1 Reduced basis model order reduction in optimal control of a nonsmooth semilinear elliptic PDE
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Marco Bernreuther
Abstract
In this paper, we consider an optimization problem governed by a nonsmooth semilinear elliptic partial differential equation. We apply a reduced order approach to obtain a computationally fast and certified numerical solution approach. Using the reduced basis method and efficient a posteriori error estimation for the primal and dual equations, we develop an adaptive algorithm and successfully test it for several numerical examples.
Abstract
In this paper, we consider an optimization problem governed by a nonsmooth semilinear elliptic partial differential equation. We apply a reduced order approach to obtain a computationally fast and certified numerical solution approach. Using the reduced basis method and efficient a posteriori error estimation for the primal and dual equations, we develop an adaptive algorithm and successfully test it for several numerical examples.
Chapters in this book
- Frontmatter I
- Preface V
- Contents VII
- 1 Reduced basis model order reduction in optimal control of a nonsmooth semilinear elliptic PDE 1
- 2 Pointwise moving control for the 1-D wave equation 33
- 3 Limits of stabilizability for a semilinear model for gas pipeline flow 59
- 4 Minimal cost-time strategies for mosquito population replacement 73
- 5 The sterile insect technique used as a barrier control against reinfestation 91
- 6 Variational discretization approach applied to an optimal control problem with bounded measure controls 113
- 7 An optimal control problem for equations with p-structure and its finite element discretization 137
- 8 Unstructured space-time finite element methods for optimal sparse control of parabolic equations 167
- 9 An adaptive finite element approach for lifted branched transport problems 189
- 10 High-order homogenization of the Poisson equation in a perforated periodic domain 237
- 11 Least-squares approaches for the 2D Navier–Stokes system 285
- 12 Numerical issues and turnpike phenomenon in optimal shape design 343
- 13 Feedback stabilization of Cahn–Hilliard phase-field systems 367
- 14 Ensemble Kalman filter for neural network-based one-shot inversion 393
- 15 Deep learning in high dimension: ReLU neural network expression for Bayesian PDE inversion 419
- Index 463
Chapters in this book
- Frontmatter I
- Preface V
- Contents VII
- 1 Reduced basis model order reduction in optimal control of a nonsmooth semilinear elliptic PDE 1
- 2 Pointwise moving control for the 1-D wave equation 33
- 3 Limits of stabilizability for a semilinear model for gas pipeline flow 59
- 4 Minimal cost-time strategies for mosquito population replacement 73
- 5 The sterile insect technique used as a barrier control against reinfestation 91
- 6 Variational discretization approach applied to an optimal control problem with bounded measure controls 113
- 7 An optimal control problem for equations with p-structure and its finite element discretization 137
- 8 Unstructured space-time finite element methods for optimal sparse control of parabolic equations 167
- 9 An adaptive finite element approach for lifted branched transport problems 189
- 10 High-order homogenization of the Poisson equation in a perforated periodic domain 237
- 11 Least-squares approaches for the 2D Navier–Stokes system 285
- 12 Numerical issues and turnpike phenomenon in optimal shape design 343
- 13 Feedback stabilization of Cahn–Hilliard phase-field systems 367
- 14 Ensemble Kalman filter for neural network-based one-shot inversion 393
- 15 Deep learning in high dimension: ReLU neural network expression for Bayesian PDE inversion 419
- Index 463
