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Optimization and Control (math.OC)

Thu, 03 Aug 2023

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1.Optimal Distributed Control for a Cahn-Hilliard-Darcy System with Mass Sources, Unmatched Viscosities and Singular Potential

Authors:Marco Abatangelo, Cecilia Cavaterra, Maurizio Grasselli, Hao Wu

Abstract: We study a Cahn-Hilliard-Darcy system in two dimensions with mass sources, unmatched viscosities and singular potential. This system is equipped with no-flux boundary conditions for the (volume) averaged velocity $\mathbf{u}$, the difference of the volume fractions $\varphi$, and the chemical potential $\mu$, along with an initial condition for $\varphi$. The resulting initial boundary value problem can be considered as a basic, though simplified, model for the evolution of solid tumor growth. The source term in the Cahn-Hilliard equation contains a control $R$ that can be thought, for instance, as a drug or a nutrient. Our goal is to study an optimal control problem with a tracking type cost functional given by the sum of three $L^2$ norms involving $\varphi(T)$ ($T>0$ is the final time), $\varphi$ and $R$. We first prove the existence and uniqueness of a global strong solution with $\varphi$ being strictly separated from the pure phases $\pm 1$. Thanks to this result, we are able to analyze the control-to-state mapping $\mathcal{S}: R \mapsto \varphi$, obtaining the existence of an optimal control, the Fr\'{e}chet differentiability of $\mathcal{S}$ and first-order necessary optimality conditions expressed through a suitable variational inequality for the adjoint variables. Finally, we show the differentiability of the control-to-costate operator and establish a second-order sufficient condition for the strict local optimality.

2.Efficiency of First-Order Methods for Low-Rank Tensor Recovery with the Tensor Nuclear Norm Under Strict Complementarity

Authors:Dan Garber, Atara Kaplan

Abstract: We consider convex relaxations for recovering low-rank tensors based on constrained minimization over a ball induced by the tensor nuclear norm, recently introduced in \cite{tensor_tSVD}. We build on a recent line of results that considered convex relaxations for the recovery of low-rank matrices and established that under a strict complementarity condition (SC), both the convergence rate and per-iteration runtime of standard gradient methods may improve dramatically. We develop the appropriate strict complementarity condition for the tensor nuclear norm ball and obtain the following main results under this condition: 1. When the objective to minimize is of the form $f(\mX)=g(\mA\mX)+\langle{\mC,\mX}\rangle$ , where $g$ is strongly convex and $\mA$ is a linear map (e.g., least squares), a quadratic growth bound holds, which implies linear convergence rates for standard projected gradient methods, despite the fact that $f$ need not be strongly convex. 2. For a smooth objective function, when initialized in certain proximity of an optimal solution which satisfies SC, standard projected gradient methods only require SVD computations (for projecting onto the tensor nuclear norm ball) of rank that matches the tubal rank of the optimal solution. In particular, when the tubal rank is constant, this implies nearly linear (in the size of the tensor) runtime per iteration, as opposed to super linear without further assumptions. 3. For a nonsmooth objective function which admits a popular smooth saddle-point formulation, we derive similar results to the latter for the well known extragradient method. An additional contribution which may be of independent interest, is the rigorous extension of many basic results regarding tensors of arbitrary order, which were previously obtained only for third-order tensors.

3.Topology Optimization for Uniform Flow Distribution in Electrolysis Cells

Authors:Leon Baeck, Sebastian Blauth, Christian Leithäuser, René Pinnau, Kevin Sturm

Abstract: In this paper we consider the topology optimization for a bipolar plate of a hydrogen electrolysis cell. We present a model for the bipolar plate using the Stokes equation with an additional drag term, which models the influence of fluid and solid regions. Furthermore, we derive a criterion for a uniform flow distribution in the bipolar plate. To obtain shapes that are well-manufacturable, we introduce a novel smoothing technique for the fluid velocity. Finally, we present some numerical results and investigate the influence of the smoothing on the obtained shapes.

4.Subspace-Constrained Continuous Methane Leak Monitoring and Optimal Sensor Placement

Authors:Kashif Rashid, Lukasz Zielinski, Junyi Yuan, Andrew Speck

Abstract: This work presents a procedure that can quickly identify and isolate methane emission sources leading to expedient remediation. Minimizing the time required to identify a leak and the subsequent time to dispatch repair crews can significantly reduce the amount of methane released into the atmosphere. The procedure developed utilizes permanently installed low-cost methane sensors at an oilfield facility to continuously monitor leaked gas concentration above background levels. The methods developed for optimal sensor placement and leak inversion in consideration of predefined subspaces and restricted zones are presented. In particular, subspaces represent regions comprising one or more equipment items that may leak, and restricted zones define regions in which a sensor may not be placed due to site restrictions by design. Thus, subspaces constrain the inversion problem to specified locales, while restricted zones constrain sensor placement to feasible zones. The development of synthetic wind models, and those based on historical data, are also presented as a means to accommodate optimal sensor placement under wind uncertainty. The wind models serve as realizations for planning purposes, with the aim of maximizing the mean coverage measure for a given number of sensors. Once the optimal design is established, continuous real-time monitoring permits localization and quantification of a methane leak source. The necessary methods, mathematical formulation and demonstrative test results are presented.

5.Energy System Optimisation using (Mixed Integer) Linear Programming

Authors:Sebastian Miehling, Andreas Hanel, Jerry Lambert, Sebastian Fendt, Hartmut Spliethoff

Abstract: Although energy system optimisation based on linear optimisation is often used for influential energy outlooks and studies for political decision-makers, the underlying background still needs to be described in the scientific literature in a concise and general form. This study presents the main equations and advanced ideas and explains further possibilities mixed integer linear programming offers in energy system optimisation. Furthermore, the equations are shown using an example system to present a more practical point of view. Therefore, this study is aimed at researchers trying to understand the background of studies using energy system optimisation and researchers building their implementation into a new framework. This study describes how to build a standard model, how to implement advanced equations using linear programming, and how to implement advanced equations using mixed integer linear programming, as well as shows a small exemplary system. - Presentation of the OpTUMus energy system optimisation framework - Set of equations for a fully functional energy system model - Example of a simple energy system model