Computational Science Laboratory
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Browsing Computational Science Laboratory by Author "Attia, Ahmed"
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- Cluster Sampling Filters for Non-Gaussian Data AssimilationAttia, Ahmed; Moosavi, Azam; Sandu, Adrian (MDPI, 2018-05-31)This paper presents a fully non-Gaussian filter for sequential data assimilation. The filter is named the “cluster sampling filter”, and works by directly sampling the posterior distribution following a Markov Chain Monte-Carlo (MCMC) approach, while the prior distribution is approximated using a Gaussian Mixture Model (GMM). Specifically, a clustering step is introduced after the forecast phase of the filter, and the prior density function is estimated by fitting a GMM to the prior ensemble. Using the data likelihood function, the posterior density is then formulated as a mixture density, and is sampled following an MCMC approach. Four versions of the proposed filter, namely C ℓ MCMC , C ℓ HMC , MC- C ℓ HMC , and MC- C ℓ HMC are presented. C ℓ MCMC uses a Gaussian proposal density to sample the posterior, and C ℓ HMC is an extension to the Hamiltonian Monte-Carlo (HMC) sampling filter. MC- C ℓ MCMC and MC- C ℓ HMC are multi-chain versions of the cluster sampling filters C ℓ MCMC and C ℓ HMC respectively. The multi-chain versions are proposed to guarantee that samples are taken from the vicinities of all probability modes of the formulated posterior. The new methodologies are tested using a simple one-dimensional example, and a quasi-geostrophic (QG) model with double-gyre wind forcing and bi-harmonic friction. Numerical results demonstrate the usefulness of using GMMs to relax the Gaussian prior assumption especially in the HMC filtering paradigm.
- The reduced-order hybrid Monte Carlo sampling smootherAttia, Ahmed; Stefanescu, Razvan; Sandu, Adrian (Wiley-Blackwell, 2017-01-10)Hybrid Monte-Carlo (HMC) sampling smoother is a fully non-Gaussian four-dimensional data assimilation algorithm that works by directly sampling the posterior distribution formulated in the Bayesian framework. The smoother in its original formulation is computationally expensive due to the intrinsic requirement of running the forward and adjoint models repeatedly. Here we present computationally efficient versions of the HMC sampling smoother based on reduced-order approximations of the underlying model dynamics. The schemes developed herein are tested numerically using the shallow-water equations model on Cartesian coordinates. The results reveal that the reduced-order versions of the smoother are capable of accurately capturing the posterior probability density, while being significantly faster than the original full order formulation.
- A Sampling Filter for Non-Gaussian Data AssimilationAttia, Ahmed; Sandu, Adrian (2014-12-09)Data assimilation combines information from models, measurements, and priors to estimate the state of a dynamical system such as the atmosphere. The Ensemble Kalman filter (EnKF) is a family of ensemble-based data assimilation approaches that has gained wide popularity due its simple formulation, ease of implementation, and good practical results. Most EnKF algorithms assume that the underlying probability distributions are Gaussian. Although this assumption is well accepted, it is too restrictive when applied to large nonlinear models, nonlinear observation operators, and large levels of uncertainty. Several approaches have been proposed in order to avoid the Gaussianity assumption. One of the most successful strategies is the maximum likelihood ensemble filter (MLEF) which computes a maximum a posteriori estimate of the state assuming the posterior distribution is Gaussian. MLEF is designed to work with nonlinear and even non-differentiable observation operators, and shows good practical performance. However, there are limits to the degree of nonlinearity that MLEF can handle. This paper proposes a new ensemble-based data assimilation method, named the "sampling filter", which obtains the analysis by sampling directly from the posterior distribution. The sampling strategy is based on a Hybrid Monte Carlo (HMC) approach that can handle non-Gaussian probability distributions. Numerical experiments are carried out using the Lorenz-96 model and observation operators with different levels of non-linearity and differentiability. The proposed filter is also tested with shallow water model on a sphere with linear observation operator. The results show that the sampling filter can perform well even in highly nonlinear situations were EnKF and MLEF filters diverge.