Algorithms for PDE-Constrained Optimization
نویسندگان
چکیده
In this paper we review a number of algorithmic approaches for solving optimization problems with PDE constraints. Most of these methods were originally developed for finite dimensional problems. When applied to optimization problems with PDE constraints, new aspects become important. For instance, (discretized) PDE-constrained problems are inherently large-scale. Some aspects, like mesh independent convergence behavior, can only be explained by incorporating the infinite dimensional point of view, which is not present in finite dimensional problems. Moreover, discretization and solution of PDE-constrained optimization problems should not be viewed as independent. Rather, they must be intertwined to yield efficient algorithms. In this article, we provide a unified treatment of algorithms, which proceeds on a formal basis and does not distinguish between infinite dimensional and discretized problem settings. To make the presentation more concrete, we consider the solution of an optimal control problem for the stationary Navier-Stokes system as a prototypical example. The volume force acts as the control function, and the corresponding state is given by the velocity and pressure components of the solution to the Navier-Stokes equation. Clearly, this is a rather artificial setting from an engineering point of view, although a certain possibility to control the Lorentz forces exists in electrically conducting fluids (see [Davidson(2001), Griesse and Kunisch(2006)]). All algorithms will be explained in this setting.
منابع مشابه
Model Problems in PDE-Constrained Optimization
This work aims to aid in introducing, experimenting and benchmarking algorithms for PDE-constrained optimization problems by presenting a set of such model problems. We specifically examine a type of PDE-constrained optimization problem, the parameter estimation problem. We present three model parameter estimation problems, each containing a different type of partial differential equation as th...
متن کاملNumerical Techniques for Optimization Problems with PDE Constraints
Optimization problems with partial differential equation (PDE) constraints arise in many science and engineering applications. Their robust and efficient solution present many mathematical challenges and requires a tight integration of properties and structures of the underlying problem, of fast numerical PDE solvers, and of numerical nonlinear optimization. This workshop brought together exper...
متن کاملOPTIMAL CONSTRAINED DESIGN OF STEEL STRUCTURES BY DIFFERENTIAL EVOLUTIONARY ALGORITHMS
Structural optimization, when approached by conventional (gradient based) minimization algorithms presents several difficulties, mainly related to computational aspects for the huge number of nonlinear analyses required, that regard both Objective Functions (OFs) and Constraints. Moreover, from the early '80s to today's, Evolutionary Algorithms have been successfully developed and applied as a ...
متن کاملThe Transport PDE and Mixed-Integer Linear Programming
Discrete, nonlinear and PDE constrained optimization are mostly considered as different fields of mathematical research. Nevertheless many real-life problems are most naturally modeled as PDE constrained mixed integer nonlinear programs. For example, nonlinear network flow problems where the flow dynamics are governed by a transport equation are of this type. We present four different applicati...
متن کامل