Constrained Optimization and Optimal Control for Partial Differential Equations

We consider optimal control-based boundary feedback stabilization of flow problems for incompressible fluids. We follow an analytical 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 Navier-Stokes equations by designing a stabilizing controller based on a corresponding linear-quadratic optimal control problem. For this purpose, algorithmic advances in solving the associated algebraic Riccati equations are needed and investigated here. The computational complexity of the new algorithms is essentially proportional to the simulation of the forward problem.

Parabolic variational inequalities of Allen-Cahn and Cahn-Hilliard type are solved using methods involving constrained optimization. Time discrete variants are formulated with the help of Lagrange multipliers for local and non-local equality and inequality constraints. Fully discrete problems resulting from finite element discretizations in space are solved with the help of a primal-dual active set approach. We show several numerical computations also involving systems of parabolic variational inequalities.

In certain applications of PDE constrained optimization one would like to base an optimization method on an already existing contractive method (solver) for the forward problem. The forward problem consists of finding a feasible point with some parts of the variables (e.g., design variables) held fixed. This approach often leads to so-called simultaneous, all-at-once, or oneshot optimization methods. If only one iteration of the forward method per optimization iteration is necessary, a simultaneous method is called one-step. We present three illustrative linear examples in four dimensions with two constraints which highlight that in general there is only little connection between contraction of forward problem method and simultaneous one-step optimization method. We analyze the asymptotics of three prototypical regularization strategies to possibly recover convergence and compare them with Griewank’s One-Step One-Shot projected Hessian preconditioners. We present de facto loss of convergence for all of these methods, which leads to the conclusion that, at least for fast contracting forward methods, the forward problem solver must be used with adaptive accuracy controlled by the optimization method.

In this work, we present an all-in-one optimization approach suitable to solve complex optimal control problems with time-dependent nonlinear partial differential algebraic equations and point-wise control constraints. A newly developed generalized SQP-method is combined with an error based multilevel strategy and the state-of-the-art software package Kardos to allow the efficient resolution of different space and time scales in an adaptive manner. The numerical performance of the method is demonstrated and analyzed for a real-life two-dimensional radiative heat transfer problem modelling the optimal boundary control for a cooling process in glass manufacturing.

The optimal design of structures and systems described by partial differential equations (PDEs) often gives rise to large-scale optimization problems, in particular if the underlying system of PDEs represents a multi-scale, multi-physics problem. Therefore, reduced order modeling techniques such as balanced truncation model reduction, proper orthogonal decomposition, or reduced basis methods are used to significantly decrease the computational complexity while maintaining the desired accuracy of the approximation. In particular, we are interested in such shape optimization problems where the design issue is restricted to a relatively small portion of the computational domain. In this case, it appears to be natural to rely on a full order model only in that specific part of the domain and to use a reduced order model elsewhere. A convenient methodology to realize this idea consists in a suitable combination of domain decomposition techniques and balanced truncation model reduction. We will consider such an approach for shape optimization problems associated with the time-dependent Stokes system and derive explicit error bounds for the modeling error. As an application in life sciences, we will be concerned with the optimal design of capillary barriers as part of a network of microchannels and reservoirs on microfluidic biochips that are used in clinical diagnostics, pharmacology, and forensics for high-throughput screening and hybridization in genomics and protein profiling in proteomics.

We study PDE-constrained optimization problems where the state equation is solved by a pseudo-time stepping or fixed point iteration. We present a technique that improves primal, dual feasibility and optimality simultaneously in each iteration step, thus coupling state and adjoint iteration and control/design update. Our goal is to obtain bounded retardation of this coupled iteration compared to the original one for the state, since the latter in many cases has only a Q-factor close to one. For this purpose and based on a doubly augmented Lagrangian, which can be shown to be an exact penalty function, we discuss in detail the choice of an appropriate control or design space preconditioner, discuss implementation issues and present a convergence analysis. We show numerical examples, among them applications from shape design in fluid mechanics and parameter optimization in a climate model.

We present an overview on recent results concerning hyperbolic systems on networks. We present a summary of theoretical results on existence, uniqueness and stability. The established theory extends previously known results on the Cauchy problem for nonlinear, 2

×

2 hyperbolic balance laws. The proofs are based on Wave-Front Tracking and therefore we present detailed results on the Riemann problem first.

