Elliptic State Equations Research Articles

Control problems in the coefficients and the domain for linear elliptic equations

In the present work we are interested in an optimal design problem for a linear elliptic state equation with a homogeneous boundary Dirichlet condition. The control variables correspond to the coefficients of the diffusion term and the open set where the equation is posed. From the application point of view these variables represent the layout of the materials composing the corresponding domain and its shape. We obtain a relaxed formulation of the problem, the optimality conditions, and we provide a numerical algorithm to solve it. Some numerical simulations are also carried out.

On Optimal Control Problems with Controls Appearing Nonlinearly in an Elliptic State Equation

An optimal control problem for a semilinear elliptic equation is discussed, where the control appears nonlinearly in the state equation but is not included in the objective functional. The existence of optimal controls is proved by a measurable selection technique. First-order necessary optimality conditions are derived and two types of second-order sufficient optimality conditions are established. A first theorem invokes a well-known assumption on the set of zeros of the switching function. A second relies on coercivity of the second derivative of the reduced objective functional. The results are applied to the convergence of optimal state functions for a finite element discretizion of the control problem.

Solving a Bernoulli type free boundary problem with random diffusion

The present article is concerned with the numerical solution of a free boundary problem for an elliptic state equation with random diffusion. The domain under consideration is represented by a level set function which is evolved by the objective’s shape gradient. The state is computed by the finite element method, where the underlying triangulation is constructed by means of a marching cubes algorithm. The high-dimensional integral, which is induced by the random diffusion, is approximated by the quasi-Monte Carlo method. By numerical experiments, we validate the feasibility of the approach.

An MHSS-like iteration method for two-by-two linear systems with application to FDE optimization problems

In this paper, we exploit the numerical solvers for a fractional differential equations (FDE) optimization problem with constraints given by fractional elliptic state equations. The discretized linear system can be rewritten as a two-by-two linear system. By making use of the strategy of modified Hermitian and skew-Hermitian splitting (MHSS) iteration method proposed by Bai, we propose an MHSS-like iteration method. Comparing with the MHSS iteration method, the MHSS-like method only requires one to solve the two-by-two linear systems by a sparse Cholesky factorization and a fast Fourier transform. Hence, the MHSS-like method has less workload than the MHSS method. The convergence properties and the quasi-optimal parameters are analyzed in detail. Numerical examples are used to testify the efficiency of the new method.

Solving a free boundary problem with nonconstant coefficients

The present article is concerned with the numerical solution of a free boundary problem for an elliptic state equation with nonconstant coefficients. We maximize the Dirichlet energy functional over all domains of fixed volume. The domain under consideration is represented by a level set function, which is driven by the objective's shape gradient. The state is computed by the finite element method where the underlying triangulation is constructed by means of a marching cubes algorithm. We show that the combination of these tools lead to an efficient solver for general shape optimization problems.

Error estimates for Neumann boundary control problems with energy regularization

AbstractA Neumann boundary control problem for a second order elliptic state equation is considered. The problem is regularized by an energy term which is equivalent to theThe state and co-state are approximated by piecewise linear finite elements. For the approximation of the control variable we use carefully designed spaces of piecewise linear or piecewise constant functions, such that an inf-sup condition is satisfied. Error estimates for the approximate solution are proved for all three variables and we show a relation between convergence rate and the opening angles at corners of the domain. As the control grows in general unboundedly near the concave corners for unconstrained problems, it becomes active and hence regular when control constraints are present. We show that in this case the convergence rates are higher than in the unbounded case. Numerical tests suggest that the estimates derived are optimal in the unconstrained case but too pessimistic in the control constrained case.

Mesh-independent convergence of penalty methods applied to optimal control with partial differential equations

In this paper, optimal control problems with elliptic state equations and constraints on controls are considered. Also state constraints are briefly discussed. Barrier-penalty methods are applied to treat the occurring restrictions. In the case of finite-dimensional optimization problems, the considered methods have a linear rate of convergence in dependence of the penalty parameter. However, in the case of infinite-dimensional problems, as studied in this article, the direct application of finite-dimensional theory, as given in Grossmann and Zadlo [A general class of penalty/barrier path-following Newton methods for nonlinear programming, Optimization 54 (2005), pp. 161–190], would lead to mesh-dependent order one estimates that deteriorate if the discretization is refined. In this article a first rigorous proof is given for inequality constrained problems that in the case of quadratic penalties a mesh-independence principle holds, i.e. the first-order convergence estimate holds for the continuous problem as well as for discretized problems independently of the discretization step size. The penalty techniques rest upon the control approximate reduction as discussed, e.g. in Grossmann et al. [C. Grossmann, H. Kunz, and R. Meischner, Elliptic control by penalty techniques with control reduction, in System Modeling and Optimization, IFIP Advances in Information and Communication Technology, Vol. 312, Springer, Berlin, 2009, pp. 251–267; M. Hinze, A variational discretization concept in control constrained optimization: the linear-quadratic case, Comput. Optim. Appl. 30 (2005), pp. 45–61]. For the discretization conforming linear element discretization is applied. Some numerical examples illustrate and confirm the theoretical results.

