9 An adaptive finite element approach for lifted branched transport problems
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Carolin Dirks
Abstract
We consider the so-called branched transport and variants thereof in two space dimensions. In these models, we seek an optimal transportation network for a given mass transportation task. In two space dimensions, they are closely connected to Mumford-Shah-type image processing problems, which in turn can be related to certain higher-dimensional convex optimization problems via so-called functional lifting. We examine the relation between these different models and exploit it to solve the branched transport model numerically via convex optimization. To this end, we develop an efficient numerical treatment based on a specifically designed class of adaptive finite elements. This method allows the computation of finely resolved optimal transportation networks despite the high dimensionality of the convex optimization problem and its complicated set of nonlocal constraints. In particular, by design of the discretization the infinite set of constraints reduces to a finite number of inequalities.
Abstract
We consider the so-called branched transport and variants thereof in two space dimensions. In these models, we seek an optimal transportation network for a given mass transportation task. In two space dimensions, they are closely connected to Mumford-Shah-type image processing problems, which in turn can be related to certain higher-dimensional convex optimization problems via so-called functional lifting. We examine the relation between these different models and exploit it to solve the branched transport model numerically via convex optimization. To this end, we develop an efficient numerical treatment based on a specifically designed class of adaptive finite elements. This method allows the computation of finely resolved optimal transportation networks despite the high dimensionality of the convex optimization problem and its complicated set of nonlocal constraints. In particular, by design of the discretization the infinite set of constraints reduces to a finite number of inequalities.
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
