This paper introduces tensorial calculus techniques in the framework of POD to reduce the computational complexity of the reduced nonlinear terms. The resulting method, named tensorial POD, can be applied to polynomial nonlinearities of any degree p. Such nonlinear terms have an online complexity of O.k pC1 /, where k is the dimension of POD basis and therefore is independent of full space dimension. However, it is efficient only for quadratic nonlinear terms because for higher nonlinearities, POD model proves to be less time consuming once the POD basis dimension k is increased. Numerical experiments are carried out with a two-dimensional SWE test problem to compare the performance of tensorial POD, POD, and POD/discrete empirical interpolation method (DEIM). Numerical results show that tensorial POD decreases by 76 the computational cost of the online stage of POD model for configurations using more than 300,000 model variables. The tensorial POD SWE model was only 2 to 8 slower than the POD/DEIM SWE model but the implementation effort is considerably increased. Tensorial calculus was again employed to construct a new algorithm allowing POD/DEIM SWE model to compute its offline stage faster than POD and tensorial POD approaches. 498 R.ŞTEFȂNESCU, A. SANDU AND I. M. NAVON truncation does not extend easily for high-order systems, and several grammians approximations were proposed leading to methods such as approximate subspace iteration [9], least squares approximation [10], Krylov subspace methods [11,12], and balanced POD [13]. Among moment matching methods, we mention partial realization [14,15], Padé approximation [16][17][18][19], and rational approximation [20].Although for linear models we are able to produce input-independent highly accurate reduced models, in the case of general nonlinear systems, the transfer function approach is not yet applicable and input-specified semi-empirical methods are usually employed. Recently, some encouraging research results using generalized transfer functions and generalized moment matching have been obtained by [21] for nonlinear model order reduction but future investigations are required.Proper orthogonal decomposition and its variants are also known as Karhunen-Loève expansions [22,23], principal component analysis [24], and empirical orthogonal functions [25] among others. It is the most prevalent basis selection method for nonlinear problems and, among other requirements, relies on the fact that the desired simulation is well simulated in the input collection. Data analysis using POD is conducted to extract basis functions from experimental data or detailed simulations of high-dimensional systems (method of snapshots introduced by [26-28]) for subsequent use in Galerkin projections that yield low-dimensional dynamical models. Unfortunately, the POD Galerkin approach displays a major disadvantage because its nonlinear reduced terms still have to be evaluated on the original state space making the simulation of the reduced order system too expensive. There exist seve...

Discrete empirical interpolation method (DEIM) Reduced-order models (ROMs) Shallow water equations Finite difference methodsThis work studies reduced order modeling (ROM) approaches to speed up the solution of variational data assimilation problems with large scale nonlinear dynamical models. It is shown that a key requirement for a successful reduced order solution is that reduced order Karush-Kuhn-Tucker conditions accurately represent their full order counterparts. In particular, accurate reduced order approximations are needed for the forward and adjoint dynamical models, as well as for the reduced gradient. New strategies to construct reduced order based are developed for proper orthogonal decomposition (POD) ROM data assimilation using both Galerkin and Petrov-Galerkin projections. For the first time POD, tensorial POD, and discrete empirical interpolation method (DEIM) are employed to develop reduced data assimilation systems for a geophysical flow model, namely, the two dimensional shallow water equations. Numerical experiments confirm the theoretical framework for Galerkin projection. In the case of Petrov-Galerkin projection, stabilization strategies must be considered for the reduced order models. The new reduced order shallow water data assimilation system provides analyses similar to those produced by the full resolution data assimilation system in one tenth of the computational time.

SUMMARYHybrid Monte Carlo 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 owing to the intrinsic requirement of running the forward and adjoint models repeatedly. Here we present computationally efficient versions of the hybrid Monte Carlo 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.

In this investigation, we propose several algorithms to recover the location and intensity of a radiation source located in a simulated 250 m × 180 m block in an urban center based on synthetic measurements. Radioactive decay and detection are Poisson random processes, so we employ likelihood functions based on this distribution. Due to the domain geometry and the proposed response model, the negative logarithm of the likelihood is only piecewise continuous differentiable, and it has multiple local minima. To address these difficulties, we investigate three hybrid algorithms comprised of mixed optimization techniques. For global optimization, we consider Simulated Annealing (SA), Particle Swarm (PS) and Genetic Algorithm (GA), which rely solely on objective function evaluations; i.e., they do not evaluate the gradient in the objective function. By employing early stopping criteria for the global optimization methods, a pseudo-optimum point is obtained. This is subsequently utilized as the initial value by the deterministic Implicit Filtering method (IF), which is able to find local extrema in non-smooth functions, to finish the search in a narrow domain. These new hybrid techniques combining global optimization and Implicit Filtering address difficulties associated with the non-smooth response, and their performances are shown to significantly decrease the computational time over the global optimization methods alone. To quantify uncertainties associated with the source location and intensity, we employ the Delayed Rejection Adaptive Metropolis (DRAM) and DiffeRential Evolution Adaptive Metropolis (DREAM) algorithms. Marginal densities of the source properties are obtained, and the means of the chains' compare accurately with the estimates produced by the hybrid algorithms.

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