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Register a new userEnsemble Riemannian Data Assimilation over the Wasserstein Space
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Added by
Sagar Kumar Tamang
Publication date
2020
fields
Mathematical Statistics
and research's language is
English
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In this paper, we present an ensemble data assimilation paradigm over a Riemannian manifold equipped with the Wasserstein metric. Unlike the Eulerian penalization of error in the Euclidean space, the Wasserstein metric can capture translation and difference between the shapes of square-integrable probability distributions of the background state and observations -- enabling to formally penalize geophysical biases in state-space with non-Gaussian distributions. The new approach is applied to dissipative and chaotic evolutionary dynamics and its potential advantages and limitations are highlighted compared to the classic variational and filtering data assimilation approaches under systematic and random errors.
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This paper presents a new variational data assimilation (VDA) approach for the formal treatment of bias in both model outputs and observations. This approach relies on the Wasserstein metric stemming from the theory of optimal mass transport to penalize the distance between the probability histograms of the analysis state and an a priori reference dataset, which is likely to be more uncertain but less biased than both model and observations. Unlike previous bias-aware VDA approaches, the new Wasserstein metric VDA (WM-VDA) dynamically treats systematic biases of unknown magnitude and sign in both model and observations through assimilation of the reference data in the probability domain and can fully recover the probability histogram of the analysis state. The performance of WM-VDA is compared with the classic three-dimensional VDA (3D-Var) scheme on first-order linear dynamics and the chaotic Lorenz attractor. Under positive systematic biases in both model and observations, we consistently demonstrate a significant reduction in the forecast bias and unbiased root mean squared error.
In this paper, we comparatively analyze the Bures-Wasserstein (BW) geometry with the popular Affine-Invariant (AI) geometry for Riemannian optimization on the symmetric positive definite (SPD) matrix manifold. Our study begins with an observation that the BW metric has a linear dependence on SPD matrices in contrast to the quadratic dependence of the AI metric. We build on this to show that the BW metric is a more suitable and robust choice for several Riemannian optimization problems over ill-conditioned SPD matrices. We show that the BW geometry has a non-negative curvature, which further improves convergence rates of algorithms over the non-positively curved AI geometry. Finally, we verify that several popular cost functions, which are known to be geodesic convex under the AI geometry, are also geodesic convex under the BW geometry. Extensive experiments on various applications support our findings.
When considering functional principal component analysis for sparsely observed longitudinal data that take values on a nonlinear manifold, a major challenge is how to handle the sparse and irregular observations that are commonly encountered in longitudinal studies. Addressing this challenge, we provide theory and implementations for a manifold version of the principal analysis by conditional expectation (PACE) procedure that produces representations intrinsic to the manifold, extending a well-established version of functional principal component analysis targeting sparsely sampled longitudinal data in linear spaces. Key steps are local linear smoothing methods for the estimation of a Frechet mean curve, mapping the observed manifold-valued longitudinal data to tangent spaces around the estimated mean curve, and applying smoothing methods to obtain the covariance structure of the mapped data. Dimension reduction is achieved via representations based on the first few leading principal components. A finitely truncated representation of the original manifold-valued data is then obtained by mapping these tangent space representations to the manifold. We show that the proposed estimates of mean curve and covariance structure achieve state-of-the-art convergence rates. For longitudinal emotional well-being data for unemployed workers as an example of time-dynamic compositional data that are located on a sphere, we demonstrate that our methods lead to interpretable eigenfunctions and principal component scores. In a second example, we analyze the body shapes of wallabies by mapping the relative size of their body parts onto a spherical pre-shape space. Compared to standard functional principal component analysis, which is based on Euclidean geometry, the proposed approach leads to improved trajectory recovery for sparsely sampled data on nonlinear manifolds.
Model uncertainty quantification is an essential component of effective data assimilation. Model errors associated with sub-grid scale processes are often represented through stochastic parameterizations of the unresolved process. Many existing Stochastic Parameterization schemes are only applicable when knowledge of the true sub-grid scale process or full observations of the coarse scale process are available, which is typically not the case in real applications. We present a methodology for estimating the statistics of sub-grid scale processes for the more realistic case that only partial observations of the coarse scale process are available. Model error realizations are estimated over a training period by minimizing their conditional sum of squared deviations given some informative covariates (e.g. state of the system), constrained by available observations and assuming that the observation errors are smaller than the model errors. From these realizations a conditional probability distribution of additive model errors given these covariates is obtained, allowing for complex non-Gaussian error structures. Random draws from this density are then used in actual ensemble data assimilation experiments. We demonstrate the efficacy of the approach through numerical experiments with the multi-scale Lorenz 96 system using both small and large time scale separations between slow (coarse scale) and fast (fine scale) variables. The resulting error estimates and forecasts obtained with this new method are superior to those from two existing methods.
Through ensemble-based data assimilation (DA), we address one of the most notorious difficulties in phase-resolved ocean wave forecast, regarding the deviation of numerical solution from the true surface elevation due to the chaotic nature of and underrepresented physics in the nonlinear wave models. In particular, we develop a coupled approach of the high-order spectral (HOS) method with the ensemble Kalman filter (EnKF), through which the measurement data can be incorporated into the simulation to improve the forecast performance. A unique feature in this coupling is the mismatch between the predictable zone and measurement region, which is accounted for through a special algorithm to modify the analysis equation in EnKF. We test the performance of the new EnKF-HOS method using both synthetic data and real radar measurements. For both cases (though differing in details), it is shown that the new method achieves much higher accuracy than the HOS-only method, and can retain the phase information of an irregular wave field for arbitrarily long forecast time with sequentially assimilated data.
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