# Optimization with Partial Differential Equations

The solution of partial differential equations has been in the focus of numerical analysis for many decades as many processes in engineering and science are described by a PDE or by systems of PDEs. In the past, the solution of the PDE, i.e., forward problem, has been at the heart of many research projects. Recently with advances in algorithms and computing technology, the solution of so-called optimal control problems with respect to PDE constraints has become increasingly popular. The goal in this field is to find the parameters/setup for a particular PDE or systems of PDEs such that an objective function is minimized. The CSC group focuses on two categories of optimal control problems.

### Open loop Control and Efficient Solution of Linear Saddle Point Systems

The first category is the so-called open loop control problem where the system input is computed purely on the basis of a-priori knowledge. Here, we focus on the efficient solution of the optimization problem with state-of-the-art numerical techniques. The task of solving optimal control problems is challenging as most realistic problems are posed in three dimensions and the discretization, often done with finite elements, results in many degrees of freedom. At the heart of the problem typically lies the solution of the first order optimality conditions or Karush–Kuhn–Tucker (KKT) conditions and the corresponding linear system in saddle point form. For realistic scenarios it is not possible to use techniques based on factorizations to solve this problem. Hence iterative Krylov subspace solvers have to be employed. These are usually only feasible if some form of preconditioning is applied; a technique that solves a modified, simpler problem. We derived preconditioners that take the structure of the KKT system and the underlying infinite-dimensional problem into account. For steady and unsteady problems we constructed efficient approximations to the Schur-complement of the saddle point problem, which involves the forward and the adjoint discretized PDE. In the case of unsteady problems we showed that multilevel techniques approximating only one matrix provide sufficient approximations for the Schur-complement and that independence with respect to the mesh-parameter is given. For a variety of problems, such as the (time-periodic) heat equation or the unsteady Stokes equation, we illustrated that only crude approximations to the discretized PDE suffice to obtain accurate results to the optimal control problem.

Matrix equations of the one kind or the other are a central tool in all kinds of applications. In optimal control the linear quadratic control problem features a feedback solution that is determined via the solution of an algebraic Riccati equation. The system Gramians of a linear time invariant dynamical system are the solutions of two adjoint Lyapunov equations. Riccati, Lyapunov and Sylvester equations play an important role in different model order reduction techniques for continuous time linear dynamical systems. They all have discrete time counterparts as e.g. the well known Stein equations. Krylov subspace and eigenvalue methods can be related to certain Sylvester equations, as well.

## Internal Projects

### Optimal Control of Chemical Processes

Chemical reactive flows lead to optimization problems governed by systems of convection diffusion partial differential equations (PDEs) with nonlinear reaction mechanisms. Such problems are strongly coupled as inaccuracies in one unknown directly affect all other unknowns. Prediction of these unknowns is very important for the safe and economical operation of biochemical and chemical engineering processes. The solution of these PDEs can exhibit boundary and/or interior layers on small regions where the solution has large gradients, when convection dominates diffusion. Hence, special numerical techniques are required, which take the structure of the convection into acount. Recently, discontinuous Galerkin (DG) methods became an alternative to the finite difference, finite volume and continuous finite element methods for solving wave dominated problems like convection diffusion equations since they possess higher accuracy. We focus here on an application of adaptive DG methods for (un)steady optimal control problems governed by copuled convection dominated PDEs with nonlinear reaction terms with state and/or control constraints in 2D and 3D.
more

### Fast Solvers for Phase Field Models (with NDS group)

Many problems in material science, image processing or chemistry are well represented using phase-field equations. We here investigate schemes that enable the fast solution by also being robust to parameter changes.
more

### Low-rank Solver (with NDS group)

Many of the discretized problems have an inherent tensor-structure that allows us to employ low-rank methods. We here develop state of the art techniques that enable an efficient treatment of such problems.
more

### Preconditioning for PDE-constrained optimization (with NDS group)

Processes across the sciences are described by differential equations. We here consider their discretized counterparts and provide accelerated methods for obtaining an accurate solution to such problems.

## Concluded Projects

### High Performance Computing and PDE-constrained optimization (with NDS group)

Employing a monolithic approach representing "all of the physics" and "all-of-the optimization" in one system, we often require many time-steps and are led to large-scale problems. We here combine modern HPC techniques with state-of-the-art algorithms.

### Closed Loop Control and Solution of Large Scale Matrix Equations

The second category of control problems studied within the research group is the one of regulator problems. These problems are described by the minimization of an objective function over a possibly infinite time horizon incorporating a feedback loop, i.e., the incorporation of certain output (observed quantities) of the current system behavior into the control input. Again, one is interested in the minimization of a functional subject to unsteady, possibly nonlinear, PDEs. In general the PDEs are rewritten as an infinite dimensional evolution equation, or Cauchy problem. In the case of a linear PDE the optimization problem can then be posed as a linear quadratic regulator (LQR) problem that we solve via a system theoretic approach leading to algebraic or differential Riccati equations. For nonlinear PDEs this approach must be embedded in a model predictive control (MPC) scheme, where local linearizations around working points (or working trajectories) are employed. In any case after semi-discretization using the method of lines, a very large system of ordinary differential equations (ODEs) and related very large matrix equations are obtained. Our special focus in these approaches is to prove the numerical feasibility of such matrix equation based approaches in the PDE context.