We present a hierarchical solution concept for optimization problems governed by the time-dependent Navier–Stokes system. Discretisation is carried out with finite elements in space and a one-step-

θ

-scheme in time. By combining a Newton solver for the treatment of the nonlinearity with a space-time multigrid solver for linear subproblems, we obtain a robust solver whose convergence behaviour is independent of the refinement level of the discrete problem. A set of numerical examples analyses the solver behaviour for various problem settings with respect to efficiency and robustness of this approach.

When combining the numerical concept of variational discretization introduced in [Hin03, Hin05] and semi-smooth Newton methods for the numerical solution of pde constrained optimization with control constraints [HIK03, Ulb03] special emphasis has to be placed on the implementation, convergence and globalization of the numerical algorithm. In the present work we address all these issues following [HV]. In particular we prove fast local convergence of the algorithm and propose a globalization strategy which is applicable in many practically relevant mathematical settings. We illustrate our analytical and algorithmical findings by numerical experiments.

This paper presents some details for the development, analysis, and implementation of efficient numerical optimization algorithms using algorithmic differentiation (AD) in the context of partial differential equation (PDE) constrained optimization. This includes an error analysis for the discrete adjoints computed with AD and a systematic structure exploitation including efficient checkpointing routines, especially multistage and online checkpointing approaches.

In the paper we show how, based on the preconditioned Krylov subspace methods, to compute the covariance matrix of parameter estimates, which is crucial for efficient methods of optimum experimental design.

This article is concerned with different approaches to elastic shape optimization under stochastic loading. The underlying stochastic optimization strategy builds upon the methodology of two-stage stochastic programming. In fact, in the case of linear elasticity and quadratic objective functionals our strategy leads to a computational cost which scales linearly in the number of

linearly independent

applied forces, even for a large set of realizations of the random loading. We consider, besides minimization of the expectation value of suitable objective functionals, also two different risk averse approaches, namely the

expected excess

and the

excess probability

. Numerical computations are performed using either a level set approach representing implicit shapes of general topology in combination with composite finite elements to resolve elasticity in two and three dimensions, or a collocation boundary element approach, where polygonal shapes represent geometric details attached to a lattice and describing a perforated elastic domain. Topology optimization is performed using the concept of topological derivatives. We generalize this concept, and derive an analytical expression which takes into account the interaction between neighboring holes. This is expected to allow efficient and reliable optimization strategies of elastic objects with a large number of geometric details on a fine scale.

The mean compliance minimization in structural topology optimization is solved with the help of a phase field approach. Two steepest descent approaches based on

L

2- and

H-

1-gradient flow dynamics are discussed. The resulting flows are given by Allen-Cahn and Cahn-Hilliard type dynamics coupled to a linear elasticity system. We finally compare numerical results obtained from the two different approaches.

We present an approach to shape optimization which is based on transformation to a reference domain with continuous adjoint computations. This method is applied to the instationary Navier-Stokes equations for which we discuss the appropriate setting and discuss Fréchet differentiability of the velocity field with respect to domain transformations. Goal-oriented error estimation is used for an adaptive refinement strategy. Finally, we give some numerical results.

In this paper the solution of a Bernoulli type free boundary problem by means of shape optimization is considered. Four different formulations are compared from an analytical and numerical point of view. By analyzing the shape Hessian in case of matching data it is distinguished between well-posed and ill-posed formulations. A nonlinear Ritz-Galerkin method is applied for the discretization of the shape optimization problem. In case of well-posedness existence and convergence of the approximate shapes is proven. In combination with a fast boundary element method efficient first and second-order shape optimization algorithms are obtained.

Numerical schemes for large scale shape optimization are considered. Exploiting the structure of shape optimization problems is shown to lead to very efficient optimization methods based on non-parametric surface gradients in Hadamard form. The resulting loss of regularity is treated using higher-order shape Newton methods where the shape Hessians are studied using operator symbols. The application ranges from shape optimization of obstacles in an incompressible Navier–Stokes fluid to super- and transonic airfoil and wing optimizations.

In this work we develop an adaptive algorithm for solving elliptic optimal control problems with simultaneously appearing state and control constraints. Building upon the concept proposed in [9] the algorithm applies a Moreau-Yosida regularization technique for handling state constraints. The state and co-state variables are discretized using continuous piecewise linear finite elements while a variational discretization concept is applied for the control. To perform the adaptive mesh refinement cycle we derive local error representations which extend the goal-oriented error approach to our setting. The performance of the overall adaptive solver is demonstrated by a numerical example.

