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 MCMC, C HMC, MC-C HMC, and MC-C HMC are presented. C MCMC uses a Gaussian proposal density to sample the posterior, and C HMC is an extension to the Hamiltonian Monte-Carlo (HMC) sampling filter. MC-C MCMC and MC-C HMC are multi-chain versions of the cluster sampling filters C MCMC and C 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.

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This paper introduces multivariate input-output models to predict the errors and bases dimensions of local parametric Proper Orthogonal Decomposition reduced-order models. We refer to these mappings as the multivariate predictions of local reduced-order model characteristics (MP-LROM) models. We use Gaussian processes and artificial neural networks to construct approximations of these multivariate mappings. Numerical results with a viscous Burgers model illustrate the performance and potential of the machine learning-based regression MP-LROM models to approximate the characteristics of parametric local reduced-order models. The predicted reduced-order models errors are compared against the multifidelity correction and reduced-order model error surrogates methods predictions, whereas the predicted reduced-order dimensions are tested against the standard method based on the spectrum of snapshots matrix. Since the MP-LROM models incorporate more features and elements to construct the probabilistic mappings, they achieve more accurate results.However, for high-dimensional parametric spaces, the MP-LROM models might suffer from the curse of dimensionality. Scalability challenges of MP-LROM models and the feasible ways of addressing them are also discussed in this study. KEYWORDSlocal reduced-order models, proper orthogonal decomposition, regression machine learning techniques 512 How to cite this article:Moosavi A,Ştefȃnescu R, Sandu A. Multivariate predictions of local reduced-order-model errors and dimensions. Int J Numer Meth Engng. 2018;113:512-533.

Physics-based computer models, such as fluid flow simulations, seek to approximate the behavior of a real system based on the physical equations that govern the evolution of that system. The model approximation of reality, however, is imperfect because it is subject to uncertainties coming from different sources: finite model resolution, uncertainty in model parameter values, uncertainty in input data such as external forgings, and uncertainty in the structure of the model itself. Many studies to date have considered the effects of parameter and data uncertainty on model outputs, and have offered solutions to obtain the best fitted parameter values for a model. However, much less effort has been devoted to the study of structural uncertainty, which is caused by our incomplete knowledge about the true physical processes, and manifests itself as missing dynamics in the model. This paper seeks to understand structural uncertainty by studying the observable errors, i.e. the discrepancies between the model solutions and measurements of the physical system. The dynamics of these errors is modeled using a state-space approach, which enables one to identify the source of uncertainty and to recognize the missing dynamics inside model. Furthermore, the model solution can be improved by correcting it with the error predicted by the state-space approach. The proposed methodology is applied to two test problems, Lorenz-96 and a stratospheric chemistry model.