Matrix equations of the one kind or the other are a central tool in all kinds of applications. In optimal control the linear quadratic control problem features a feedback solution that is determined via the solution of an algebraic Riccati equation. The system Gramians of a linear time invariant dynamical system are the solutions of two adjoint Lyapunov equations. Riccati, Lyapunov and Sylvester equations play an important role in different model order reduction techniques for continuous time linear dynamical systems. They all have discrete time counterparts as e.g. the well known Stein equations. Krylov subspace and eigenvalue methods can be related to certain Sylvester equations, as well.

## Internal Projects

### Optimal Control-Based Feedback Stabilization in Multi-Field Flow Problems.

The goal of this project is to derive and investigate numerical algorithms for optimal control-based boundary feedback stabilization of multi-field flow problems.
We will follow an approach laid out during the last years in a series of papers by Barbu, Lasiecka, Triggiani, Raymond, and others. They have shown that it is possible to stabilize perturbed flows described by the Navier-Stokes equation by designing a stabilizing controller based on a corresponding linear-quadratic optimal control problem.
more

### Solution of the Inverse Heat Conduction Problem (IHCP) using LQR Techniques

With respect to the reconstruction of a heat source the inverse heat conduction problem (IHCP) is an ill-posed problem. Nevertheless such kind of problems often arises in fields of process engineering and an efficient and accurate solution method is required. The task of this project now is to handle the inverse problem via an optimal control approach. The main goal is to develop and implement an efficient algorithm to reconstruct the heat source. In particular the arising temperature from a drilling process has to be computed. To solve this problem a few measurements are taken from the outer boundary of the working piece. To keep the first tests computable with respect to their computation times a simple breadboard construction is used.
more

## Concluded Projects

### Boundary Feedback Stabilization Using Non-Conforming Finite Elements

We are investigating a new finite element method to improve boundary feedback stabilization techniques of instationary, incompressible flow problems. Since standard finite elements do not fulfill divergence freeness condition by themselves we cannot guarantee the validity of this condition after solving the arising linear systems by iterative solvers. We, now, formulate the boundary feedback approach in operator terms and solve the underlying PDE in each step, where the divergence freeness condition is handled inside the solver. By using special finite elements we can improve the solver and end up with a fast robust algorithm.
more

### Numerical Solution of Optimal Control Problems with Instationary Diffusion-Convection and Diffusion-Reaction Equations

In this project we want to develop numerical algorithms of optimal control problems for instationary convection-diffusion and diffusion-reaction equations by using methods of state and output feedback. Linear problems with quadratic cost functional can be interpreted as a linear-quadratic regulator (LQR) problem. For the solutions of LQR problems new efficient methods were developed where Peter Benner was involved. These methods are coupled with solvers for the underlying stationary forward problem by appropriate interfaces.
We obtain nonlinear problems if nonlinear differential operators or nonlinear boundary conditions occur. The solution of nonlinear problems can be found by solving a class of optimal control problems which are called tracking-problems by means of state or output feedback. Since in general the optimal control cannot be computed directly or with untenable effort as done in the lineare case we will use sub-optimal strategies. We will focus on the development of numerical methods for the application of Model Predictive Control (MPC) for 2D and 3D problems. In doing so the whole time horizon will be covered by shorter time frames at which a sub-problem is solved by using an LQR or LQG design. The LQR and LQG designs for the parabolic problems arising after linearization can be solved by the numerical methods mentioned above.
more ### Parallel algorithms for large-scale sparse algebraic Riccati equations and application in control

The project has run as part of the DAAD program "Acciones Integradas Hispano-Alemanas" ### Parallel numerical solution of optimal control problems with instationary diffusion-convection-reaction equations

This project considers boundary value problems with linear partial differential equations and boundary control. Previous work: In his diploma thesis Jens Saak worked on the controlled cooling of steel profiles during production process. He was using rail profiles as an example. It was taken into concern that different areas of the surface of the material should be cooled differently intense. The starting point, from the view of an engineer, was the fact that an equally distributed temperature profile on some cross-section of the profile (before the final rolling), established by selective cooling, leads to better mechanical material characteristics of the end product than using conventional rolling without selective cooling. The rising mathematical problem is that of optimal control using boundary control for a (linearized) heat-equation. The variational formulation leads to a linear quadratic optimal control problem in classical formulation which in contrary to classical problems is formulated in an infinite dimensional Hilbert-space. This problem, due to Gibson 1979, can be solved by a feedback approach. The numerical implementation uses finite element semidiscretisation in space. This lead to an LQR-problem for an ordinary differential equation. This equation has a very high dimension (>1000) which leads to additional difficulties for the numerical treatment.
Go to Editor View