A semismooth Newton method for a class of semilinear optimal control problems with box and volume constraints

In this paper we consider optimal control problems subject to a semilinear elliptic state equation together with the control constraints 0≤u≤1 and ∫u=m. Optimality conditions for this problem are derived and reformulated as a nonlinear, nonsmooth equation which is solved using a semismooth Newton method. A regularization of the nonsmooth equation is necessary to obtain the superlinear convergence of the semismooth Newton method. We prove that the solutions of the regularized problems converge to a solution of the original problem and a path-following technique is used to ensure a constant decrease rate of the residual. We show that, in certain situations, the optimal controls take 0–1 values, which amounts to solving a topology optimization problem with volume constraint.

Iteration–discretization methods for variational inequalities over fixed point sets

In the last few years a wide range of iterative methods has been developed to treat variational inequalities over fixed point sets in Hilbert spaces. As a rule computational handling of problems in infinite dimensional Hilbert spaces in addition requires some discretization. Any useful discretization of the original leads to families of variational inequalities over families of fixed point sets over finite dimensional spaces. Thus, two infinite techniques, namely discretization and iteration, are embedded into each other. In the present paper this task is addressed by an iterative method with only a finite number of steps of the proposed iterative method in each of the discrete spaces. From the algorithmic point of view these methods are of iteration–discretization type. The major aim here is to provide the convergence analysis for the introduced abstract iteration–discretization methods. As an illustration we later apply the method to a simple control problem with elliptic state equations and some bound on the controls. As the discretization technique for the state equation a nested family of piecewise linear C0-elements conforming finite element discretizations is used. These discretizations provide inner approximations of the underlying fixed point problems for the variational inequalities. The abstract convergence analysis given in the first part covers the considered illustrative example.

A Mesh-Independence Principle for Quadratic Penalties Applied to Semilinear Elliptic Boundary Control

The quadratic loss penalty is a well known technique for opti- mization and control problems to treat constraints. In the present paper they are applied to handle control bounds in a boundary control problems with semilinear elliptic state equations. Unlike in the case of finite dimensional optimization for infinite dimensional problems the order of convergence could only be roughly estimated, but numerical experiments revealed a clearly better convergence behavior with constants independent of the dimension of the used discretization. The main result in the present paper is the proof of sharp con- vergence bounds for both, the finite und infinite dimensional problem with a mesh-independence in case of the discretization. Further, to achieve an efficient realization of penalty methods the principle of control reduction is applied, i.e. the control variable is represented by the adjoint state variable by means of some nonlinear function. The resulting optimality system this way depends only on the state and adjoint state. This system is discretized by conforming linear finite elements. Numerical experiments show exactly the theoretically predicted behavior of the studied penalty technique.

Second Order Analysis for Bang-Bang Control Problems of PDEs

In this paper, we derive some sufficient second order optimality conditions for control problems of partial differential equations (PDEs) when the cost functional does not involve the usual quadratic term for the control or higher nonlinearities for it. Though not always, in this situation the optimal control is typically bang-bang. Two different control problems are studied. The second differs from the first in the presence of the $L^1$ norm of the control. This term leads to optimal controls that are sparse and usually take only three different values (we call them bang-bang-bang controls). Though the proofs are detailed in the case of a semilinear elliptic state equation, the approach can be extended to parabolic control problems. Some hints are provided in the last section to extend the results.

OPTIMAL DESIGN OF LOW-CONTRAST TWO-PHASE STRUCTURES FOR THE WAVE EQUATION

This paper is concerned with the following optimal design problem: find the distribution of two phases in a given domain that minimizes an objective function evaluated through the solution of a wave equation. This type of optimization problem is known to be ill-posed in the sense that it generically does not admit a minimizer among classical admissible designs. Its relaxation could be found, in principle, through homogenization theory but, unfortunately, it is not always explicit, in particular for objective functions depending on the solution gradient. To circumvent this difficulty, we make the simplifying assumption that the two phases have a low constrast. Then, a second-order asymptotic expansion with respect to the small amplitude of the phase coefficients yields a simplified optimal design problem which is amenable to relaxation by means of H-measures. We prove a general existence theorem in a larger class of composite materials and propose a numerical algorithm to compute minimizers in this context. As in the case of an elliptic state equation, the optimal composites are shown to be rank-one laminates. However, the proof that relaxation and small-amplitude limit commute is more delicate than in the elliptic case.

