Finite element discretization and efficient numerical solution of elliptic and parabolic sparse control problems

This thesis is concerned with the numerical analysis of sparse control problems for elliptic and parabolic state equations. A focus is set on controls which are measures in space, where the instationary problem formulation favors fixed-in-space point sources. A general optimization framework based on a sparsity-preserving regularization and a semismooth Newton method is developed. A priori error estimates for a suitable finite element discretization of two model problems are derived. An algorithm for adaptive mesh refinement is proposed.

[1]  Michael Ulbrich,et al.  Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces , 2011, MOS-SIAM Series on Optimization.

[2]  Barbara Kaltenbacher,et al.  Adaptive discretizations for the choice of a Tikhonov regularization parameter in nonlinear inverse problems , 2011 .

[3]  Kazufumi Ito,et al.  On a semi-smooth Newton method and its globalization , 2009, Math. Program..

[4]  Stefan Ulbrich,et al.  Optimization with PDE Constraints , 2008, Mathematical modelling.

[5]  Herbert Amann,et al.  Nonautonomous Parabolic Equations Involving Measures , 2005 .

[6]  G. M. Troianiello,et al.  Elliptic Differential Equations and Obstacle Problems , 1987 .

[7]  Boris Vexler,et al.  Adaptive Finite Elements for Elliptic Optimization Problems with Control Constraints , 2008, SIAM J. Control. Optim..

[8]  Arnd Rösch,et al.  Error estimates for joint Tikhonov and Lavrentiev regularization of constrained control problems , 2009, 0909.4648.

[9]  Roland Herzog,et al.  Approximation of sparse controls in semilinear equations by piecewise linear functions , 2012, Numerische Mathematik.

[10]  Alain Prignet,et al.  Remarks on existence and uniqueness of solutions of elliptic problems with right-hand side measures , 2007 .

[11]  L. Qi,et al.  A Survey of Some Nonsmooth Equations and Smoothing Newton Methods , 1999 .

[12]  Wolfgang Hensgen.  A simple proof of Singer’s representation theorem , 1996 .

[13]  Roland Herzog,et al.  Optimality Conditions and Error Analysis of Semilinear Elliptic Control Problems with L1 Cost Functional , 2012, SIAM J. Optim..

[14]  Stephen M. Robinson,et al.  Normal Maps Induced by Linear Transformations , 1992, Math. Oper. Res..

[15]  Karl Kunisch,et al.  Optimal Control of the Undamped Linear Wave Equation with Measure Valued Controls , 2016, SIAM J. Control. Optim..

[16]  Lutz Recke,et al.  Linear Elliptic Boundary Value Problems with Non – Smooth Data: Normal Solvability on Sobolev – Campanato Spaces , 2001 .

[17]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[18]  Karl Kunisch,et al.  A measure space approach to optimal source placement , 2012, Comput. Optim. Appl..

[19]  Anton Schiela,et al.  Barrier Methods for Optimal Control Problems with State Constraints , 2009, SIAM J. Optim..

[20]  T. Steihaug.  The Conjugate Gradient Method and Trust Regions in Large Scale Optimization , 1983 .

[21]  Michael Hinze,et al.  Convergence of a Finite Element Approximation to a State-Constrained Elliptic Control Problem , 2007, SIAM J. Numer. Anal..

[22]  Fredi Tröltzsch,et al.  On Finite Element Error Estimates for Optimal Control Problems with Elliptic PDEs , 2009, LSSC.

[23]  Rolf Rannacher,et al.  An optimal control approach to a posteriori error estimation in finite element methods , 2001, Acta Numerica.

[24]  Emmanuel J. Candès,et al.  Super-Resolution from Noisy Data , 2012, Journal of Fourier Analysis and Applications.

[25]  Jérôme Droniou,et al.  Solving convection-diffusion equations with mixed, Neumann and Fourier boundary conditions and measures as data, by a duality method , 2000, Advances in Differential Equations.

