Algorithms for PDE‐constrained optimization

We study the convergence properties of an algorithm for the inverse problem of electrical impedance tomography, which can be reduced to a partial differential equation (PDE) constrained optimization problem. The direct problem consists of the potential equation div( u) = 0 in a circle, with Neumann condition describing the behavior of the electrostatic potential in a medium with conductivity given by the function (x, y). We suppose that at each time a current i is applied to the boundary of the circle (Neumann's data), and that it is possible to measure the corresponding potential i (Dirichlet data). The inverse problem is to find (x, y) given a finite number of Cauchy pairs measurements ( i , i ), i = 1, , N . The problem is formulated as a least squares problem, and the developed algorithm solves the continuous problem using descent iterations in its corresponding finite element approximations. Wolfe's conditions are used to ensure the global convergence of the optimization algorithm for the continuous problem. Although exact data are assumed, measurement errors in data and regularization methods shall be considered in a future work.

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“…This is a consistent and undetermined system. On the other hand, condition (14) generates the additional equation:…”

Section: Least Squares Formulation and Discretizationmentioning

confidence: 99%

“…From finite element theory, it is well known (see [26]) that we have convergence conditions, i.e., if ∈ L ∞ ( ), ≥ K , and if u i h is the solution of (13)- (14) for fixed h = , ∀h > 0, then:…”

Section: Remark 3 the Last Equation Is A Consequence Of Replacingmentioning

confidence: 99%

A Globally Convergent Algorithm for a PDE-Constrained Optimization Problem Arising in Electrical Impedance Tomography

Carrillo

Gómez

2015

Numerical Functional Analysis and Optimization

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“…The proposed model-based methodology for robust optimal control implies formulating and solving a large-scale dynamic optimization problem (DOP) constrained by PDEs [28]. Optimal design, optimal control, and parameter estimation of systems governed by PDE give rise to a class of problems known as PDE-constrained optimization [29,30]. The size and complexity of the discretized PDEs often pose significant challenges for contemporary optimization methods.…”

Section: Pde-constrained Dynamic Optimizationmentioning

confidence: 99%

Methods and Tools for Robust Optimal Control of Batch Chromatographic Separation Processes

et al. 2015

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This contribution concerns the development of generic methods and tools for robust optimal control of high-pressure liquid chromatographic separation processes. The proposed methodology exploits a deterministic robust formulation, that employs a linearization of the uncertainty set, based on Lyapunov differential equations to generate optimal elution trajectories in the presence of uncertainty. Computational tractability is obtained by casting the robust counterpart problem in the framework of bilevel optimal control where the upper level concerns forward simulation of the Lyapunov differential equation, and the nominal open-loop optimal control problem augmented with the robustified target component purity inequality constraint margin is considered in the lower level. The lower-level open-loop optimal control problem, constrained by spatially discretized partial differential equations, is transcribed into a finite dimensional nonlinear program using direct collocation, which is then solved by a primal-dual interior point method. The advantages of the robustification strategy are highlighted through the solution of a challenging ternary complex mixture separation problem for a hydrophobic interaction chromatography system. The study shows that penalizing the changes in the zero-order hold control gives optimal solutions with low sensitivity to uncertainty. A key result is that the robustified general elution trajectories outperformed the conventional linear trajectories both in terms of recovery yield and robustness.

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“…This is necessary in order to use a black box-type method like the steepest descent algorithm as indicated below. We refer to [12,13] for the theoretical background, and a recent overview about algorithms used for solving the optimization problem.…”

Section: Computational Studiesmentioning

confidence: 99%

Optimal control in evolutionary micromagnetism

et al. 2014

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Abstract. We consider an optimal control problem subject to the Landau-Lifshitz-Gilbert equation which describes the evolution of magnetizations in S 2 . The problem is motivated in order to control switching processes of ferromagnets. Existence of an optimum and the first order necessary optimality system are derived. We show (up to subsequences) convergence of state, adjoint and control variables of a time discretization (semi-implicit Euler method) for vanishing time step size. A main step here is to verify corresponding stability properties for the semi-discrete state, which is nontrivial since the iterates take values which only approximate S 2 . We use a perturbation argument within a variational discretization in order to show error bounds for the semi-discrete state variables, from which we may then infer uniform bounds for the semi-discrete state and also adjoint variables. Numerical studies underline these results and compare this discretization with a further variant which bases on a projection strategy for the state equation to enhance iterates to better approximate S 2 .

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Algorithms for PDE‐constrained optimization

Cited by 114 publications

References 31 publications

A Globally Convergent Algorithm for a PDE-Constrained Optimization Problem Arising in Electrical Impedance Tomography

A Globally Convergent Algorithm for a PDE-Constrained Optimization Problem Arising in Electrical Impedance Tomography

Methods and Tools for Robust Optimal Control of Batch Chromatographic Separation Processes

Optimal control in evolutionary micromagnetism

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