7,207 research outputs found
A space-time pseudospectral discretization method for solving diffusion optimal control problems with two-sided fractional derivatives
We propose a direct numerical method for the solution of an optimal control
problem governed by a two-side space-fractional diffusion equation. The
presented method contains two main steps. In the first step, the space variable
is discretized by using the Jacobi-Gauss pseudospectral discretization and, in
this way, the original problem is transformed into a classical integer-order
optimal control problem. The main challenge, which we faced in this step, is to
derive the left and right fractional differentiation matrices. In this respect,
novel techniques for derivation of these matrices are presented. In the second
step, the Legendre-Gauss-Radau pseudospectral method is employed. With these
two steps, the original problem is converted into a convex quadratic
optimization problem, which can be solved efficiently by available methods. Our
approach can be easily implemented and extended to cover fractional optimal
control problems with state constraints. Five test examples are provided to
demonstrate the efficiency and validity of the presented method. The results
show that our method reaches the solutions with good accuracy and a low CPU
time.Comment: This is a preprint of a paper whose final and definite form is with
'Journal of Vibration and Control', available from
[http://journals.sagepub.com/home/jvc]. Submitted 02-June-2018; Revised
03-Sept-2018; Accepted 12-Oct-201
Pointwise-in-time error estimates for an optimal control problem with subdiffusion constraint
In this work, we present numerical analysis for a distributed optimal control
problem, with box constraint on the control, governed by a subdiffusion
equation which involves a fractional derivative of order in
time. The fully discrete scheme is obtained by applying the conforming linear
Galerkin finite element method in space, L1 scheme/backward Euler convolution
quadrature in time, and the control variable by a variational type
discretization. With a space mesh size and time stepsize , we
establish the following order of convergence for the numerical solutions of the
optimal control problem: in the
discrete norm and
in the discrete
norm, with any small and
. The analysis relies essentially on the maximal
-regularity and its discrete analogue for the subdiffusion problem.
Numerical experiments are provided to support the theoretical results.Comment: 20 pages, 6 figure
An Analysis of Galerkin Proper Orthogonal Decomposition for Subdiffusion
In this work, we develop a novel Galerkin-L1-POD scheme for the subdiffusion
model with a Caputo fractional derivative of order in time,
which is often used to describe anomalous diffusion processes in heterogeneous
media. The nonlocality of the fractional derivative requires storing all the
solutions from time zero. The proposed scheme is based on continuous piecewise
linear finite elements, L1 time stepping, and proper orthogonal decomposition
(POD). By constructing an effective reduced-order scheme using problem-adapted
basis functions, it can significantly reduce the computational complexity and
storage requirement. We shall provide a complete error analysis of the scheme
under realistic regularity assumptions by means of a novel energy argument.
Extensive numerical experiments are presented to verify the convergence
analysis and the efficiency of the proposed scheme.Comment: 25 pp, 5 figure
A FEM for an optimal control problem of fractional powers of elliptic operators
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
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
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