[26]  Jean-Pierre Raymond,et al.  Optimal pointwise control of semilinear parabolic equations , 2000 .

[27]  Roland Becker,et al.  Optimal control of the convection-diffusion equation using stabilized finite element methods , 2007, Numerische Mathematik.

[28]  H. Brezis.  Functional Analysis, Sobolev Spaces and Partial Differential Equations , 2010 .

[29]  Karl Kunisch,et al.  Optimal Control of Semilinear Elliptic Equations in Measure Spaces , 2014, SIAM J. Control. Optim..

[30]  S. M. Robinson.  Newton's method for a class of nonsmooth functions , 1994 .

[31]  Andreas Günther,et al.  An interior point algorithm with inexact step computation in function space for state constrained optimal control , 2011, Numerische Mathematik.

[32]  Patricia Brunner,et al.  A deterministic approach to the adapted optode placement for illumination of highly scattering tissue , 2012, Biomedical optics express.

[33]  W. Wollner,et al.  A posteriori error estimates for a finite element discretization of interior point methods for an elliptic optimization problem with state constraints , 2010, Comput. Optim. Appl..

[34]  M. Fornasier,et al.  Mean-Field Optimal Control , 2013, 1306.5913.

[35]  J. Dieudonne.  Foundations of Modern Analysis , 1969 .

[36]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[37]  Jürgen Appell,et al.  Nonlinear superposition operators: Preface , 1990 .

[38]  Enrique Zuazua,et al.  Sparse initial data identification for parabolic PDE and its finite element approximations , 2015 .

[39]  Stephen J. Gardiner,et al.  Classical Potential Theory , 2000 .

[40]  Daniel Wachsmuth,et al.  Update strategies for perturbed nonsmooth equations , 2008, Optim. Methods Softw..

[41]  Herbert Amann,et al.  Linear and Quasilinear Parabolic Problems , 2019, Monographs in Mathematics.

[42]  Otmar Scherzer,et al.  Sparsity Regularization for Radon Measures , 2009, SSVM.

[43]  Martin Weiser,et al.  A control reduced primal interior point method for a class of control constrained optimal control problems , 2008, Comput. Optim. Appl..

[44]  Michael Hintermüller,et al.  Goal-Oriented Adaptivity in Pointwise State Constrained Optimal Control of Partial Differential Equations , 2010, SIAM J. Control. Optim..

[45]  Enrique Zuazua,et al.  Spike controls for elliptic and parabolic PDEs , 2013, Syst. Control. Lett..

[46]  Carlos E. Kenig,et al.  The Inhomogeneous Dirichlet Problem in Lipschitz Domains , 1995 .

[47]  V. Girault,et al.  A Local Regularization Operator for Triangular and Quadrilateral Finite Elements , 1998 .

[48]  Michael Ulbrich,et al.  Semismooth Newton Methods for Operator Equations in Function Spaces , 2002, SIAM J. Optim..

[49]  F. Henneke.  Sparse Time-Frequency Control of Bilinear Quantum Systems , 2015 .

[50]  Karl Kunisch,et al.  Approximation of Elliptic Control Problems in Measure Spaces with Sparse Solutions , 2012, SIAM J. Control. Optim..

[51]  Daniel Ralph,et al.  Global Convergence of Damped Newton's Method for Nonsmooth Equations via the Path Search , 1994, Math. Oper. Res..

[52]  Joachim Rehberg,et al.  Maximal parabolic regularity for divergence operators including mixed boundary conditions , 2008, 0903.0239.

[53]  Michèle Vanmaele,et al.  Some results in lumped mass finite-element approximation of eigenvalue problems using numerical quadrature formulas , 1992 .

[54]  Ralf Kornhuber,et al.  Nonsmooth Newton Methods for Set-Valued Saddle Point Problems , 2009, SIAM J. Numer. Anal..

[55]  Michael Hintermüller,et al.  Moreau-Yosida Regularization in State Constrained Elliptic Control Problems: Error Estimates and Parameter Adjustment , 2009, SIAM J. Numer. Anal..

