In this article, the denoising of smooth (H 1-regular) images is considered. To reach this objective, we introduce a simple and highly efficient over-relaxation technique for solving the convex, ...non-smooth optimization problems resulting from the denoising formulation. We describe the algorithm, discuss its convergence and present the results of numerical experiments, which validate the methods under consideration with respect to both efficiency and denoising capability. Several issues concerning the convergence of an Uzawa algorithm for the solution of the same problem are also discussed.
In this paper, we study the solution of a certain non-smooth eigenvalue problem, using operator-splitting methods to solve an equivalent, constrained minimization problem. We present the ...Marchuk-Yanenko and Peaceman-Rachford schemes for solving the problem and compare their performance numerically on some model problems. The Peaceman-Rachford scheme turns out to be superior to the Marchuk-Yanenko scheme in terms of accuracy and computational efficiency.
Dedicated to J. Douglas, G. Marchuk, D. Peaceman, H. Rachford, and G. Strang for their contributions to
operator-splitting methods.
This article is concerned with the numerical solution of multiobjective control problems associated with linear partial differential equations. More precisely, for such problems, we look for the Nash ...equilibrium, which is the solution to a noncooperative game. First, we study the continuous case. Then, to compute the solution of the problem, we combine finite-difference methods for the time discretization, finite-element methods for the space discretization, and conjugate-gradient algorithms for the iterative solution of the discrete control problems. Finally, we apply the above methodology to the solution of several tests problems.
We discuss in this article the application of controllability techniques to the computation of the time-periodic solutions of evolution equations. The basic principles of the computational methods ...are presented in a fairly general context where the time discretization aspect is also discussed. Then this general methodology is applied to the solution of scattering problems for harmonic planar waves by two- and three-dimensional purely reflecting nonconvex obstacles. Numerical results obtained by the above method and comparisons with the results obtained by more classical methods show the superiority of the former ones.
In this paper an operator-splitting method is applied to find the micro-structure of a liquid crystal model with a simplified Oseen–Frank energy functional. Both projection and penalty methods are ...used to deal with the constant length constraint. The methods are implemented to compute director fields of liquid crystal slabs of various shapes and with various boundary data. The computational results verify researcher expectations. Some new singularity patterns are observed as well.
This article is concerned with the numerical solution of multiobjective control problems associated with nonlinear partial differential equations and more precisely the Burgers equation. For this ...kind of problems, we look for the Nash equilibrium, which is the solution to a noncooperative game. To compute the solution of the problem, we use a combination of finite-difference methods for the time discretization, finite-element methods for the space discretization, and a quasi-Newton BFGS algorithm for the iterative solution of the discrete control problem. Finally, we apply the above methodology to the solution of several tests problems. To be able to compare our results with existing results in the literature, we discuss first a single-objective control problem, already investigated by other authors. Finally, we discuss the multiobjective case.
This article discusses computational methods for the numerical simulation of unsteady Bing-ham visco-plastic flow. These methods are based on time-discretization by operator-splitting and take ...advantage of a characterization of the solutions involving some kind of Lagrange multipliers. The full discretization is achieved by combining the above operator-splitting meth-ods with finite element approximations, the advection being treated by a wave-like equation“equivalent” formulation easier to implement than the method of characteristics or high order upwinding methods. The authors illustrate the methodology discussed in this article with the results of numerical experiments concerning the simulation of wall driven cavity Bingham flowin two dimensions.
There are very important results by Enrique Zuazua on the subject of the convection-diffusion equation
u
t
−div(
a
(
x
)∇
u
)=−
d
⋅∇(|
u
|
q
−1
u
), in (0,+∞) × ℝ
N
.
In some sense this paper deals ...with a linear (
i.e. q
= 1) elliptic counterpart of the above equation if
d
is not constant.
We prove regularizing results on the solutions, under assumptions of interplay between the datum and the coefficient of the zero order term or between the modulus of the drift and the coefficient of the zero order term.
We study the controllability of a coupled system of linear parabolic equations, with nonnegativity constraint on the state. We establish two results of controllability to trajectories in large time: ...one for diagonal diffusion matrices with an “approximate” nonnegativity constraint, and a another stronger one, with “exact” nonnegativity constraint, when all the diffusion coefficients are equal and the eigenvalues of the coupling matrix have nonnegative real part. The proofs are based on a “staircase” method. Finally, we show that state-constrained controllability admits a positive minimal time, even with weaker unilateral constraint on the state.
PDF
