Adaptive finite element discretization in PDE‐based optimization

Rannacher, Rolf; Vexler, Boris

doi:10.1002/gamm.201010014

Cited by 11 publications

(4 citation statements)

References 58 publications

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“…The development of numerical methods for multi-population optimal control problems is a topic that originated a large literature in the last years. Besides the well-established methodology of the discretization of PDE constrained optimal control problems by means of finite element methods, mainly applied for elliptic and parabolic type of equations (see for instance [26]), a particularly promising approach is based on their kinetic description using Boltzmann models, see [2]. In [1], the implementation of such methods to solve a control problem similar to Problem 2 successfully produced nontrivial optimal strategies, one of which was shown in Figure 1.…”

Section: Discussionmentioning

confidence: 99%

Optimal control problems in transport dynamics

Bongini

Buttazzo

2017

Math. Models Methods Appl. Sci.

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In the present paper we deal with an optimal control problem related to a model in population dynamics; more precisely, the goal is to modify the behavior of a given density of individuals via another population of agents interacting with the first. The cost functional to be minimized to determine the dynamics of the second population takes into account the desired target or configuration to be reached as well as the quantity of control agents. Several applications may fall into this framework, as for instance driving a mass of pedestrian in (or out of) a certain location; influencing the stock market by acting on a small quantity of key investors; controlling a swarm of unmanned aerial vehicles by means of few piloted drones.

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Section: Discussionmentioning

confidence: 99%

Optimal control problems in transport dynamics

Bongini

Buttazzo

2017

Math. Models Methods Appl. Sci.

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“…We seek to minimize the Lagrangian corresponding to problem (P). A similar approach, in the context of adaptive finite elements, can be found in [3,53]. We utilize (10b) to extend the primal-dual RB approach of [27, Section 2.4] to quadratic output functionals and define the NCD-corrected RB reduced functional…”

Section: Rom For the Optimality System -Ncd-corrected Approachmentioning

confidence: 99%

A non-conforming dual approach for adaptive Trust-Region Reduced Basis approximation of PDE-constrained optimization

Keil,

Mechelli,

Ohlberger

et al. 2020

Preprint

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In this contribution we propose and rigorously analyze new variants of adaptive Trust-Region methods for parameter optimization with PDE constraints and bilateral parameter constraints. The approach employs successively enriched Reduced Basis surrogate models that are constructed during the outer optimization loop and used as model function for the Trust-Region method. Each Trust-Region sub-problem is solved with the projected BFGS method. Moreover, we propose a non-conforming dual (NCD) approach to improve the standard RB approximation of the optimality system. Rigorous improved a posteriori error bounds are derived and used to prove convergence of the resulting NCD-corrected adaptive Trust-Region Reduced Basis algorithm. Numerical experiments demonstrate that this approach enables to reduce the computational demand for large scale or multi-scale PDE constrained optimization problems significantly.

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“…The dual weighted residual (DWR) method [84,85] now seeks to refine the mesh used to discretize the PDE by weighting (local) residuals with information about their global influence on the goal functional J [86]. These weights are computed by the dual problem…”

Section: Goal-oriented Error Estimationmentioning

confidence: 99%

Compression Challenges in Large Scale Partial Differential Equation Solvers

Götschel

Weiser

2019

Algorithms

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Solvers for partial differential equations (PDE) are one of the cornerstones of computational science. For large problems, they involve huge amounts of data that needs to be stored and transmitted on all levels of the memory hierarchy. Often, bandwidth is the limiting factor due to relatively small arithmetic intensity, and increasingly so due to the growing disparity between computing power and bandwidth. Consequently, data compression techniques have been investigated and tailored towards the specific requirements of PDE solvers during the last decades. This paper surveys data compression challenges and discusses examples of corresponding solution approaches for PDE problems, covering all levels of the memory hierarchy from mass storage up to main memory. Exemplarily, we illustrate concepts at particular methods, and give references to alternatives.

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Adaptive finite element discretization in PDE‐based optimization

Cited by 11 publications

References 58 publications

Optimal control problems in transport dynamics

Optimal control problems in transport dynamics

A non-conforming dual approach for adaptive Trust-Region Reduced Basis approximation of PDE-constrained optimization

Compression Challenges in Large Scale Partial Differential Equation Solvers

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