156 research outputs found
All-at-once preconditioning in PDE-constrained optimization
The optimization of functions subject to partial differential equations (PDE) plays an important role in many areas of science and industry. In this paper we introduce the basic concepts of PDE-constrained optimization and show how the all-at-once approach will lead to linear systems in saddle point form. We will discuss implementation details and different boundary conditions. We then show how these system can be solved efficiently and discuss methods and preconditioners also in the case when bound constraints for the control are introduced. Numerical results will illustrate the competitiveness of our techniques
Implementing a smooth exact penalty function for equality-constrained nonlinear optimization
We develop a general equality-constrained nonlinear optimization algorithm
based on a smooth penalty function proposed by Fletcher (1970). Although it was
historically considered to be computationally prohibitive in practice, we
demonstrate that the computational kernels required are no more expensive than
other widely accepted methods for nonlinear optimization. The main kernel
required to evaluate the penalty function and its derivatives is solving a
structured linear system. We show how to solve this system efficiently by
storing a single factorization each iteration when the matrices are available
explicitly. We further show how to adapt the penalty function to the class of
factorization-free algorithms by solving the linear system iteratively. The
penalty function therefore has promise when the linear system can be solved
efficiently, e.g., for PDE-constrained optimization problems where efficient
preconditioners exist. We discuss extensions including handling simple
constraints explicitly, regularizing the penalty function, and inexact
evaluation of the penalty function and its gradients. We demonstrate the merits
of the approach and its various features on some nonlinear programs from a
standard test set, and some PDE-constrained optimization problems
Path-following primal-dual interior-point methods for shape optimization of stationary flow problems
We consider shape optimization of Stokes flow in channels where the objective is to design the lateral walls of the channel in such a way that a desired velocity profile is achieved. This amounts to the solution of a PDE constrained optimization problem with the state equation given by the Stokes system and the design variables being the control points of a BĂ©zier curve representation of the lateral walls subject to bilateral constraints. Using a finite element discretization of the problem by Taylor-Hood elements, the shape optimization problem is solved numerically by a path-following primal-dual interior-point method applied to the parameter dependent nonlinear system representing the optimality conditions. The method is an all-at-once approach featuring an adaptive choice of the continuation parameter, inexact Newton solves by means of right-transforming iterations, and a monotonicity test for convergence monitoring. The performance of the adaptive continuation process is illustrated by several numerical examples
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