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1. Numerical efficacy study of data assimilation for the 2D magnetohydrodynamic equations

   https://doi.org/https://doi.org/10.3934/jcd.2019006  

   Hudson, Joshua; Jolly, Michael (July 2019, Journal of computational dynamics)

   We study the computational efficiency of several nudging data assimilation algorithms for the 2D magnetohydrodynamic equations, using varying amounts and types of data. We find that the algorithms work with much less resolution in the data than required by the rigorous estimates in [7]. We also test other abridged nudging algorithms to which the analytic techniques in [7] do not seem to apply. These latter tests indicate, in particular, that velocity data alone is sufficient for synchronization with a chaotic reference solution, while magnetic data alone is not. We demonstrate that a new nonlinear nudging algorithm, which is adaptive in both time and space, synchronizes at a super exponential rate. [7] A. Biswas, J. Hudson, A. Larios and Y. Pei, Continuous data assimilation for the 2D magnetohydrodynamic equations using one component of the velocity and magnetic fields, Asymptot. Anal., 108 (2018), 1-43. 

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2. Efficient quantum algorithm for dissipative nonlinear differential equations

   https://doi.org/10.1073/pnas.2026805118  

   Liu, Jin-Peng; Kolden, Herman Øie; Krovi, Hari K.; Loureiro, Nuno F.; Trivisa, Konstantina; Childs, Andrew M. (August 2021, Proceedings of the National Academy of Sciences)

   null (Ed.)

   Nonlinear differential equations model diverse phenomena but are notoriously difficult to solve. While there has been extensive previous work on efficient quantum algorithms for linear differential equations, the linearity of quantum mechanics has limited analogous progress for the nonlinear case. Despite this obstacle, we develop a quantum algorithm for dissipative quadratic n-dimensional ordinary differential equations. Assuming R < 1 , where R is a parameter characterizing the ratio of the nonlinearity and forcing to the linear dissipation, this algorithm has complexity T 2 q   poly ( log ⁡ T , log ⁡ n , log ⁡ 1 / ϵ ) / ϵ , where T is the evolution time, ϵ is the allowed error, and q measures decay of the solution. This is an exponential improvement over the best previous quantum algorithms, whose complexity is exponential in T. While exponential decay precludes efficiency, driven equations can avoid this issue despite the presence of dissipation. Our algorithm uses the method of Carleman linearization, for which we give a convergence theorem. This method maps a system of nonlinear differential equations to an infinite-dimensional system of linear differential equations, which we discretize, truncate, and solve using the forward Euler method and the quantum linear system algorithm. We also provide a lower bound on the worst-case complexity of quantum algorithms for general quadratic differential equations, showing that the problem is intractable for R ≥ 2 . Finally, we discuss potential applications, showing that the R < 1 condition can be satisfied in realistic epidemiological models and giving numerical evidence that the method may describe a model of fluid dynamics even for larger values of R. 

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3. Super-Exponential Convergence Rate of a Nonlinear Continuous Data Assimilation Algorithm: The 2D Navier–Stokes Equation Paradigm

   https://doi.org/10.1007/s00332-024-10014-w  

   Carlson, Elizabeth; Larios, Adam; Titi, Edriss S (April 2024, Journal of Nonlinear Science)

   We study a nonlinear-nudging modification of the Azouani–Olson–Titi continuous data assimilation (downscaling) algorithm for the 2D incompressible Navier–Stokes equations. We give a rigorous proof that the nonlinear-nudging system is globally well posed and, moreover, that its solutions converge to the true solution exponentially fast in time. Furthermore, we also prove that once the error has decreased below a certain order one threshold, the convergence becomes double exponentially fast in time, up until a precision determined by the sparsity of the observed data. In addition, we demonstrate the applicability of the analytical and sharpness of the results computationally. 

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4. Distributed optimal resource allocation using transformed primal-dual method

   https://doi.org/10.23919/ACC55779.2023.10156601  

   Kia, Solmaz S.; Wei, Jingrong; Chen, Long (May 2023, American Control Conference)

   We consider an in-network optimal resource allocation problem in which a group of agents interacting over a connected graph want to meet a demand while minimizing their collective cost. The contribution of this paper is to design a distributed continuous-time algorithm for this problem inspired by a recently developed first-order transformed primal-dual method. The solution applies to cluster-based setting where each agent may have a set of subagents, and its local cost is the sum of the cost of these subagents. The proposed algorithm guarantees an exponential convergence for strongly convex costs and asymptotic convergence for convex costs. Exponential convergence when the local cost functions are strongly convex is achieved even when the local gradients are only locally Lipschitz. For convex local cost functions, our algorithm guarantees asymptotic convergence to a point in the minimizer set. Through numerical examples, we show that our proposed algorithm delivers a faster convergence compared to existing distributed resource allocation algorithms. 

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5. An Exponentially Convergent Primal-Dual Algorithm for Nonsmooth Composite Minimization

   https://doi.org/10.1109/CDC.2018.8619760  

   Ding, Dongsheng; Hu, Bin; Dhingra, Neil K.; Jovanovic, Mihailo R. (December 2018, 2018 IEEE Conference on Decision and Control (CDC))

   We consider a class of nonsmooth convex composite optimization problems, where the objective function is given by the sum of a continuously differentiable convex term and a potentially non-differentiable convex regularizer. In [1], the authors introduced the proximal augmented Lagrangian method and derived the resulting continuous-time primal-dual dynamics that converge to the optimal solution. In this paper, we extend these dynamics from continuous to discrete time via the forward Euler discretization. We prove explicit bounds on the exponential convergence rates of our proposed algorithm with a sufficiently small step size. Since a larger step size can improve the convergence speed, we further develop a linear matrix inequality (LMI) condition which can be numerically solved to provide rate certificates with general step size choices. In addition, we prove that a large range of step size values can guarantee exponential convergence. We close the paper by demonstrating the performance of the proposed algorithm via computational experiments. 

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