[56]  Michael Hintermüller,et al.  Globalization of SQP-Methods in Control of the Instationary Navier-Stokes Equations , 2002 .

[57]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[58]  Michael Hinze,et al.  Hamburger Beiträge zur Angewandten Mathematik The semi-smooth Newton method for variationally discretized control constrained elliptic optimal control problems ; implementation , convergence and globalization , 2022 .

[59]  Wei Gong,et al.  A Priori Error Analysis for Finite Element Approximation of Parabolic Optimal Control Problems with Pointwise Control , 2014, SIAM J. Control. Optim..

[60]  Rolf Rannacher,et al.  A Posteriori Error Estimation in PDE-constrained Optimization with Pointwise Inequality Constraints , 2012, Constrained Optimization and Optimal Control for Partial Differential Equations.

[61]  Massimo Fornasier,et al.  Recovery Algorithms for Vector-Valued Data with Joint Sparsity Constraints , 2008, SIAM J. Numer. Anal..

[62]  Anton Schiela.  A Simplified Approach to Semismooth Newton Methods in Function Space , 2008, SIAM J. Optim..

[63]  Alexandre Ern,et al.  A Posteriori Control of Modeling Errors and Discretization Errors , 2003, Multiscale Model. Simul..

[64]  Michael Hintermüller,et al.  A SQP-Semismooth Newton-type Algorithm applied to Control of the instationary Navier--Stokes System Subject to Control Constraints , 2006, SIAM J. Optim..

[65]  H. Alt.  Lineare Funktionalanalysis : eine anwendungsorientierte Einführung , 2002 .

[66]  Georg Stadler,et al.  Elliptic optimal control problems with L1-control cost and applications for the placement of control devices , 2009, Comput. Optim. Appl..

[67]  Arnd Rösch,et al.  Superconvergence Properties of Optimal Control Problems , 2004, SIAM J. Control. Optim..

[68]  E. Casas.  Control of an elliptic problem with pointwise state constraints , 1986 .

[69]  Kazufumi Ito,et al.  The Primal-Dual Active Set Strategy as a Semismooth Newton Method , 2002, SIAM J. Optim..

[70]  Gerd Wachsmuth,et al.  Convergence and regularization results for optimal control problems with sparsity functional , 2011 .

[71]  W. Ziemer.  Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation , 1989 .

[72]  Karl Kunisch,et al.  Parabolic Control Problems in Measure Spaces with Sparse Solutions , 2013, SIAM J. Control. Optim..

[73]  J. Daniel.  The Conjugate Gradient Method for Linear and Nonlinear Operator Equations , 1967 .

[74]  Eduardo Casas,et al.  Using piecewise linear functions in the numerical approximation of semilinear elliptic control problems , 2007, Adv. Comput. Math..

[75]  N. Meyers.  An $L^p$-estimate for the gradient of solutions of second order elliptic divergence equations , 1963 .

[76]  K. Kunisch,et al.  Primal-Dual Strategy for Constrained Optimal Control Problems , 1999 .

[77]  Karl Kunisch,et al.  Path-following Methods for a Class of Constrained Minimization Problems in Function Space , 2006, SIAM J. Optim..

[78]  Patrick L. Combettes,et al.  Signal Recovery by Proximal Forward-Backward Splitting , 2005, Multiscale Model. Simul..

[79]  C. Carstensen.  QUASI-INTERPOLATION AND A POSTERIORI ERROR ANALYSIS IN FINITE ELEMENT METHODS , 1999 .

[80]  Arnd Rösch,et al.  Optimal Control of the Stokes Equations: A Priori Error Analysis for Finite Element Discretization with Postprocessing , 2006, SIAM J. Numer. Anal..

[81]  J. Frédéric Bonnans,et al.  Perturbation Analysis of Optimization Problems , 2000, Springer Series in Operations Research.

