Paper Info
Title
A FEM for an Optimal Control Problem of Fractional Powers of Elliptic Operators.
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 Dirichlet-to-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 the 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 semidiscrete based on the so-called variational approach, where the control is not discretized, and the other one that 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 degrees of freedom. Numerical experiments validate the derived error estimates and reveal a competitive performance of anisotropic over quasi-uniform refinement.
Year
DOI
Venue
2015
10.1137/140975061
SIAM JOURNAL ON CONTROL AND OPTIMIZATION
Keywords
Field
DocType
linear-quadratic optimal control problem,fractional derivatives,fractional diffusion,weighted Sobolev spaces,finite elements,stability,anisotropic estimates
Tensor product,Discretization,Mathematical optimization,Optimal control,Mathematical analysis,Elliptic operator,Finite element method,Operator (computer programming),Fractional calculus,Mathematics,Piecewise
Journal
Volume
Issue
ISSN
53
6
0363-0129
Citations 
PageRank 
References 
8
0.91
13
Authors
2
Name
Order
Citations
PageRank
H. Antil17110.56
Enrique Otárola28613.91