A FEM for an Optimal Control Problem of Fractional Powers of Elliptic Operators

Abstract

Abstract. We study solution techniques for a linear-quadratic optimal control problem involving fractional powers of elliptic operators. These fractional operators can be realized as the Dirichletto-Neumann map for a nonuniformly elliptic problem posed on a semi-infinite cylinder in one more spatial dimension. Thus, we consider an equivalent formulation with a nonuniformly elliptic operator as state equation. The rapid decay of the solution to this problem suggests a truncation that is suitable for numerical approximation. We discretize the proposed truncated state equation using first degree tensor product finite elements on anisotropic meshes. For the control problem we analyze two approaches: one that is semi-discrete based on the so-called variational approach, where the control is not discretized, and the other one is fully discrete via the discretization of the control by piecewise constant functions. For both approaches, we derive a priori error estimates with respect to the degrees of freedom. Numerical experiments validate the derived error estimates and reveal a competitive performance of anisotropic over quasi-uniform refinement.

A solution for fractional PDE constrained optimization problems using reduced basis method

“…for more detail, see (Antil and Otarola , 2015). As a result, the right Caputo fractional derivative can be written as a left Caputo fractional derivative.…”

Numerical approximations for fractional elliptic equationsviathe method of semigroups

“…Similarly to what we do in this work with the heat semigroup e t∆ B f , in [52] the fast decay of the solution of (1.11) in the extension variable y ∈ [0, ∞) is exploited to obtain a convenient truncation of the extended unbounded domain. Numerical methods for optimal control problems related to (1.1)-(1.2) have also been developed using characterization (1.11) in [4] and (1.10) in [33]. We refer to the very recent work by Bonito et al [13] for a general review of numerical methods for fractional diffusion.…”

A Priori Error Estimates for the Optimal Control of the Integral Fractional Laplacian

“…PDE-constrained optimization problems involving fractional and nonlocal equations are not new in the literature; we mention, e.g., the works by Antil and Otárola [4], Otárola [41], and D'Elia and Gunzburger [18,19]. In [4], the authors consider a linear-quadratic optimal control problem for the spectral definition of the fractional Laplacian; control constraints are also considered. The authors also propose and study solution techniques to approximate the underlying solution.…”