[82]  Boris Vexler,et al.  A Priori Error Estimates for Space-Time Finite Element Discretization of Parabolic Optimal Control Problems , 2019, Constrained Optimization and Optimal Control for Partial Differential Equations.

[83]  Rolf Rannacher,et al.  A Feed-Back Approach to Error Control in Finite Element Methods: Basic Analysis and Examples , 1996 .

[84]  Rolf Rannacher,et al.  Some Optimal Error Estimates for Piecewise Linear Finite Element Approximations , 1982 .

[85]  J. Hiriart-Urruty,et al.  Generalized Hessian matrix and second-order optimality conditions for problems withC1,1 data , 1984 .

[86]  J. Lions.  Optimal Control of Systems Governed by Partial Differential Equations , 1971 .

[87]  J. Rehberg,et al.  Hölder Continuity and Optimal Control for Nonsmooth Elliptic Problems , 2009 .

[88]  H. Triebel.  Interpolation Theory, Function Spaces, Differential Operators , 1978 .

[89]  R. Rannacher,et al.  Zur L∞-Konvergenz linearer finiter Elemente beim Dirichlet-Problem , 1976 .

[90]  D. Lorenz,et al.  Elastic-net regularization: error estimates and active set methods , 2009, 0905.0796.

[91]  Ivan P. Gavrilyuk,et al.  Lagrange multiplier approach to variational problems and applications , 2010, Math. Comput..

[92]  Fredi Tröltzsch,et al.  Error Estimates for the Numerical Approximation of a Semilinear Elliptic Control Problem , 2002, Comput. Optim. Appl..

[93]  S. Dirkse,et al.  The path solver: a nommonotone stabilization scheme for mixed complementarity problems , 1995 .

[94]  Boris Vexler,et al.  Numerical Sensitivity Analysis for the Quantity of Interest in PDE-Constrained Optimization , 2007, SIAM J. Sci. Comput..

[95]  G. Stampacchia,et al.  Inverse Problem for a Curved Quantum Guide , 2012, Int. J. Math. Math. Sci..

[96]  Boris Vexler,et al.  A Priori Error Analysis for Discretization of Sparse Elliptic Optimal Control Problems in Measure Space , 2013, SIAM J. Control. Optim..

[97]  Herbert Amann,et al.  Compact embeddings of vector valued Sobolev and Besov spaces , 2000 .

[98]  Arnd Rösch,et al.  Primal-Dual Active Set Strategy for a General Class of Constrained Optimal Control Problems , 2002, SIAM J. Optim..

[99]  Dominik Meidner,et al.  Adaptive Space-Time Finite Element Methods for Optimization Problems Governed by Nonlinear Parabolic Systems , 2007 .

[100]  F. Gamboa,et al.  Spike detection from inaccurate samplings , 2013, 1301.5873.

[101]  H. Maurer,et al.  On L1‐minimization in optimal control and applications to robotics , 2006 .

[102]  Kazufumi Ito,et al.  On the Choice of the Regularization Parameter in Nonlinear Inverse Problems , 1992, SIAM J. Optim..

[103]  Eduardo Casas,et al.  New regularity results and improved error estimates for optimal control problems with state constraints , 2014 .

[104]  Carsten Gräser.  Globalization of Nonsmooth Newton Methods for Optimal Control Problems , 2008 .

[105]  I. Ekeland,et al.  Convex analysis and variational problems , 1976 .

[106]  Rolf Rannacher,et al.  Adaptive Finite Element Methods for Optimal Control of Partial Differential Equations: Basic Concept , 2000, SIAM J. Control. Optim..

[107]  Wen Shen,et al.  A Multidimensional Optimal-Harvesting Problem with Measure-Valued Solutions , 2013, SIAM J. Control. Optim..

[108]  A. Mart,et al.  THEORETICAL AND NUMERICAL ANALYSIS OF AN OPTIMAL CONTROL PROBLEM RELATED TO WASTEWATER TREATMENT , 2000 .

