Adaptive finite element methods for sparse PDE-constrained optimization

Abstract

We propose and analyze reliable and efficient a posteriori error estimators for an optimal control problem that involves a nondifferentiable cost functional, the Poisson problem as state equation and control constraints. To approximate the solution to the state and adjoint equations we consider a piecewise linear finite element method whereas three different strategies are used to approximate the control variable: piecewise constant discretization, piecewise linear discretization and the so-called variational discretization approach. For the first two aforementioned solution techniques we devise an error estimator that can be decomposed as the sum of four contributions: two contributions that account for the discretization of the control variable and the associated subgradient, and two contributions related to the discretization of the state and adjoint equations. The error estimator for the variational discretization approach is decomposed only in two contributions that are related to the discretization of the state and adjoint equations. On the basis of the devised a posteriori error estimators, we design simple adaptive strategies that yield optimal rates of convergence for the numerical examples that we perform. PDE-constrained optimization, nondifferentiable objectives, sparse controls, a posteriori error analysis, adaptive finite elements.

An adaptive finite element method for the sparse optimal control of fractional diffusion

“…When s = 0.8 (Figure 3c) and s = 0.9 (Figure 3d), the incompatibility of the desired data is not longer active and then the optimal adjoint state do not exhibits boundary layers. For the case s = 1 such a theory is available in [55]. • Sparse optimal controls: The fact that the cost functional J involves the term ‖z‖ L 1 (Ω) leads to sparsely supported optimal controls.…”

“…We refer the reader to [48][49][50] for an up to date discussion that also includes the design of AFEMs, convergence results, and optimal complexity. To the best of our knowledge the only works that provide an advance concerning this matter are [35,55]. Starting with the pioneering work [51], several authors have contributed to its advancement.…”

“…Starting with the pioneering work [51], several authors have contributed to its advancement. These results have been recently extended in [55], where the authors consider three different strategies to approximate the control variable: piecewise constant discretization, piecewise linear discretization, and the so-called variational discretization approach. In contrast to these advances, the theory for optimal control problems involving a sparsity functional, as (1.2), is much less developed.…”

“…In contrast to these advances, the theory for optimal control problems involving a sparsity functional, as (1.2), is much less developed. To the best of our knowledge the only works that provide an advance concerning this matter are [35,55]. In [35], the authors propose a residual-type a posteriori error estimator, for the piecewise constant discretization of the optimal control, and prove that it yields an upper bound for the approximation error of the state and control variables ( [35], Theorem 6.2).…”

“…In [35], the authors propose a residual-type a posteriori error estimator, for the piecewise constant discretization of the optimal control, and prove that it yields an upper bound for the approximation error of the state and control variables ( [35], Theorem 6.2). These results have been recently extended in [55], where the authors consider three different strategies to approximate the control variable: piecewise constant discretization, piecewise linear discretization, and the so-called variational discretization approach. The authors propose a posteriori error estimators, for each scheme, and derive reliability and efficiency estimates.…”