In this paper we summerize recent results on a posteriori error estimation and adaptivity for space-time finite element discretizations of parabolic optimization problems. The provided error estimates assess the discretization error with respect to a given quantity of interest and separate the influences of different parts of the discretization (time, space, and control discretization). This allows us to set up an efficient adaptive strategy producing economical (locally) refined meshes for each time step and an adapted time discretization. The space and time discretization errors are equilibrated, leading to an efficient method.

This article summarizes several recent results on goal-oriented error estimation and mesh adaptation for the solution of elliptic PDE-constrained optimization problems with additional inequality constraints. The first part is devoted to the control constrained case. Then some emphasis is given to pointwise inequality constraints on the state variable and on its gradient. In the last part of the article regularization techniques for state constraints are considered and the question is addressed, how the regularization parameter can adaptively be linked to the discretization error.

This paper deals with

L

2-error estimates for finite element approximations of control constrained distributed optimal control problems governed by linear partial differential equations. First, general assumptions are stated that allow to prove second-order convergence in control, state and adjoint state. Afterwards these assumptions are verified for problems where the solution of the state equation has singularities due to corners or edges in the domain or nonsmooth coefficients in the equation. In order to avoid a reduced convergence order, graded finite element meshes are used.

Solutions to optimization problems with pde constraints inherit special properties; the associated state solves the pde which in the optimization problem takes the role of a equality constraint, and this state together with the associated control solves an optimization problem, i.e., together with multipliers satisfies first- and second-order necessary optimality conditions. In this note we review the state of the art in designing discrete concepts for optimization problems with pde constraints with emphasis on structure conservation of solutions on the discrete level, and on error analysis for the discrete variables involved. As model problem for the state we consider an elliptic pde which is well understood from the analytical point of view. This allows to focus on structural aspects in discretization. We discuss the approaches

First discretize, then optimize

and

First optimize, then discretize

, and consider in detail two variants of the

First discretize, then optimize

approach, namely variational discretization, a discrete concept which avoids explicit discretization of the controls, and piecewise constant control approximations. We consider general constraints on the control, and also consider pointwise bounds on the state. We outline the basic ideas for providing optimal error analysis and accomplish our analytical findings with numerical examples which confirm our analytical results. Furthermore we present a brief review on recent literature which appeared in the field of discrete techniques for optimization problems with pde constraints.

In this note we present a framework for the a posteriori error analysis of control constrained optimal control problems with linear PDE constraints. It is solely based on reliable and efficient error estimators for the corresponding

linear

state and adjoint

equations

. We show that the sum of these estimators gives a reliable and efficient estimator for the optimal control problem.

In this article we summarize recent results on a priori error estimates for space-time finite element discretizations of linear-quadratic parabolic optimal control problems. We consider the following three cases: problems without inequality constraints, problems with pointwise control constraints, and problems with state constraints pointwise in time. For all cases, error estimates with respect to the temporal and to the spatial discretization parameters are derived. The results are illustrated by numerical examples.

We survey the results of SPP 1253 project “Numerical Analysis of State-constrained Optimal Control Problems for PDEs”. In the first part, we consider Lavrentiev-type regularization of both distributed and boundary control. In the second part, we present a priori error estimates for elliptic control problems with finite-dimensional control space and state-constraints both in finitely many points and in all points of a subdomain with nonempty interior.

The treatment of hepatic lesions with radio-frequency (RF) ablation has become a promising minimally invasive alternative to surgical resection during the last decade. In order to achieve treatment qualities similar to surgical R0 resections, patient specific mathematical modeling and simulation of the biophysical processes during RF ablation are valuable tools. They allow for an a priori estimation of the success of the therapy as well as an optimization of the therapy parameters. In this report we discuss our recent efforts in this area: a model of partial differential equations (PDEs) for the patient specific numerical simulation of RF ablation, the optimization of the probe placement under the constraining PDE system and the identification of material parameters from temperature measurements. A particular focus lies on the uncertainties in the patient specific tissue properties. We discuss a stochastic PDE model, allowing for a sensitivity analysis of the optimal probe location under variations in the material properties. Moreover, we optimize the probe location under uncertainty, by considering an objective function, which is based on the expectation of the stochastic distribution of the temperature distribution. The application of our models and algorithms to data from real patient’s CT scans underline their applicability.

We present topology optimization of piezoelectric loudspeakers using the SIMP method and topology gradient based methods along with analytical and numerical results.

