Numerical Approximation of a One-Dimensional Elliptic Optimal Design Problem

Casado–Díaz, Juan; Castro, Carlos; Luna-Laynez, Manuel; Zuazua, Enrique

doi:10.1137/10081928x

Cited by 13 publications

(8 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We refer to [4] for estimates in the numerical study of some optimal design problems for two-phase materials in dimension one. In the present paper, we observe that (1.9) has important consequences with respect to the existence of a solution for the unrelaxed problem, i.e., where θ is a characteristic function.…”

Section: Introductionmentioning

confidence: 99%

Smoothness Properties for the Optimal Mixture of Two Isotropic Materials: The Compliance and Eigenvalue Problems

Casado–Díaz¹

2015

SIAM J. Control Optim.

View full text Add to dashboard Cite

Abstract. The present paper is devoted to obtaining some smoothness results for the solution of two classical control problems relative to the optimal mixture of two isotropic materials. In the first one, the goal is to maximize the energy. In the second one, we want to minimize the first eigenvalue of the corresponding elliptic operator. At least for the first problem it is well known that it does not have a solution in general. Thus, we deal with a relaxed formulation. One of the applications of our results is in fact the nonexistence of a solution for the unrelaxed problem. In this sense, we improve a classical nonexistence result by Murat and Tartar for the maximization of the energy which was obtained assuming the solution smooth. We also get a counterexample to the existence of a solution for the eigenvalue problem which, to our knowledge, was an open problem.

show abstract

Section: Introductionmentioning

confidence: 99%

Smoothness Properties for the Optimal Mixture of Two Isotropic Materials: The Compliance and Eigenvalue Problems

Casado–Díaz¹

2015

SIAM J. Control Optim.

View full text Add to dashboard Cite

show abstract

“…Error estimates and numerical analysis for inverse problems involving BV-functions are studied in [5,6]. Related discussion of ODE-constrained control problems involving discontinuous functions and their numerical analysis can be found in, e.g., [1,2,10,21,25,26,37,38].The main difficulty in deriving error estimates for the above problem is given by the fact that it lacks certain coercivity properties that are usually employed to obtain error estimates for the controls, for instance by suitably testing the first order necessary optimality conditions. Hence, only error estimates for the state and the adjoint state can be proven in a rather direct manner; these are, however, suboptimal.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

Finite element error estimates for one-dimensional elliptic optimal control by BV-functions

Hafemeyer¹,

Mannel²,

Neitzel³

et al. 2019

Mathematical Control &Amp; Related Fields

View full text Add to dashboard Cite

Here, u ∈ V := H 1 0 (Ω) is the state and q ∈ Q := BV (Ω) is the control, where BV (Ω) denotes the space of functions of bounded variation (BV) on the interval Ω := (0, 1). The operator A is elliptic and α is a positive real number. The two finite element schemes that will be analyzed are identical in regard to the discretization of state and adjoint state, but they differ in the treatment of the control. In the variational discretization the control is not discretized, while in the second scheme the control is discretized by piecewise constant functions.The significance of the above control problem is given by the use of the BVseminorm q ′ M(Ω) in the objective. This favors piecewise constant controls with only a limited number of jumps, which makes this problem type interesting in many practical applications. The precise functional analytic setting will be provided in the next section.Optimal control problems with BV-controls defined in one space dimension are strongly related to control problems with measures as controls. Both BV optimal control problems and optimal control problems with measures have attracted significant research interest in the recent past, see, e.g. [8,13,14,17,22,23] for the former and [11,12,15,16,29,30] for the latter.Error estimates for PDE-constrained optimal control problems involving measures have been presented in [11,30,31,34,35]. For error estimates of further sparsity promoting optimal control problems with PDEs see for example [19,30]. The literature on error estimates for optimal control problems with controls in BV is rather limited. We are only aware of [14,18]. Error estimates and numerical analysis for inverse problems involving BV-functions are studied in [5,6]. Related discussion of ODE-constrained control problems involving discontinuous functions and their numerical analysis can be found in, e.g., [1,2,10,21,25,26,37,38].The main difficulty in deriving error estimates for the above problem is given by the fact that it lacks certain coercivity properties that are usually employed to obtain error estimates for the controls, for instance by suitably testing the first order necessary optimality conditions. Hence, only error estimates for the state and the adjoint state can be proven in a rather direct manner; these are, however, suboptimal. To obtain an error estimate for the control and also to improve the error estimates for state and adjoint state, we make use of a structural assumption on the Lagrange multiplierΦ arising from the convex subdifferential of the term q ′ M(Ω) . Specifically, we assume thatΦ, which is a C 2 function inΩ, has only finitely many global extreme points and that it exhibits quadratic growth near those points (i.e.,Φ ′′ = 0 near those points; see Assumption 4.4 and Assumption 4.5). Since the jump set of the optimal control is contained in the set of global extreme points ofΦ, see Corollary 1, this assumption implies that the optimal control admits only finitely many jumps, which is a rather typical situation in practice. In addition, it ensures ...

show abstract

“…It is also possible to take simultaneously ε → 0, h → 0 to show the approximation of (R). In the one dimensional case, for two materials, rates of convergence are established in [4].…”

Section: The Discretized Optimization Problem Ismentioning

confidence: 99%

A Penalization and Regularization Technique in Shape Optimization Problems

Philip¹,

Tiba²

2013

SIAM J. Control Optim.

View full text Add to dashboard Cite

We consider shape optimization problems, where the state is governed by elliptic partial differential equations (PDE). Using a regularization technique, unknown shapes are encoded via shape functions, turning the shape optimization into optimal control problems for the unknown functions. The method is studied for elliptic PDE to be solved in an unknown region (to be optimized), where the regularization technique together with a penalty method extends the PDE to a larger fixed domain. Additionally, the method is studied for the optimal layout problem, where the unknown regions determine the coefficients of the state equation. In both cases and in arbitrary dimension, the existence of optimal shapes is established for the regularized and the original problem, with convergence of optimal shapes if the regularization parameter tends to zero. Error estimates are proved for the layout problem. In the context of finite element approximations, convergence and differentiability properties are shown. A series of numerical experiments demonstrate the method computationally for an industrially relevant elliptic PDE with two unknown shapes, one giving the region where the PDE is solved, and the other determining the PDE's coefficients.

show abstract

Numerical Approximation of a One-Dimensional Elliptic Optimal Design Problem

Cited by 13 publications

References 17 publications

Smoothness Properties for the Optimal Mixture of Two Isotropic Materials: The Compliance and Eigenvalue Problems

Smoothness Properties for the Optimal Mixture of Two Isotropic Materials: The Compliance and Eigenvalue Problems

Finite element error estimates for one-dimensional elliptic optimal control by BV-functions

A Penalization and Regularization Technique in Shape Optimization Problems

Contact Info

Product

Resources

About