A priori error estimates for elliptic optimal control problems with a bilinear state equation

In this paper a priori error analysis for the finite element discretization of an optimal control problem governed by an elliptic state equation is considered. The control variable enters the state equation as a coefficient and is subject to pointwise inequality constraints. We derive a priori error estimates for the discretization error in the control variable and confirm our theoretical results by numerical examples.

A Posteriori Error Estimates for Optimal Control Problems

We present new a posteriori error estimates for approximate solutions of optimal control problems. Our approach is based on so-called functional a posteriori error estimates that were derived in the last decade for many elliptic and parabolic boundary value problems. They provide guaranteed two-sided bounds of approximation errors for any conforming approximation of a problem considered. In the paper, we consider optimal control problems with distributed control functions that enter right-hand sides of elliptic type state equations and apply functional a posteriori estimates to approximate solutions of state equations. This procedure leads to guaranteed two-sided estimates for the cost functional and also allows to obtain guaranteed a posteriori estimates for state and control functions.

Optimal design under the one-dimensional wave equation

An optimal design problem governed by the wave equation is examined in detail. Specifically, we seek the time-dependent optimal layout of two isotropic materials on a 1-d domain by minimizing a functional depending quadratically on the gradient of the state with coefficients that may depend on space, time and design. As it is typical in this kind of problems, they are ill-posed in the sense that there is not an optimal design. We therefore examine relaxation by using the representation of two-dimensional ($(x,t)\\in\\bkR^{2}$) divergence free vector fields as rotated gradients. By means of gradient Young measures, we transform the original optimal design problem into a non-convex vector variational problem, for which we can compute an explicit form of the \`\`\\emph{constrained quasiconvexification }" of the cost density. Moreover, this quasiconvexification is recovered by first or second-order laminates which give us the optimal microstructure at every point. Finally, we analyze the relaxed problem and some numerical experiments are performed. The perspective is similar to the one developed in previous papers for linear elliptic state equations. The novelty here lies in the state equation (the wave equation), and our contribution consists in understanding the differences with respect to elliptic cases.

Discretization of elliptic control problems with time dependent parameters

We consider linear-quadratic problems of optimal control with an elliptic state equation and control constraints. For a discretization of the state equation by the method of Finite Differences and a piecewise approximation of the control we develop error estimates for the solution of the discrete problem and, based on the optimality conditions, we construct a new feasible control for which we derive error estimates of quadratic order.

Symmetric Indefinite Preconditioners for Saddle Point Problems with Applications to PDE-Constrained Optimization Problems

We consider large scale sparse linear systems in saddle point form. A natural property of such indefinite 2-by-2 block systems is the positivity of the (1,1) block on the kernel of the (2,1) block. Many solution methods, however, require that the positivity of the (1,1) block is satisfied everywhere. To enforce the positivity everywhere, an augmented Lagrangian approach is usually chosen. However, the adjustment of the involved parameters is a critical issue. We will present a different approach that is not based on such an explicit augmentation technique. For the considered class of symmetric and indefinite preconditioners, assumptions are presented that lead to symmetric and positive definite problems with respect to a particular scalar product. Therefore, conjugate gradient acceleration can be used. An important class of applications are optimal control problems. It is typical for such problems that the cost functional contains an extra regularization parameter. For control problems with elliptic state equations and distributed control, a special indefinite preconditioner for the discretized problem is constructed, which leads to convergence rates of the preconditioned conjugate gradient method that are not only independent of the mesh size but also independent of the regularization parameter. Numerical experiments are presented for illustrating the theoretical results.

Study of the Optimal Harvesting Control and the Optimality System for an Elliptic Problem

An optimal harvesting problem with a concave nonquadratic cost functional and a diffusive degenerate elliptic logistic state equation type is investigated. Under certain assumptions, we prove the existence and uniqueness of an optimal control. A characterization of the optimal control via the optimality system is also derived, which leads to approximating the optimal control.

Optimization of conducting structures by using the homogenization method

Approximation and numerical realization of a class of optimization problems with control variables represented by coefficients of linear elliptic state equations is considered. Convergence analysis of well-posed problems is performed by using one- and two-level approximation strategies. The latter is utilized in an optimization layout problem for two conductive constituents, for which the necessary steps to transfer the well-posed problem into a computational form are described and some numerical experiments are given.

Reliable Solution for a 1D Quasilinear Elliptic Equation with Uncertain Coefficients

The solution of a quasilinear elliptic state equation depends on the coefficient function belonging to an admissible set. The solution is evaluated by a cost functional the value of which is to be maximized over the admissible set, i.e., the reliable (safe) solution is searched for. Due to the nature of the equation, the Kirchhoff transformation can be applied to obtain both the existence of the true state solution and a cost sensitivity formula. In many cases, the latter makes it possible to determine the reliable solution immediately. The problem is approximated by means of the finite element method, and some convergence results are proven. Numerical examples illustrate the theory which can be directly generalized to spatial problems.