC) methods for estimating the expectation of discretized random fields. Specifically, we consider a setting in which the input and output vectors of numerical simulators have inconsistent dimensions across the multilevel hierarchy. This motivates the introduction of grid transfer operators borrowed from multigrid methods. By adapting mathematical tools from multigrid methods, we perform a theoretical spectral analysis of the MLMC estimator of the expectation of discretized random fields, in the specific case of linear, symmetric and circulant simulators. We then propose filtered MLMC (F-MLMC) estimators based on a filtering mechanism similar to the smoothing process of multigrid methods, and we show that the filtering operators improve the estimation of both the small- and large-scale components of the variance, resulting in a reduction of the total variance of the estimator. Next, the conclusions of the spectral analysis are experimentally verified with a one-dimensional illustration. Finally, the proposed F-MLMC estimator is applied to the problem of estimating the discretized variance field of a diffusion-based covariance operator, which amounts to estimating the expectation of a discretized random field. The numerical experiments support the conclusions of the theoretical analysis even with non-linear simulators, and demonstrate the improvements brought by the F-MLMC estimator compared to both a crude MC and an unfiltered MLMC estimator.

math.NA↗

arXiv2024

Impact of an Ensemble of Ocean Data Assimilations in ECMWF's next generation ocean reanalysis system

An Ensemble of Data Assimilations (EDA) can provide valuable information on the analysis and short-range forecast uncertainties. The present ECMWF operational ocean analysis and reanalysis system, called ORAS5, produces an ensemble but does not exploit it for the specification of the background-error covariance matrix $\mathbf{B}$, a key component of the data assimilation system. In this article, we describe EDA developments for the ocean, which take advantage of the short-range forecast ensemble for specifying, in two distinct ways, parameters of a covariance model representation of $\mathbf{B}$. First, we generate a climatological ensemble over an extended period to produce seasonally varying climatological estimates of background-error variances and horizontal correlation length-scales. Second, on each assimilation cycle, we diagnose flow-dependent variances from the ensemble and blend them with the climatological estimates to form hybrid variances. We also use the ensemble to diagnose flow-dependent vertical correlation length-scales. We demonstrate for the Argo-rich period that this new, hybrid formulation of $\mathbf{B}$ results in a significant reduction of background errors compared to the parameterized formulation of $\mathbf{B}$ used in ORAS5. The new ocean EDA system will be employed in ORAS6, ECMWF's next generation ocean reanalysis system.

physics.ao-ph↗

arXiv2022

Impact of correlated observation errors on the convergence of the conjugate gradient algorithm in variational data assimilation

An important class of nonlinear weighted least-squares problems arises from the assimilation of observations in atmospheric and ocean models. In variational data assimilation, inverse error covariance matrices define the weighting matrices of the least-squares problem. For observation errors, a diagonal matrix (i.e., uncorrelated errors) is often assumed for simplicity even when observation errors are suspected to be correlated. While accounting for observationerror correlations should improve the quality of the solution, it also affects the convergence rate of the minimization algorithms used to iterate to the solution. If the minimization process is stopped before reaching full convergence, which is usually the case in operational applications, the solution may be degraded even if the observation-error correlations are correctly accounted for. In this article, we explore the influence of the observation-error correlation matrix (R) on the convergence rate of a preconditioned conjugate gradient (PCG) algorithm applied to a one-dimensional variational data assimilation (1D-Var) problem. We design the idealised 1D-Var system to include two key features used in more complex systems: we use the background error covariance matrix (B) as a preconditioner (B-PCG); and we use a diffusion operator to model spatial correlations in B and R. Analytical and numerical results with the 1D-Var system show a strong sensitivity of the convergence rate of B-PCG to the parameters of the diffusion-based correlation models. Depending on the parameter choices, correlated observation errors can either speed up or slow down the convergence. In practice, a compromise may be required in the parameter specifications of B and R between staying close to the best available estimates on the one hand and ensuring an adequate convergence rate of the minimization algorithm on the other.

math.NA↗