606 research outputs found
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
Optimal control of fractional semilinear PDEs
In this paper we consider the optimal control of semilinear fractional PDEs
with both spectral and integral fractional diffusion operators of order
with . We first prove the boundedness of solutions to both
semilinear fractional PDEs under minimal regularity assumptions on domain and
data. We next introduce an optimal growth condition on the nonlinearity to show
the Lipschitz continuity of the solution map for the semilinear elliptic
equations with respect to the data. We further apply our ideas to show
existence of solutions to optimal control problems with semilinear fractional
equations as constraints. Under the standard assumptions on the nonlinearity
(twice continuously differentiable) we derive the first and second order
optimality conditions
Sobolev spaces with non-Muckenhoupt weights, fractional elliptic operators, and applications
We propose a new variational model in weighted Sobolev spaces with
non-standard weights and applications to image processing. We show that these
weights are, in general, not of Muckenhoupt type and therefore the classical
analysis tools may not apply. For special cases of the weights, the resulting
variational problem is known to be equivalent to the fractional Poisson
problem. The trace space for the weighted Sobolev space is identified to be
embedded in a weighted space. We propose a finite element scheme to solve
the Euler-Lagrange equations, and for the image denoising application we
propose an algorithm to identify the unknown weights. The approach is
illustrated on several test problems and it yields better results when compared
to the existing total variation techniques
Optimal Control Of Surface Shape
Controlling the shapes of surfaces provides a novel way to direct
self-assembly of colloidal particles on those surfaces and may be useful for
material design. This motivates the investigation of an optimal control problem
for surface shape in this paper. Specifically, we consider an objective
(tracking) functional for surface shape with the prescribed mean curvature
equation in graph form as a state constraint. The control variable is the
prescribed curvature. We prove existence of an optimal control, and using
improved regularity estimates, we show sufficient differentiability to make
sense of the first order optimality conditions. This allows us to rigorously
compute the gradient of the objective functional for both the continuous and
discrete (finite element) formulations of the problem. Moreover, we provide
error estimates for the state variable and adjoint state. Numerical results are
shown to illustrate the minimizers and optimal controls on different domains
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