4,335 research outputs found
Optimal Control of Convective FitzHugh-Nagumo Equation
We investigate smooth and sparse optimal control problems for convective
FitzHugh-Nagumo equation with travelling wave solutions in moving excitable
media. The cost function includes distributed space-time and terminal
observations or targets. The state and adjoint equations are discretized in
space by symmetric interior point Galerkin (SIPG) method and by backward Euler
method in time. Several numerical results are presented for the control of the
travelling waves. We also show numerically the validity of the second order
optimality conditions for the local solutions of the sparse optimal control
problem for vanishing Tikhonov regularization parameter. Further, we estimate
the distance between the discrete control and associated local optima
numerically by the help of the perturbation method and the smallest eigenvalue
of the reduced Hessian
Preconditioning of Active-Set Newton Methods for PDE-constrained Optimal Control Problems
We address the problem of preconditioning a sequence of saddle point linear
systems arising in the solution of PDE-constrained optimal control problems via
active-set Newton methods, with control and (regularized) state constraints. We
present two new preconditioners based on a full block matrix factorization of
the Schur complement of the Jacobian matrices, where the active-set blocks are
merged into the constraint blocks. We discuss the robustness of the new
preconditioners with respect to the parameters of the continuous and discrete
problems. Numerical experiments on 3D problems are presented, including
comparisons with existing approaches based on preconditioned conjugate
gradients in a nonstandard inner product
Solving optimal control problems governed by random Navier-Stokes equations using low-rank methods
Many problems in computational science and engineering are simultaneously
characterized by the following challenging issues: uncertainty, nonlinearity,
nonstationarity and high dimensionality. Existing numerical techniques for such
models would typically require considerable computational and storage
resources. This is the case, for instance, for an optimization problem governed
by time-dependent Navier-Stokes equations with uncertain inputs. In particular,
the stochastic Galerkin finite element method often leads to a prohibitively
high dimensional saddle-point system with tensor product structure. In this
paper, we approximate the solution by the low-rank Tensor Train decomposition,
and present a numerically efficient algorithm to solve the optimality equations
directly in the low-rank representation. We show that the solution of the
vorticity minimization problem with a distributed control admits a
representation with ranks that depend modestly on model and discretization
parameters even for high Reynolds numbers. For lower Reynolds numbers this is
also the case for a boundary control. This opens the way for a reduced-order
modeling of the stochastic optimal flow control with a moderate cost at all
stages.Comment: 29 page
On the Virtual Element Method for Topology Optimization on polygonal meshes: a numerical study
It is well known that the solution of topology optimization problems may be
affected both by the geometric properties of the computational mesh, which can
steer the minimization process towards local (and non-physical) minima, and by
the accuracy of the method employed to discretize the underlying differential
problem, which may not be able to correctly capture the physics of the problem.
In light of the above remarks, in this paper we consider polygonal meshes and
employ the virtual element method (VEM) to solve two classes of paradigmatic
topology optimization problems, one governed by nearly-incompressible and
compressible linear elasticity and the other by Stokes equations. Several
numerical results show the virtues of our polygonal VEM based approach with
respect to more standard methods
Error estimates for the numerical approximation of a distributed optimal control problem governed by the von K\'arm\'an equations
In this paper, we discuss the numerical approximation of a distributed
optimal control problem governed by the von Karman equations, defined in
polygonal domains with point-wise control constraints. Conforming finite
elements are employed to discretize the state and adjoint variables. The
control is discretized using piece-wise constant approximations. A priori error
estimates are derived for the state, adjoint and control variables under
minimal regularity assumptions on the exact solution. Numerical results that
justify the theoretical results are presented
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