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Data assimilation provides algorithms for widespread applications in various fields. It is of practical use to deal with a large amount of information in the complex system that is hard to estimate. Weather forecasting is one of the applications, where the prediction of meteorological data are corrected given the observations. Numerous approaches are contained in data assimilation. One specific sequential method is the Kalman Filter. The core is to estimate unknown information with the new data that is measured and the prior data that is predicted. As a matter of fact, there are different improved methods in the Kalman Filter. In this project, the Ensemble Kalman Filter with perturbed observations is considered. It is achieved by Monte Carlo simulation. In this method, the ensemble is involved in the calculation instead of the state vectors. In addition, the measurement with perturbation is viewed as the suitable observation. These changes compared with the Linear Kalman Filter make it more advantageous in that applications are not restricted in linear systems any more and less time is taken when the data are calculated by computers. The thesis seeks to develop the Ensemble Kalman Filter with perturbed observation gradually. With the Mathematical preliminaries including the introduction of dynamical systems, the Linear Kalman Filter is built. Meanwhile, the prediction and analysis processes are derived. After that, we use the analogy thoughts to lead in the non-linear Ensemble Kalman Filter with perturbed observations. Lastly, a classic Lorenz 63 model is illustrated by MATLAB. In the example, we experiment on the number of ensemble members and seek to investigate the relationships between the error of variance and the number of ensemble members. We reach the conclusion that on a limited scale the larger number of ensemble members indicates the smaller error of prediction.

The well-known Kalman filters model dynamical systems by relying on state-space representations with the next state updated, and its uncertainty controlled, by fresh information associated with newly observed system outputs. This paper generalizes, for the first time in the literature, Kalman and extended Kalman filters to discrete-time settings where inputs, states, and outputs are represented as attributed graphs whose topology and attributes can change with time. The setup allows us to adapt the framework to cases where the output is a vector or a scalar too (node/graph level tasks). Within the proposed theoretical framework, the unknown state-transition and the readout functions are learned end-to-end along with the downstream prediction task.

We detail how to use Newton's method for distortion-based curved $r$-adaption to a discrete high-order metric field while matching a target geometry. Specifically, we combine two terms: a distortion measuring the deviation from the target metric; and a penalty term measuring the deviation from the target boundary. For this combination, we consider four ingredients. First, to represent the metric field, we detail a log-Euclidean high-order metric interpolation on a curved (straight-edged) mesh. Second, for this metric interpolation, we detail the first and second derivatives in physical coordinates. Third, to represent the domain boundaries, we propose an implicit representation for 2D and 3D NURBS models. Fourth, for this implicit representation, we obtain the first and second derivatives. The derivatives of the metric interpolation and the implicit representation allow minimizing the objective function with Newton's method. For this second-order minimization, the resulting meshes simultaneously match the curved features of the target metric and boundary. Matching the metric and the geometry using second-order optimization is an unprecedented capability in curved (straight-edged) $r$-adaption. This capability will be critical in global and cavity-based curved (straight-edged) high-order mesh adaption.

An algorithm based on a deep probabilistic architecture referred to as a tree-structured sum-product network (t-SPN) is considered for cell classification. The t-SPN is constructed such that the unnormalized probability is represented as conditional probabilities of a subset of most similar cell classes. The constructed t-SPN architecture is learned by maximizing the margin, which is the difference in the conditional probability between the true and the most competitive false label. To enhance the generalization ability of the architecture, L2-regularization (REG) is considered along with the maximum margin (MM) criterion in the learning process. To highlight cell features, this paper investigates the effectiveness of two generic high-pass filters: ideal high-pass filtering and the Laplacian of Gaussian (LOG) filtering. On both HEp-2 and Feulgen benchmark datasets, the t-SPN architecture learned based on the max-margin criterion with regularization produced the highest accuracy rate compared to other state-of-the-art algorithms that include convolutional neural network (CNN) based algorithms. The ideal high-pass filter was more effective on the HEp-2 dataset, which is based on immunofluorescence staining, while the LOG was more effective on the Feulgen dataset, which is based on Feulgen staining.

We develop a unifying framework for interpolatory $\mathcal{L}_2$-optimal reduced-order modeling for a wide classes of problems ranging from stationary models to parametric dynamical systems. We first show that the framework naturally covers the well-known interpolatory necessary conditions for $\mathcal{H}_2$-optimal model order reduction and leads to the interpolatory conditions for $\mathcal{H}_2 \otimes \mathcal{L}_2$-optimal model order reduction of multi-input/multi-output parametric dynamical systems. Moreover, we derive novel interpolatory optimality conditions for rational discrete least-squares minimization and for $\mathcal{L}_2$-optimal model order reduction of a class of parametric stationary models. We show that bitangential Hermite interpolation appears as the main tool for optimality across different domains. The theoretical results are illustrated on two numerical examples.