Two injuring effects of cryopreservation of living cells are under study. First, stresses arising due to non-simultaneous freezing of water inside and outside of cells are modeled and controlled. Second, dehydration of cells caused by earlier ice building in the extracellular liquid compared to the intracellular one is simulated. A low-dimensional mathematical model of competitive ice formation inside and outside of living cells during freezing is derived by applying an appropriate averaging technique to partial differential equations describing the dynamics of water-to-ice phase change. This reduces spatially distributed relations to a few ordinary differential equations with control parameters and uncertainties. Such equations together with an objective functional that expresses the difference between the amount of ice inside and outside of a cell are considered as a differential game. The aim of the control is to minimize the objective functional, and the aim of the disturbance is opposite. A stable finite-difference scheme for computing the value function is applied to the problem. On the base of the computed value function, optimal cooling protocols ensuring simultaneous freezing of water inside and outside of living cells are designed. Thus, balancing the inner and outer pressures prevents cells from injuring. Another mathematical model describes shrinkage and swelling of cells caused by their osmotic dehydration and rehydration during freezing and thawing. The model is based on the theory of ice formation in porous media and Stefan-type conditions describing the osmotic inflow/outflow related to the change of the salt concentration in the extracellular liquid. The cell shape is searched as a level set of a function which satisfies a Hamilton-Jacobi equation resulting from a Stefan-type condition for the normal velocity of the cell boundary. Hamilton-Jacobi equations are numerically solved using finite-difference schemes for finding viscosity solutions as well as by computing reachable sets of an associated conflict control problem. Examples of the shape evolution computed in two and three dimensions are presented

The production of nanoscaled particulate products with exactly pre-defined characteristics is of enormous economic relevance. Although there are different particle formation routes they may all be described by one class of equations. Therefore, simulating such processes comprises the solution of nonlinear, hyperbolic integro-partial differential equations. In our project we aim to study this class of equations in order to develop efficient tools for the identification of optimal process conditions to achieve desired product properties. This objective is approached by a joint effort of the mathematics and the engineering faculty. Two model-processes are chosen for this study, namely a precipitation process and an innovative aerosol process allowing for a precise control of residence time and temperature. Since the overall problem is far too complex to be solved directly a hierarchical sequence of simplified problems has been derived which are solved consecutively. In particular, the simulation results are finally subject to comparison with experiments.

Geometric evolution equations, such as mean curvature flow and surface diffusion, play an important role in mathematical modeling in various fields, ranging from materials to life science. Controlling the surface or interface evolution would be desirable for many of these applications. We attack this problem by considering the bulk contribution, which defines a driving force for the geometric evolution equation, as a distributed control. In order to solve the control problem we use a phase-field approximation and demonstrate the applicability of the approach on various examples. In the first example the effect of an electric field on the evolution of nanostructures on crystalline surfaces is considered. The mathematical problem corresponds to surface diffusion or a Cahn-Hilliard model. In the second example we consider mean curvature flow or a Allen-Cahn model

In this contribution, the optimization of periodic chromatographic simulated moving bed SMB processes is discussed. The rigorous optimization is based on a nonlinear pde model which incorporates rigorous models of the chromatographic columns and the discrete shifts of the inlet and outlet ports. The potential of the optimization is demonstrated for a separation problem with nonlinear isotherm of the Langmuir type for an SMB process and the ModiCon process. Here, an efficient numerical approach based on multiple shooting is employed. An overview of established optimization approaches for SMB processes is given.

We discuss the derivation and investigation of efficient mathematical methods for the solution of optimization and identification problems for radiation dominant processes, which are described by a nonlinear integrodifferential system or diffusive type approximations. These processes are for example relevant in glass production or in the layout of gas turbine combustion chambers. The main focus is on the investigation of optimization algorithms based on the adjoint variables, which are applied to the full radiative heat transfer system as well as to diffusive type approximations. In addition to the optimization we also study new approaches to the reconstruction of the initial temperature from boundary measurements, since its precise knowledge is mandatory for any satisfactory simulation. In particular, we develop a fast, derivative-free method for the solution of the inverse problem, such that we can use many different models for the simulation of the radiative process.

We report on a shape optimization framework that couples a highlyparallel finite element solver with a geometric kernel and different optimization algorithms. The entire optimization framework is transformed with automatic differentiation techniques, and the derivative code is employed to compute derivatives of the optimal shapes with respect to viscosity. This methodology provides a powerful tool to investigate the necessity of intricate constitutive models by taking derivatives with respect to model parameters