Inexact-Newton method for solving operator equations in infinite-dimensional spaces

In this paper, we develop an inexact-Newton method for solving nonsmooth operator equations in infinite-dimensional spaces. The linear convergence and superlinear convergence of inexact-Newton method under some conditions are shown. Then, we characterize the order of convergence in terms of the rate of convergence of the relative residuals. The present inexact-Newton method could be viewed as the extensions of previous ones with same convergent results in finite-dimensional spaces.     

On Hukuhara’s differentiable iteration semigroups of linear set-valued functions

In this paper, we investigate the uniform convergence of continuous linear set-valued functions on compact sets. We also consider conditions under which the family of continuous linear extensions of a differential iteration semigroup of continuous linear set-valued functions is a differentiable iteration semigroup. In particular, since the cones and normed spaces are not supposed to be complete our main results generalize some recent results on Hukuhara’s derivative of set-valued functions.     

Wavelet Approximation in Weighted Sobolev Spaces of Mixed Order with Applications to the Electronic Schr?dinger Equation

We study the approximation of functions in weighted Sobolev spaces of mixed order by anisotropic tensor products of biorthogonal, compactly supported wavelets. As a main result, we characterize these spaces in terms of wavelet coefficients, which also enables us to explicitly construct approximations. In particular, we derive approximation rates for functions in exponentially weighted Sobolev spaces discretized on optimized general sparse grids. Under certain regularity assumptions, the rate of convergence is independent of the number of dimensions. We apply these results to the electronic Schr?dinger equation and obtain a convergence rate which is independent of the number of electrons; numerical results for the helium atom are presented.     

Convergence of a finite element method for non-parametric mean curvature flow

Summary. Convergence for the spatial discretization by linear finite elements of the non-parametric mean curvature flow is proved under natural regularity assumptions on the continuous solution. Asymptotic convergence is also obtained for the time derivative which is proportional to mean curvature. An existence result for the continuous problem in adequate spaces is included. Received September 30, 1993     

Local convergence of quasi-Newton methods under metric regularity

We consider quasi-Newton methods for generalized equations in Banach spaces under metric regularity and give a sufficient condition for q-linear convergence. Then we show that the well-known Broyden update satisfies this sufficient condition in Hilbert spaces. We also establish various modes of q-superlinear convergence of the Broyden update under strong metric subregularity, metric regularity and strong metric regularity. In particular, we show that the Broyden update applied to a generalized equation in Hilbert spaces satisfies the Dennis–Moré condition for q-superlinear convergence. Simple numerical examples illustrate the results.     

Hölder estimates of p-harmonic extension operators

It is now a well-known fact that for 1<p<∞ the p-harmonic functions on domains in metric measure spaces equipped with a doubling measure supporting a (1,p)-Poincaré inequality are locally Hölder continuous. In this note we provide a characterization of domains in such metric spaces for which p-harmonic extensions of Hölder continuous boundary data are globally Hölder continuous. We also provide a link between this regularity property of the domain and the uniform p-fatness of the complement of the domain.     

Maximal regularity in continuous interpolation spaces and quasilinear parabolic equations

In this paper we establish a geometric theory for abstract quasilinear parabolic equations. In particular, we study existence, uniqueness, and continuous dependence of solutions. Moreover, we give conditions for global existence and establish smoothness properties of solutions. The results are based on maximal regularity estimates in continuous interpolation spaces. An important new ingredient is that we are able to show that quasilinear parabolic evolution equations generate a smooth semiflow on the trace spaces associated with maximal regularity, which are the natural phase spaces in this framework. Received August 10, 2000; accepted September 20, 2000.     

Semi-classical limit of the bottom of spectrum of a Schrödinger operator on a path space over a compact Riemannian manifold

We determine the limit of the bottom of spectrum of Schrödinger operators with variable coefficients on Wiener spaces and path spaces over finite-dimensional compact Riemannian manifolds in the semi-classical limit. These are extensions of the results in [S. Aida, Semiclassical limit of the lowest eigenvalue of a Schrödinger operator on a Wiener space, J. Funct. Anal. 203 (2) (2003) 401-424]. The problem on path spaces over Riemannian manifolds is considered as a problem on Wiener spaces by using Ito's map. However the coefficient operator is not a bounded linear operator and the dependence on the path is not continuous in the uniform convergence topology if the Riemannian curvature tensor on the underling manifold is not equal to 0. The difficulties are solved by using unitary transformations of the Schrödinger operators by approximate ground state functions and estimates in the rough path analysis.     

Banach Space-Valued Ornstein–Uhlenbeck Processes with General Drift Coefficients

We construct Ornstein–Uhlenbeck processes with values in Banach space and with continuous paths. The drift coefficient must only generate a strongly continuous semigroup on the Hilbert space which determines the Brownian motion. We admit arbitrary starting points and consider also invariant measures for the process, generalizing earlier work in many directions. A price for the generality is that sometimes one has to enlarge the phase space but most previously known results are covered.The constructions are based on abstract Wiener space methods, more precisely on images of abstract Wiener spaces under suitable linear transformations of the Cameron–Martin space. The image abstract Wiener measures are then given by stochastic extensions. We present the basic spaces and operators and the most important results on image spaces and stochastic extensions in some detail.     

