# TU Darmstadt ULB TUbiblio

# Peer Methods in Optimal Control

## Abstract

In this thesis we analyze implicit and linearly implicit peer methods in the context of optimization problems with ordinary or partial differential equations as constraints.

In many practical applications, like the cooling of glass, the propagation of a flame front in a cooled channel or the hardening of steel, the underlying physical process can be modeled by ordinary differential equations (ODE) or partial differential equations (PDE). The wish to optimize these processes leads to the field of ODE- and PDE-constrained optimization.

The constraints, in this case an ODE or PDE system, have to be evaluated several times in an optimization algorithm. Therefore it is very important to use efficient discretization methods for the arising differential equations. Runge-Kutta and Rosenbrock methods are a popular choice for ODEs and the time discretization of parabolic PDEs. However they suffer from order reduction when applied to stiff problems.

A promising alternative are peer methods. These methods construct several approximations to the solution in one time step like one-step methods and use the approximations of the last time step like multistep methods. Peer methods are proven to show no order reduction when applied to stiff problems. More details on peer methods are presented in Chapter 3. In this thesis we analyze peer methods within the optimal control with differential equations.

There are two popular approaches when solving optimal control problems. The first approach is called first-discretize-then-optimize, while the other is the first-optimize-then-discretize approach. In Chapter 4 we analyze the interchangeability of these two approaches when using peer methods. We find that the two approaches give quite different results for peer methods and especially conclude, that peer methods are not well suited for the first-discretize-then-optimize approach.

Therefore, we concentrate then on the first-optimize-then-discretize approach and especially want to employ peer methods within a multilevel optimization approach. To this end we derive a fully adaptive, that is adaptive in time and space, discretization for parabolic PDEs in Chapter 5.

We follow the Rothe approach and discretize first in time by a linearly implicit peer method leading to several linear elliptic problems. These are then discretized by multilevel linear finite elements. We derive a spatial error estimator based on hierarchical bases. The time error is estimated by comparing the computed solution with a solution of lower order. We look at the efficiency of the spatial error estimator both analytically and numerically. Finally we compare the performance of peer methods to that of Rosenbrock methods for three PDE test examples in 2D. We see that peer methods are competitive to Rosenbrock methods.

This fully adaptive scheme is then used within a multilevel optimization in Chapter 6. We first introduce the optimization algorithm and especially look at the points where the time integration plays a role. Finally, we present results for three PDE constrained control problems. Again the peer methods are competitive to Rosenbrock methods.

Item Type: Ph.D. Thesis
Erschienen: 2016
Creators: Schröder, Dirk
Title: Peer Methods in Optimal Control
Language: English
Abstract:

In this thesis we analyze implicit and linearly implicit peer methods in the context of optimization problems with ordinary or partial differential equations as constraints.

In many practical applications, like the cooling of glass, the propagation of a flame front in a cooled channel or the hardening of steel, the underlying physical process can be modeled by ordinary differential equations (ODE) or partial differential equations (PDE). The wish to optimize these processes leads to the field of ODE- and PDE-constrained optimization.

The constraints, in this case an ODE or PDE system, have to be evaluated several times in an optimization algorithm. Therefore it is very important to use efficient discretization methods for the arising differential equations. Runge-Kutta and Rosenbrock methods are a popular choice for ODEs and the time discretization of parabolic PDEs. However they suffer from order reduction when applied to stiff problems.

A promising alternative are peer methods. These methods construct several approximations to the solution in one time step like one-step methods and use the approximations of the last time step like multistep methods. Peer methods are proven to show no order reduction when applied to stiff problems. More details on peer methods are presented in Chapter 3. In this thesis we analyze peer methods within the optimal control with differential equations.

There are two popular approaches when solving optimal control problems. The first approach is called first-discretize-then-optimize, while the other is the first-optimize-then-discretize approach. In Chapter 4 we analyze the interchangeability of these two approaches when using peer methods. We find that the two approaches give quite different results for peer methods and especially conclude, that peer methods are not well suited for the first-discretize-then-optimize approach.

Therefore, we concentrate then on the first-optimize-then-discretize approach and especially want to employ peer methods within a multilevel optimization approach. To this end we derive a fully adaptive, that is adaptive in time and space, discretization for parabolic PDEs in Chapter 5.

We follow the Rothe approach and discretize first in time by a linearly implicit peer method leading to several linear elliptic problems. These are then discretized by multilevel linear finite elements. We derive a spatial error estimator based on hierarchical bases. The time error is estimated by comparing the computed solution with a solution of lower order. We look at the efficiency of the spatial error estimator both analytically and numerically. Finally we compare the performance of peer methods to that of Rosenbrock methods for three PDE test examples in 2D. We see that peer methods are competitive to Rosenbrock methods.

This fully adaptive scheme is then used within a multilevel optimization in Chapter 6. We first introduce the optimization algorithm and especially look at the points where the time integration plays a role. Finally, we present results for three PDE constrained control problems. Again the peer methods are competitive to Rosenbrock methods.

Divisions: 04 Department of Mathematics > Numerical Analysis and Scientific Computing
04 Department of Mathematics
Date Deposited: 03 Jul 2016 19:55 Send an inquiry View Item