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1. A Symmetric Interior Penalty Method for an Elliptic Distributed Optimal Control Problem with Pointwise State Constraints

   https://doi.org/10.1515/cmam-2022-0148  

   Brenner, Susanne C; Gedicke, Joscha; Sung, Li-Yeng (December 2022, Computational Methods in Applied Mathematics)

   Abstract We construct a symmetric interior penalty method for an elliptic distributed optimal control problem with pointwise state constraints on general polygonal domains.The resulting discrete problems are quadratic programs with simple box constraints that can be solved efficiently by a primal-dual active set algorithm.Both theoretical analysis and corroborating numerical results are presented. 

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2. Structure Detection Method for Solving State-Variable Inequality Path Constrained Optimal Control Problems

   Byczkowski, Cale A.; Rao, Anil V. (January 2023, American Astronautical Society)

   A structure detection method is developed for solving state-variable inequality path con- strained optimal control problems. The method obtains estimates of activation and deactiva- tion times of active state-variable inequality path constraints (SVICs), and subsequently al- lows for the times to be included as decision variables in the optimization process. Once the identification step is completed, the method partitions the problem into a multiple-domain formulation consisting of constrained and unconstrained domains. Within each domain, Legendre-Gauss-Radau (LGR) orthogonal direct collocation is used to transcribe the infinite- dimensional optimal control problem into a finite-dimensional nonlinear programming (NLP) problem. Within constrained domains, the corresponding time derivative of the active SVICs that are explicit in the control are enforced as equality path constraints, and at the beginning of the constrained domains, the necessary tangency conditions are enforced. The accuracy of the proposed method is demonstrated on a well-known optimal control problem where the analytical solution contains a state constrained arc. 

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3. A C1 virtual element method for an elliptic distributed optimal control problem with pointwise state constraints

   https://doi.org/10.1142/S0218202521500640  

   Brenner, Susanne C.; Sung, Li-Yeng; Tan, Zhiyu (December 2021, Mathematical Models and Methods in Applied Sciences)

   We design and analyze a [Formula: see text] virtual element method for an elliptic distributed optimal control problem with pointwise state constraints. Theoretical estimates and corroborating numerical results are presented. 

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4. A Circuit-Theoretic Approach to State Estimation

   https://doi.org/10.1109/ISGT-Europe47291.2020.9248872  

   Li, Shimiao; Pandey, Amritanshu; Kar, Soummya; Pileggi, Larry (October 2020, IEEE PES Innovative Smart Grid Technologies Europe (ISGT-Europe))

   null (Ed.)

   Traditional state estimation (SE) methods that are based on nonlinear minimization of the sum of localized measurement error functionals are known to suffer from nonconvergence and large residual errors. In this paper we propose an equivalent circuit formulation (ECF)-based SE approach that inherently considers the complete network topology and associated physical constraints. We analyze the mathematical differences between the two approaches and show that our approach produces a linear state-estimator that is mathematically a quadratic programming (QP) problem with closed-form solution. Furthermore, this formulation imposes additional topology-based constraints that provably shrink the feasible region and promote convergence to a more physically meaningful solution. From a probabilistic viewpoint, we show that our method applies prior knowledge into the estimate, thus converging to a more physics-based estimate than the traditional observation-driven maximum likelihood estimator (MLE). Importantly, incorporation of the entire system topology and underlying physics, while being linear, makes ECF-based SE advantageous for large-scale systems. 

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5. Streaming Algorithms with Few State Changes

   https://doi.org/10.1145/3651145  

   Jayaram, Rajesh; Woodruff, David P; Zhou, Samson (May 2024, Proceedings of the ACM on Management of Data)

   In this paper, we study streaming algorithms that minimize the number of changes made to their internal state (i.e., memory contents). While the design of streaming algorithms typically focuses on minimizing space and update time, these metrics fail to capture the asymmetric costs, inherent in modern hardware and database systems, of reading versus writing to memory. In fact, most streaming algorithms write to their memory on every update, which is undesirable when writing is significantly more expensive than reading. This raises the question of whether streaming algorithms with small space and number of memory writes are possible. We first demonstrate that, for the fundamental F_pmoment estimation problem with p ≥ 1, any streaming algorithm that achieves a constant factor approximation must make Ω(n^1-1/p) internal state changes, regardless of how much space it uses. Perhaps surprisingly, we show that this lower bound can be matched by an algorithm which also has near-optimal space complexity. Specifically, we give a (1+ε)-approximation algorithm for F_pmoment estimation that use a near-optimal ~O_ε(n^1-1/p) number of state changes, while simultaneously achieving near-optimal space, i.e., for p∈[1,2), our algorithm uses poly(log n,1/ε) bits of space for, while for p>2, the algorithm uses ~O_ε(n^1-1/p) space. We similarly design streaming algorithms that are simultaneously near-optimal in both space complexity and the number of state changes for the heavy-hitters problem, sparse support recovery, and entropy estimation. Our results demonstrate that an optimal number of state changes can be achieved without sacrificing space complexity. 

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