In this paper, a new theory is developed for firstorder stochastic convex optimization, showing that the global convergence rate is sufficiently quantified by a local growth rate of the objective function in a neighborhood of the optimal solutions. In particular, if the objective function F (w) in the -sublevel set grows as fast as ‖w − w∗‖ 2 , where w∗ represents the closest optimal solution t...

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In this paper, we study the stability of backward stochastic diierential equations (BSDEs for short) w.r.t. the Brownian motion; more precisely, we will show that if W n is a martingale approximation of a Brownian motion W then the solution to the BSDE driven by the mar-tingale W n converges to the solution of the classical BSDE, namely the BSDE driven by W. The particular case of the scaled ra...

Certain implicit time stepping procedures for the incompressible Stokes or Navier-Stokes equations lead to a singular-perturbed Stokes type problem at each type step. The paper presents a convergence analysis of a geometric multigrid solver for the system of linear algebraic equations resulting from the disretization of the problem using a finite element method. Several smoothing iterative meth...

We discuss a mortar-type P 1 /P0 element method for the incompressible Stokes problem. We prove the inf-sup condition and obtain the optimal error estimate. Meanwhile, we propose a W-cycle multigrid for solving this discrete problem and prove the optimal convergence of the multigrid method, that is, the convergence rate is independent of the mesh size and mesh level. Finally, numerical experime...

In this paper, we introduce a new iterative scheme by viscosity approximation methods for strict pseudo-contractive nonself-mappings in Hilbert spaces and obtain a strong convergence theorem. The theorem improves and extends the result of Shimizu and Takahashi [T. Shimizu, W. Takahashi, Strong convergence to common fixed points of families of nonexpansive mappings, J. Math. Anal. Appl. 211 (199...

where 1 sp < co. This definition is far from being intrinsic. For an intrinsic definition of WrTp(Mm, N”) see [ 11. In this space, beside the standard topology induced by the norm (1. ]ll,p, we also have weak topology and weak convergence. Let fk, f E W’~p(Mm), where 1 < p < co. We say that fk converges to f in weak topology iff fk + f in Lp and the set (lIVfk[lp)k is bounded. Weak convergence ...

In this paper we consider the regularization of the inverse problem of diffuse optical tomography by standard regularization methods with quadratic penalty terms. We therefore investigate in detail the properties of the associated forward operators, and derive continuity and differentiability results, which are based on derivation of W 1,p regularity results for the governing elliptic boundary ...