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Alternating direction methods (ADMs) have been well studied in the literature, and they have found many efficient applications in various fields. In this note, we focus on the Douglas-Rachford ADM scheme proposed by Glowinski and Marrocco, and we aim at providing a simple approach to estimating its convergence rate in terms of the iteration number. The linearized version of this ADM scheme, which is known as the split inexact Uzawa method in the image processing literature, is also discussed.

This paper introduces a novel algorithm to approximate the matrix with minimum nuclear norm among all matrices obeying a set of convex constraints. This problem may be understood as the convex relaxation of a rank minimization problem and arises in many important applications as in the task of recovering a large matrix from a small subset of its entries (the famous Netflix problem). The algorithm is iterative, produces a sequence of matrices ..., and at each step mainly performs a soft-thresholding operation on the singular values of the matrix ... On the theoretical side, the paper provides a convergence analysis showing that the sequence of iterates converges. On the practical side, it provides numerical examples in which 1,000 x 1,000 matrices are recovered in less than a minute on a modest desktop computer. (ProQuest: ... denotes formulae/symbols omitted.)

The evidence for the United States points to balanced growth despite falling investment-good prices and a less-than-unitary elasticity of substitution between capital and labor. This is inconsistent with the Uzawa Growth Theorem. We extend Uzawïs theorem to show that the introduction of human capital accumulation in the standard way does not resolve the puzzle. However, balanced growth is possible if education is endogenous and capital is more complementary with schooling than with raw labor. We present a class of aggregate production functions for which a neoclassical growth model with capital-augmenting technological progress and endogenous schooling converges to a balanced growth path.

This study introduces Stackelberg-Nash equilibrium to neoclassical growth theory. It attempts to make neoclassical economic growth theory more robust in modelling the complexity of market structures. The model is constructed within the framework of the Solow-Uzawa two-sector model. The economy is composed of two sectors. The final goods sector is the same as in the Solow one-sector growth model which is characterized by perfect competition. The consumer goods sector is the same as the consumer goods sector in the Uzawa model but is characterized by Stackelberg duopoly. We model household behavior with Zhang’s concept of disposable income and utility. The model endogenously determines profits of duopoly which are equally distributed among the homogeneous population. We build the model and then identify the existence of an equilibrium point through simulation. We conduct comparative static analyses of some parameters. We also compare the economic performance of the traditional Uzawa model and the model with the Stackelberg-Nash equilibrium. We conclude that the imperfect competition increases national output, national wealth, and utility level in comparison to perfect competition. 

In this paper, we propose a unified primal-dual algorithm framework for two classes of problems that arise from various signal and image processing applications. We also show the connections to existing methods, in particular Bregman iteration (Osher et al., Multiscale Model. Simul. 4(2):460–489, 2005 ) based methods, such as linearized Bregman (Osher et al., Commun. Math. Sci. 8(1):93–111, 2010 ; Cai et al., SIAM J. Imag. Sci. 2(1):226–252, 2009 , CAM Report 09-28, UCLA, March 2009 ; Yin, CAAM Report, Rice University, 2009 ) and split Bregman (Goldstein and Osher, SIAM J. Imag. Sci., 2, 2009 ). The convergence of the general algorithm framework is proved under mild assumptions. The applications to ℓ 1 basis pursuit, TV− L 2 minimization and matrix completion are demonstrated. Finally, the numerical examples show the algorithms proposed are easy to implement, efficient, stable and flexible enough to cover a wide variety of applications.

We consider a fictitious domain formulation of an elliptic partial differential equation and approximate the resulting saddle-point system using a nested inexact preconditioned Uzawa iterative algorithm, which consists of three nested loops. In the outer loop the trial space for the Galerkin approximation of the Lagrange multiplier is enlarged. The intermediate loop solves this Galerkin system by a damped preconditioned Richardson iteration. Each iteration of the latter involves solving an elliptic problem on the fictitious domain whose solution is approximated by an adaptive finite element method in the inner loop. We prove that the overall method converges with the best possible rate and illustrate numerically our theoretical findings.

The purpose of this paper is to construct optimal error estimates of second-order stabilized scheme for the incompressible magnetohydrodynamic(MHD) system. For this purpose, we first construct first- and second-order semi-discrete schemes in which the time derivative term is treated by the first-order backward Euler method and the second-order backward difference formulation, respectively. Moreover, the nonlinear terms are treated by semi-implicit method, and the coupling of velocity and pressure is decoupled by a Gauge–Uzawa method. Thus, the schemes achieve decoupling of velocity, magnetic field and pressure. Most importantly, the proposed schemes do not need to deal with the artificial boundary conditions on pressure and do not need to give an initial value of pressure. Then, the unconditional stability of the two schemes is demonstrated. Furthermore, through rigorous error analysis, we provide optimal convergence orders for all unknowns. Finally, some numerical experiments demonstrate the accuracy and effectiveness of the proposed schemes.

We consider symmetric saddle point matrices. We analyze block preconditioners based on the knowledge of a good approximation for both the top left block and the Schur complement resulting from its elimination. We obtain bounds on the eigenvalues of the preconditioned matrix that depend only of the quality of these approximations, as measured by the related condition numbers. Our analysis applies to indefinite block diagonal preconditioners, block triangular preconditioners, inexact Uzawa preconditioners, block approximate factorization preconditioners, and a further enhancement of these preconditioners based on symmetric block Gauss--Seidel-type iterations. The analysis is unified and allows the comparison of these different approaches. In particular, it reveals that block triangular and inexact Uzawa preconditioners lead to identical eigenvalue distributions. These theoretical results are illustrated on the discrete Stokes problem. It turns out that the provided bounds allow one to localize accurately both real and nonreal eigenvalues. The relative quality of the different types of preconditioners is also as expected from the theory. [PUBLICATION ABSTRACT]