Value iteration can find the optimal replenishment policy for a perishable inventory problem, but is computationally demanding due to the large state spaces that are required to represent the age profile of stock. The parallel processing capabilities of modern GPUs can reduce the wall time required to run value iteration by updating many states simultaneously. The adoption of GPU-accelerated approaches has been limited in operational research relative to other fields like machine learning, in which new software frameworks have made GPU programming widely accessible. We used the Python library JAX to implement value iteration and simulators of the underlying Markov decision processes in a high-level API, and relied on this library's function transformations and compiler to efficiently utilize GPU hardware. Our method can extend use of value iteration to settings that were previously considered infeasible or impractical. We demonstrate this on example scenarios from three recent studies which include problems with over 16 million states and additional problem features, such as substitution between products, that increase computational complexity. We compare the performance of the optimal replenishment policies to heuristic policies, fitted using simulation optimization in JAX which allowed the parallel evaluation of multiple candidate policy parameters on thousands of simulated years. The heuristic policies gave a maximum optimality gap of 2.49%. Our general approach may be applicable to a wide range of problems in operational research that would benefit from large-scale parallel computation on consumer-grade GPU hardware.

Recent developments in counter-adversarial system research have led to the development of inverse stochastic filters that are employed by a defender to infer the information its adversary may have learned. Prior works addressed this inverse cognition problem by proposing inverse Kalman filter (I-KF) and inverse extended KF (I-EKF), respectively, for linear and non-linear Gaussian state-space models. However, in practice, many counter-adversarial settings involve highly non-linear system models, wherein EKF's linearization often fails. In this paper, we consider the efficient numerical integration techniques to address such nonlinearities and, to this end, develop inverse cubature KF (I-CKF) and inverse quadrature KF (I-QKF). We derive the stochastic stability conditions for the proposed filters in the exponential-mean-squared-boundedness sense. Numerical experiments demonstrate the estimation accuracy of our I-CKF and I-QKF with the recursive Cram\'{e}r-Rao lower bound as a benchmark.

The flock-guidance problem enjoys a challenging structure where multiple optimization objectives are solved simultaneously. This usually necessitates different control approaches to tackle various objectives, such as guidance, collision avoidance, and cohesion. The guidance schemes, in particular, have long suffered from complex tracking-error dynamics. Furthermore, techniques that are based on linear feedback strategies obtained at equilibrium conditions either may not hold or degrade when applied to uncertain dynamic environments. Pre-tuned fuzzy inference architectures lack robustness under such unmodeled conditions. This work introduces an adaptive distributed technique for the autonomous control of flock systems. Its relatively flexible structure is based on online fuzzy reinforcement learning schemes which simultaneously target a number of objectives; namely, following a leader, avoiding collision, and reaching a flock velocity consensus. In addition to its resilience in the face of dynamic disturbances, the algorithm does not require more than the agent position as a feedback signal. The effectiveness of the proposed method is validated with two simulation scenarios and benchmarked against a similar technique from the literature.

An algorithm based on a deep probabilistic architecture referred to as a tree-structured sum-product network (t-SPN) is considered for cell classification. The t-SPN is constructed such that the unnormalized probability is represented as conditional probabilities of a subset of most similar cell classes. The constructed t-SPN architecture is learned by maximizing the margin, which is the difference in the conditional probability between the true and the most competitive false label. To enhance the generalization ability of the architecture, L2-regularization (REG) is considered along with the maximum margin (MM) criterion in the learning process. To highlight cell features, this paper investigates the effectiveness of two generic high-pass filters: ideal high-pass filtering and the Laplacian of Gaussian (LOG) filtering. On both HEp-2 and Feulgen benchmark datasets, the t-SPN architecture learned based on the max-margin criterion with regularization produced the highest accuracy rate compared to other state-of-the-art algorithms that include convolutional neural network (CNN) based algorithms. The ideal high-pass filter was more effective on the HEp-2 dataset, which is based on immunofluorescence staining, while the LOG was more effective on the Feulgen dataset, which is based on Feulgen staining.

Dynamic programming (DP) solves a variety of structured combinatorial problems by iteratively breaking them down into smaller subproblems. In spite of their versatility, DP algorithms are usually non-differentiable, which hampers their use as a layer in neural networks trained by backpropagation. To address this issue, we propose to smooth the max operator in the dynamic programming recursion, using a strongly convex regularizer. This allows to relax both the optimal value and solution of the original combinatorial problem, and turns a broad class of DP algorithms into differentiable operators. Theoretically, we provide a new probabilistic perspective on backpropagating through these DP operators, and relate them to inference in graphical models. We derive two particular instantiations of our framework, a smoothed Viterbi algorithm for sequence prediction and a smoothed DTW algorithm for time-series alignment. We showcase these instantiations on two structured prediction tasks and on structured and sparse attention for neural machine translation.