Unstructured Space-Time Finite Element Methods for Optimal Control of Parabolic Equations
02 Mar 2021-SIAM Journal on Scientific Computing (Society for Industrial and Applied Mathematics)-Vol. 43, Iss: 2
TL;DR: In this paper, a space-time finite element method was proposed for the numerical solution of parabolic optimal control problems using Babuska's polynomial-time method on fully unstructured simplicial space time meshes.
Abstract: This work presents and analyzes space-time finite element methods on fully unstructured simplicial space-time meshes for the numerical solution of parabolic optimal control problems. Using Babuska'...
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TL;DR: It is emphasized that the energy regularization results in a more localized control with sharper contours for discontinuous target functions, which is demonstrated by a comparison with an $L^2$ regularization and with a sparse optimal control approach.
Abstract: In this paper, we analyze space-time finite element methods for the numerical solution of distributed parabolic optimal control problems with energy regularization in the Bochner space $L^2(0,T;H^{...
19 citations
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TL;DR: This approach combines the mathematical Lagrangian approach for differentiating PDE-constrained shape functions with the automated differentiation capabilities of NGSolve to allow for either a more custom-like or black-box–like behaviour of the software.
Abstract: In this paper we present a framework for automated shape differentiation in the finite element software NGSolve. Our approach combines the mathematical Lagrangian approach for differentiating PDE constrained shape functions with the automated differentiation capabilities of NGSolve. The user can decide which degree of automatisation is required and thus allows for either a more custom-like or black-box-like behaviour of the software.
We discuss the automatic generation of first and second order shape derivatives for unconstrained model problems as well as for more realistic problems that are constrained by different types of partial differential equations. We consider linear as well as nonlinear problems and also problems which are posed on surfaces. In numerical experiments we verify the accuracy of the computed derivatives via a Taylor test. Finally we present first and second order shape optimisation algorithms and illustrate them for several numerical optimisation examples ranging from nonlinear elasticity to Maxwell's equations.
6 citations
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TL;DR: This work provides temporal antiderivatives of the heat kernel necessary for the assembly of BEM matrices and the evaluation of the representation formula allowing researchers to reuse the formulae and BEM routines straightaway.
Abstract: The presented paper concentrates on the boundary element method (BEM) for the heat equation in three spatial dimensions. In particular, we deal with tensor product space-time meshes allowing for quadrature schemes analytic in time and numerical in space. The spatial integrals can be treated by standard BEM techniques known from three dimensional stationary problems. The contribution of the paper is twofold. First, we provide temporal antiderivatives of the heat kernel necessary for the assembly of BEM matrices and the evaluation of the representation formula. Secondly, the presented approach has been implemented in a publicly available library besthea allowing researchers to reuse the formulae and BEM routines straightaway. The results are validated by numerical experiments in an HPC environment.
6 citations
TL;DR: This work solves Maxwell's eigenvalue problem via isogeometric boundary elements and a contour integral method and discusses the analytic properties of the discretisation, and outlines the implementation.
Abstract: We solve Maxwell's eigenvalue problem via isogeometric boundary elements and a contour integral method. We discuss the analytic properties of the discretisation, outline the implementation, and showcase numerical examples.
6 citations
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TL;DR: In this paper, the authors analyzed space-time finite element methods for the numerical solution of distributed parabolic optimal control problems with energy regularization in the Bochner space and proved unique solvability in the continuous case.
Abstract: We analyze space-time finite element methods for the numerical solution of distributed parabolic optimal control problems with energy regularization in the Bochner space $L^2(0,T;H^{-1}(\Omega))$. By duality, the related norm can be evaluated by means of the solution of an elliptic quasi-stationary boundary value problem. When eliminating the control, we end up with the reduced optimality system that is nothing but the variational formulation of the coupled forward-backward primal and adjoint equations. Using Babuska's theorem, we prove unique solvability in the continuous case. Furthermore, we establish the discrete inf-sup condition for any conforming space-time finite element discretization yielding quasi-optimal discretization error estimates. Various numerical examples confirm the theoretical findings. We emphasize that the energy regularization results in a more localized control with sharper contours for discontinuous target functions, which is demonstrated by a comparison with an $L^2$ regularization and with a sparse optimal control approach.
3 citations
References
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TL;DR: Van der Pol's equation for a relaxation oscillator is generalized by the addition of terms to produce a pair of non-linear differential equations with either a stable singular point or a limit cycle, which qualitatively resembles Bonhoeffer's theoretical model for the iron wire model of nerve.
5,430 citations
31 Dec 1968
TL;DR: In this article, the authors introduce a system of linear and quasi-linear equations with principal part in divergence (PCI) in the form of systems of linear, quasilinear and general systems.
Abstract: Introductory material Auxiliary propositions Linear equations with discontinuous coefficients Linear equations with smooth coefficients Quasi-linear equations with principal part in divergence form Quasi-linear equations of general form Systems of linear and quasi-linear equations Bibliography.
3,986 citations
01 Oct 1962
TL;DR: In this paper, an active pulse transmission line using tunnel diodes was made to electronically simulate an animal nerve axon, and the equation of propagation for this line is the same as that for a simplified model of nerve membrane treated elsewhere.
Abstract: To electronically simulate an animal nerve axon, the authors made an active pulse transmission line using tunnel diodes. The equation of propagation for this line is the same as that for a simplified model of nerve membrane treated elsewhere. This line shapes the signal waveform during transmission, that is, there being a specific pulse-like waveform peculiar to this line, smaller signals are amplified, larger ones are attenuated, narrower ones are widened and those which are wider are shrunk, all approaching the above-mentioned specific waveform. In addition, this line has a certain threshold value in respect to the signal height, and signals smaller than the threshold or noise are eliminated in the course of transmission. Because of the above-mentioned shaping action and the existence of a threshold, this line makes possible highly reliable pulse transmission, and will be useful for various kinds of information-processing systems.
3,516 citations
Book•
01 Jan 2004
TL;DR: Theoretical Foundations for Finite Element Interpolation and Banach Spaces by Galerkin Methods are given in this article, along with a discussion of the application of the Banach and Hilbert spaces in data-structuring and mesh generation.
Abstract: I Theoretical Foundations.- 1 Finite Element Interpolation.- 2 Approximation in Banach Spaces by Galerkin Methods.- II Approximation of PDEs.- 3 Coercive Problems.- 4 Mixed Problems.- 5 First-Order PDEs.- 6 Time-Dependent Problems.- III Implementation.- 7 Data Structuring and Mesh Generation.- 8 Quadratures, Assembling, and Storage.- 9 Linear Algebra.- 10 A Posteriori Error Estimates and Adaptive Meshes.- IV Appendices.- A Banach and Hilbert Spaces.- A.1 Basic Definitions and Results.- A.2 Bijective Banach Operators.- B Functional Analysis.- B.1 Lebesgue and Lipschitz Spaces.- B.2 Distributions.- B.3 Sobolev Spaces.- Nomenclature.- References.- Author Index.
2,108 citations
Book•
11 Dec 1989
1,810 citations