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1. Extension of switch point algorithm to boundary-value problems

   https://doi.org/10.1007/s10589-023-00530-y  

   Hager, William W. ( December 2023 , Computational Optimization and Applications)

   In an earlier paper (https://doi.org/10.1137/21M1393315), the switch point algorithm was developed for solving optimal control problems whose solutions are either singular or bang-bang or both singular and bang-bang, and which possess a finite number of jump discontinuities in an optimal control at the points in time where the solution structure changes. The class of control problems that were considered had a given initial condition, but no terminal constraint. The theory is now extended to include problems with both initial and terminal constraints, a structure that often arises in boundary-value problems. Substantial changes to the theory are needed to handle this more general setting. Nonetheless, the derivative of the cost with respect to a switch point is again the jump in the Hamiltonian at the switch point. 

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2. Connecting Hamilton-Jacobi Partial Differential Equations with Maximum a Posteriori and Posterior Mean Estimators for Some Non-convex Priors.

   https://doi.org/10.1007/978-3-030-03009-4_56-1  

   Darbon, J. ; Langlois, G.P. ; Meng, T. ( May 2021 , Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging. Springer,)

   null ; null ; null ; null (Ed.)

   Many imaging problems can be formulated as inverse problems expressed as finite-dimensional optimization problems. These optimization problems generally consist of minimizing the sum of a data fidelity and regularization terms. In Darbon (SIAM J. Imag. Sci. 8:2268–2293, 2015), Darbon and Meng, (On decomposition models in imaging sciences and multi-time Hamilton-Jacobi partial differential equations, arXiv preprint arXiv:1906.09502, 2019), connections between these optimization problems and (multi-time) Hamilton-Jacobi partial differential equations have been proposed under the convexity assumptions of both the data fidelity and regularization terms. In particular, under these convexity assumptions, some representation formulas for a minimizer can be obtained. From a Bayesian perspective, such a minimizer can be seen as a maximum a posteriori estimator. In this chapter, we consider a certain class of non-convex regularizations and show that similar representation formulas for the minimizer can also be obtained. This is achieved by leveraging min-plus algebra techniques that have been originally developed for solving certain Hamilton-Jacobi partial differential equations arising in optimal control. Note that connections between viscous Hamilton-Jacobi partial differential equations and Bayesian posterior mean estimators with Gaussian data fidelity terms and log-concave priors have been highlighted in Darbon and Langlois, (On Bayesian posterior mean estimators in imaging sciences and Hamilton-Jacobi partial differential equations, arXiv preprint arXiv:2003.05572, 2020). We also present similar results for certain Bayesian posterior mean estimators with Gaussian data fidelity and certain non-log-concave priors using an analogue of min-plus algebra techniques. 

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3. Highly accurate operator factorization methods for the integral fractional Laplacian and its generalization

   https://doi.org/10.3934/dcdss.2022016  

   Wu, Yixuan ; Zhang, Yanzhi ( January 2022 , Discrete & Continuous Dynamical Systems - S)

   In this paper, we propose a new class of operator factorization methods to discretize the integral fractional Laplacian \begin{document}$ (- \Delta)^\frac{{ \alpha}}{{2}} $\end{document} for \begin{document}$ \alpha \in (0, 2) $\end{document}. One main advantage is that our method can easily increase numerical accuracy by using high-degree Lagrange basis functions, but remain its scheme structure and computer implementation unchanged. Moreover, it results in a symmetric (multilevel) Toeplitz differentiation matrix, enabling efficient computation via the fast Fourier transforms. If constant or linear basis functions are used, our method has an accuracy of \begin{document}$ {\mathcal O}(h^2) $\end{document}, while \begin{document}$ {\mathcal O}(h^4) $\end{document} for quadratic basis functions with \begin{document}$ h $\end{document} a small mesh size. This accuracy can be achieved for any \begin{document}$ \alpha \in (0, 2) $\end{document} and can be further increased if higher-degree basis functions are chosen. Numerical experiments are provided to approximate the fractional Laplacian and solve the fractional Poisson problems. It shows that if the solution of fractional Poisson problem satisfies \begin{document}$ u \in C^{m, l}(\bar{ \Omega}) $\end{document} for \begin{document}$ m \in {\mathbb N} $\end{document} and \begin{document}$ 0 < l < 1 $\end{document}, our method has an accuracy of \begin{document}$ {\mathcal O}(h^{\min\{m+l, \, 2\}}) $\end{document} for constant and linear basis functions, while \begin{document}$ {\mathcal O}(h^{\min\{m+l, \, 4\}}) $\end{document} for quadratic basis functions. Additionally, our method can be readily applied to approximate the generalized fractional Laplacians with symmetric kernel function, and numerical study on the tempered fractional Poisson problem demonstrates its efficiency.

    

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4. Optimal scheduling of multiple sensors which transmit measurements over a dynamic lossy network

   https://doi.org/10.1109/CDC40024.2019.9029779  

   Carroll, Johnson ; Hmedi, Hassan ; Arapostathis, Ari ( December 2019 , 2019 IEEE 58th Conference on Decision and Control (CDC))

   Motivated by various distributed control applications, we consider a linear system with Gaussian noise observed by multiple sensors which transmit measurements over a dynamic lossy network. We characterize the stationary optimal sensor scheduling policy for the finite horizon, discounted, and long-term average cost problems and show that the value iteration algorithm converges to a solution of the average cost problem. We further show that the suboptimal policies provided by the rolling horizon truncation of the value iteration also guarantee geometric ergodicity and provide near-optimal average cost. Lastly, we provide qualitative characterizations of the multidimensional set of measurement loss rates for which the system is stabilizable for a static network, significantly extending earlier results on intermittent observations. 

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5. Finite-horizon covariance control of linear time-varying systems

   https://doi.org/10.1109/CDC.2017.8264189  

   Goldshtein, Maxim ; Tsiotras, Panagiotis ( December 2017 , Conference on Decision and Control)

   We consider the problem of finite-horizon optimal control of a discrete linear time-varying system subject to a stochastic disturbance and fully observable state. The initial state of the system is drawn from a known Gaussian distribution, and the final state distribution is required to reach a given target Gaussian distribution, while minimizing the expected value of the control effort. We derive the linear optimal control policy by first presenting an efficient solution for the diffusion-less case, and we then solve the case with diffusion by reformulating the system as a superposition of diffusion-less systems. We show that the resulting solution coincides with a LQG problem with particular terminal cost weight matrix. 

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