Optimal control of the convection-diffusion equation using stabilized finite element methods
Roland Becker,Boris Vexler +1 more
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A stabilization scheme which leads to improved approximate solutions even on corse meshes in the convection dominated case and the in general different approaches “optimize-then- discretize” and "discretize- then-optimize" coincide for the proposed discretization scheme.Abstract:
In this paper we analyze the discretization of optimal control problems governed by convection-diffusion equations which are subject to pointwise control constraints. We present a stabilization scheme which leads to improved approximate solutions even on corse meshes in the convection dominated case. Moreover, the in general different approaches “optimize-then- discretize” and “discretize-then-optimize” coincide for the proposed discretization scheme. This allows for a symmetric optimality system at the discrete level and optimal order of convergence.read more
Citations
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A Priori Error Estimates for Space-Time Finite Element Discretization of Parabolic Optimal Control Problems Part I: Problems Without Control Constraints
Dominik Meidner,Boris Vexler +1 more
TL;DR: A priori error analysis for Galerkin finite element discretizations of optimal control problems governed by linear parabolic equations and error estimates of optimal order with respect to both space and time discretization parameters are developed.
Journal ArticleDOI
A unified convergence analysis for local projection stabilisations applied to the Oseen problem
TL;DR: In this article, the results of Braack and Burman for the standard two-level version of the local projection stabilisation were extended to discretisations of arbitrary order on simplices, quadrilaterals, and hexahedra.
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Constrained Dirichlet Boundary Control in $L^2$ for a Class of Evolution Equations
Karl Kunisch,Boris Vexler +1 more
TL;DR: A discretization based on space-time finite elements is proposed and numerical examples are included and its global and local superlinear convergences are shown.
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Optimal control of the convective Cahn–Hilliard equation
Xiaopeng Zhao,Changchun Liu +1 more
TL;DR: In this article, the authors studied the problem of optimal control of the convective Cahn-Hilliard equation in one-space dimension and proved the existence of an optimal solution to the equation.
Posted Content
OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport.
TL;DR: The proposed OT-Flow approach tackles two critical computational challenges that limit a more widespread use of CNFs, and leverages optimal transport (OT) theory to regularize the CNF and enforce straight trajectories that are easier to integrate.
References
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Book
The Finite Element Method for Elliptic Problems
Philippe G. Ciarlet,J. T. Oden +1 more
TL;DR: The finite element method has been applied to a variety of nonlinear problems, e.g., Elliptic boundary value problems as discussed by the authors, plate problems, and second-order problems.
Book
Finite Element Method for Elliptic Problems
TL;DR: In this article, Ciarlet presents a self-contained book on finite element methods for analysis and functional analysis, particularly Hilbert spaces, Sobolev spaces, and differential calculus in normed vector spaces.
Book
Optimal Control of Systems Governed by Partial Differential Equations
TL;DR: In this paper, the authors consider the problem of minimizing the sum of a differentiable and non-differentiable function in the context of a system governed by a Dirichlet problem.
Book
Numerical Approximation of Partial Differential Equations
Alfio Quarteroni,Alberto Valli +1 more
TL;DR: In this article, the authors provide a thorough illustration of numerical methods, carry out their stability and convergence analysis, derive error bounds, and discuss the algorithmic aspects relative to their implementation.
Book
Numerical Solution of Partial Differential Equations by the Finite Element Method
TL;DR: In this article, the authors present an easily accessible introduction to one of the most important methods used to solve partial differential equations, which they call finite element methods for integral equations (FEME).
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