Recently, various papers have given consideration to how fuzzy sets might be useful in geometric computing and solid modelling. This paper reviews this previous work and related topics, and adds further observations on the use of fuzzy sets for inexact shape modelling. It is noted that many serious problems remain to be solved. The wide range of solid modelling applications requiring the genera...

First-order proximal methods that solve linear and bilinear elliptic optimal control problems with a sparsity cost functional are discussed. In particular, fast convergence of these methods is proved. For benchmarking purposes, inexact proximal schemes are compared to an inexact semismooth Newton method. Results of numerical experiments are presented to demonstrate the computational effectivene...

In this paper, we consider a framework of inexact proximal point methods for convex optimization that allows a relative error tolerance in the approximate solution of each proximal subproblem and establish its convergence rate. We then show that the well-known forward-backward splitting algorithm for convex optimization belongs to this framework. Finally, we propose and establish the iteration-...

Given F : IR n ! IR m and a closed and convex set, the problem of nding x 2 IR n such that x 2 and F (x) = 0 is considered. For solving this problem an algorithm of Inexact-Newton type is de-ned. Global and local convergence proofs are presented. As a practical application, the Horizontal Nonlinear Complementarity Problem is introduced. It is shown that the Inexact-Newton algorithm can be appli...

In this work we evaluate the performance of inexact Newton-type schemes to solve the nonlinear equations arising from the SUPG/PSPG finite element formulation of steady viscoplastic incompressible fluid flows. The flow through an abrupt contraction and the rotational flow in eccentric annulus with power law and Bingham rheologies are employed as benchmarks. Our results have shown that inexact s...

We consider the local behaviour of inexact Newton methods for the solution of a semis-mooth system of equations. In particular, we give a complete characterization of the Q-superlinear and Q-quadratic convergence of inexact Newton methods. We then apply these results to a particular semismooth system of equations arising from variational inequality problems, and present a globally and locally f...

As an extension of the inexact Uzawa algorithm addressed in 1], we propose an improved inexact Uzawa-type iterative method for solving saddle point problems. The convergence rate is given in terms of the rates of the two basic iterations as deened in 1], and it is shown that the new algorithm always converges as long as the two basic iterations converge. A running title: Uzawa algorithm for sad...

Inexact algorithms allow certain operations (typically matrix–vector products or certain function evaluations) to be performed inexactly, either out of necessity or deliberately, in view of trading accuracy for speed. We review recent findings that show the impact of the inexact approach in the uniformization and generalized minimal residual (GMRES) methods when computing the transient, station...

This paper focuses on the convergence of the modified nonlinear inexact Uzawa algorithm (MNIU) for solving the saddle point problem. We improve the sufficient conditions for convergence and the convergence rate shown in Bramble et al. [J. Bramble, J. Pasciak, and A. Vassilev, Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal., 34 (1997), pp. 1072–1092] and Y...

We study the computational complexity certification of inexact gradient augmented Lagrangian methods for solving convex optimization problems with complicated constraints. We solve the augmented Lagrangian dual problem that arises from the relaxation of complicating constraints with gradient and fast gradient methods based on inexact first order information. Moreover, since the exact solution o...