[109]  Boris Vexler,et al.  Optimal A Priori Error Estimates of Parabolic Optimal Control Problems with Pointwise Control , 2013, SIAM J. Numer. Anal..

[110]  S. Lang.  Real and Functional Analysis , 1983 .

[111]  Roland Becker,et al.  Efficient numerical solution of parabolic optimization problems by finite element methods , 2007, Optim. Methods Softw..

[112]  William W. Hager,et al.  Second-Order Runge-Kutta Approximations in Control Constrained Optimal Control , 2000, SIAM J. Numer. Anal..

[113]  J. L. Lions.  1. Pointwise Control for Distributed Systems , 1992 .

[114]  Michael Hintermüller,et al.  The Length of the Primal-Dual Path in Moreau-Yosida-Based Path-Following Methods for State Constrained Optimal Control , 2014, SIAM J. Optim..

[115]  Karl Kunisch,et al.  Time optimal control for a reaction diffusion system arising in cardiac electrophysiology – a monolithic approach , 2016 .

[116]  Emmanuel J. Cand.  Towards a Mathematical Theory of Super-Resolution , 2012 .

[117]  Yurii Nesterov,et al.  On first-order algorithms for l1/nuclear norm minimization , 2013, Acta Numerica.

[118]  Robert Sandboge,et al.  Adaptive Finite Element Methods for Reactive Flow Problems , 1996 .

[119]  Heinz H. Bauschke,et al.  Convex Analysis and Monotone Operator Theory in Hilbert Spaces , 2011, CMS Books in Mathematics.

[120]  F. Clarke.  Functional Analysis, Calculus of Variations and Optimal Control , 2013 .

[121]  R. Rockafellar.  Integrals which are convex functionals. II , 1968 .

[122]  Karl Kunisch,et al.  Feasible and Noninterior Path-Following in Constrained Minimization with Low Multiplier Regularity , 2006, SIAM J. Control. Optim..

[123]  Fredi Tröltzsch,et al.  Error estimates for linear-quadratic elliptic control problems , 2002, Analysis and Optimization of Differential Systems.

[124]  Michael Hinze,et al.  A Variational Discretization Concept in Control Constrained Optimization: The Linear-Quadratic Case , 2005, Comput. Optim. Appl..

[125]  R. Winther.  Some Superlinear Convergence Results for the Conjugate Gradient Method , 1980 .

[126]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[127]  Andreas Springer.  Efficient Higher Order Discontinuous Galerkin Time Discretizations for Parabolic Optimal Control Problems , 2015 .

[128]  Arnd Rösch,et al.  Error estimates for linear-quadratic control problems with control constraints , 2006, Optim. Methods Softw..

[129]  K. Bredies,et al.  Inverse problems in spaces of measures , 2013 .

[130]  Jean-Pierre Raymond,et al.  Pontryagin's Principle for State-Constrained Control Problems Governed by Parabolic Equations with Unbounded Controls , 1998 .

[131]  C. Meyer.  Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints , 2008 .

[132]  C. Meyer,et al.  Uniqueness Criteria for the Adjoint Equation in State-Constrained Elliptic Optimal Control , 2011 .

[133]  Kazufumi Ito,et al.  The Primal-Dual Active Set Method for Nonlinear Optimal Control Problems with Bilateral Constraints , 2004, SIAM J. Control. Optim..

[134]  W. Rudin.  Real and complex analysis , 1968 .

[135]  E. Casas.  Pontryagin's Principle for State-Constrained Boundary Control Problems of Semilinear Parabolic Equations , 1997 .

[136]  J. Lions.  Quelques méthodes de résolution de problèmes aux limites non linéaires , 1969 .

[137]  Boris Vexler,et al.  A posteriori error estimation and adaptivity for elliptic optimal control problems with state constraints , 2009, Comput. Optim. Appl..

[138]  E. Casas.  L2 estimates for the finite element method for the Dirichlet problem with singular data , 1985 .

[139]  J. Moreau.  Proximité et dualité dans un espace hilbertien , 1965 .