Unconditional convergence of DIRK schemes applied to dissipative evolution equations

In this paper we prove the convergence of algebraically stable DIRK schemes applied to dissipative evolution equations on Hilbert spaces. The convergence analysis is unconditional as we do not impose any restrictions on the initial value or assume any extra regularity of the solution. The analysis is based on the observation that the schemes are linear combinations of the Yosida approximation, which enables the usage of an abstract approximation result for dissipative maps. The analysis is also extended to the case where the dissipative vector field is perturbed by a locally Lipschitz continuous map. The efficiency and robustness of these schemes are finally illustrated by applying them to a nonlinear diffusion equation.     

Geometry of optimality conditions and constraint qualifications

Certain types of necessary optimality conditions for mathematical programming problems are equivalent to corresponding regularity conditions on the constraint set. For any problem, a certain natural optimality condition, dependent upon the particular constraint set, is always satisfied. This condition can be strengthened in numerous ways by invoking appropriate regularity assumptions on the constraint set. Results are presented for Euclidean spaces and some extensions to Banach spaces are given.This work was supported in part by the Office of Naval Research, Contract No. N00014-67-A-0321-0003 (NR-047-095).     

级联算法在Besov和Triebel-Lizorkin空间上的有界性和收敛性(Ⅱ)(英文)

本文利用联合谱半径刻画了级联算法在Ｂｅｓｏｖ和Ｔｈｉｅｂｅｌ－Ｌｉｚｏｒｋｉｎ空间上的收敛性,给出了级联算法初值函数矩条件的新证明,并利用到细分分布的光滑性和非齐次细分方程解的存在性等方面．特别地,在某些条件下,我们证明了级联算法的有界性和收敛性相互等价．     

The Law of Large Numbers and the Ito-Nisio Theorem for Vector Valued Random Fields

Strong convergence results are obtained for vector-valued random fields. Substantial development of Banach-valued random fields and summability results is needed to provide the framework for the major results since many plausible extensions fail for multi-indexed Banach-valued random variables. This development yields general convergence results for random fields in Banach spaces, including an Ito-Nisio theorem and strong laws of large numbers.     

Boundary-contact problems for domains with edge singularities

We study boundary-contact problems for elliptic equations (and systems) with interfaces that have edge singularities. Such problems represent continuous operators between weighted edge spaces and subspaces with asymptotics. Ellipticity is formulated in terms of a principal symbolic hierarchy, containing interior, transmission, and edge symbols. We construct parametrices, show regularity with asymptotics of solutions in weighted edge spaces and illustrate the results by boundary-contact problems for the Laplacian with jumping coefficients.     

PAIRWISE ALMOST COMPACT SPACES

ABSTRACT

The purpose of this paper is to investigate pairwise almost compact bitopological spaces. These spaces satisfy a bitopological compactness criterion which is strictly weaker than pairwise C-compactness and is independent of other well-known bitopological compactness notions. Pairwise continuous maps from such spaces to pairwise Hausdorff spaces are pairwise almost closed, the property is invariant under suitably continuous maps, is inherited by regularly closed subspaces and may be characterized in terms of certain covers as well as the adherent convergence of certain open filter bases. Some new natural bitopological separation axioms are introduced and in conjunction with pairwise almost compactness yield interesting results, including a sufficient condition for the bitopological complete separation of disjoint regularly closed sets by semi-continuous functions.     

Besov-Morrey spaces: Function space theory and applications to non-linear PDE

This paper is devoted to the analysis of function spaces modeled on Besov spaces and their applications to non-linear partial differential equations, with emphasis on the incompressible, isotropic Navier-Stokes system and semi-linear heat equations. Specifically, we consider the class, introduced by Hideo Kozono and Masao Yamazaki, of Besov spaces based on Morrey spaces, which we call Besov-Morrey or BM spaces. We obtain equivalent representations in terms of the Weierstrass semigroup and wavelets, and various embeddings in classical spaces. We then establish pseudo-differential and para-differential estimates. Our results cover non-regular and exotic symbols. Although the heat semigroup is not strongly continuous on Morrey spaces, we show that its action defines an equivalent norm. In particular, homogeneous BM spaces belong to a larger class constructed by Grzegorz Karch to analyze scaling in parabolic equations. We compare Karch's results with those of Kozono and Yamazaki and generalize them by obtaining short-time existence and uniqueness of solutions for arbitrary data with subcritical regularity. We exploit pseudo-differential calculus to extend the analysis to compact, smooth, boundaryless, Riemannian manifolds. BM spaces are defined by means of partitions of unity and coordinate patches, and intrinsically in terms of functions of the Laplace operator.

     

A calculus of direct limits of probability spaces

In this paper, several results on probability spaces have been extended to direct limits of such. The calculus includes convergence sequences, conditional expectations, completeness questions and extensions of probability spaces.     

A ROBUST TRUST REGION ALGORITHM FOR SOLVING GENERAL NONLINEAR PROGRAMMING

1. Introductioncrust region methods are iterative. As a strategy of globalization, the trust region approach was introduced into solving unconstrained optimization and proved to be efficient androbust. An excellent survey was given by Mor6(1983). The associated research with trustregion methods for unconstrained optimization can be found in Fletcher(1980), Powell(1975),Sorensen(1981), Shultz, Schnabel and Byrd(1985), Yuan(1985). The solution of the trust region subproblem is still an activ…     

A dual method to the collocation method

In this article we consider methods which are related to the collocation method by interchanging the test and the trial spaces. Error estimates are derived. As a by-product we obtain some extensions to the known convergence results for the collocation method.