Visco-penalization of the sum of two monotone operators

A new type of approximating curve for finding a particular zero of the sum of two maximal monotone operators in a Hilbert space is investigated. This curve consists of the zeros of perturbed problems in which one operator is replaced with its Yosida approximation and a viscosity term is added. As the perturbation vanishes, the curve is shown to converge to the zero of the sum that solves a particular strictly monotone variational inequality. As an off-spring of this result, we obtain an approximating curve for finding a particular zero of the sum of several maximal monotone operators. Applications to convex optimization are discussed.     

Approximation by Bézier variants of the BBHK operators

In this work a new type of approximation operator—the Bézier variant of the BBHK operator—is introduced. Its approximation properties are studied. A convergence theorem for such approximation operators for locally bounded functions is established by means of some techniques of probability theory and analysis methods. This convergence theorem subsumes the approximation of functions of bounded variation as a special case.     

Multiple weighted estimates for maximal vector-valued commutator of multilinear Calderón-Zygmund singular integrals

This paper concerns with multiple weighted norm inequalities for maximal vector-valued multilinear singular operator and maximal commutators. The Cotlar-type inequality of maximal vector-valued multilinear singular integrals operator is obtained. On the other hand, pointwise estimates for sharp maximal function of two kinds of maximal vector-valued multilinear singular integrals and maximal vector-valued commutators are also established. By the weighted estimates of a class of new variant maximal operator, Cotlar's inequality and the sharp maximal function estimates, multiple weighted strong estimates and weak estimates for maximal vector-valued singular integrals of multilinear operators and those for maximal vector-valued commutator of multilinear singular integrals are obtained.     

Compact adjoint operators and approximation properties

This paper is concerned with the space of all compact adjoint operators from dual spaces of Banach spaces into dual spaces of Banach spaces and approximation properties. For some topology on the space of all bounded linear operators from separable dual spaces of Banach spaces into dual spaces of Banach spaces, it is shown that if a bounded linear operator is approximated by a net of compact adjoint operators, then the operator can be approximated by a sequence of compact adjoint operators whose operator norms are less than or equal to the operator norm of the operator. Also we obtain applications of the theory and, in particular, apply the theory to approximation properties.     

Maximal Monotone Operators, Convex Functions and a Special Family of Enlargements

This work establishes new connections between maximal monotone operators and convex functions. Associated to each maximal monotone operator, there is a family of convex functions, each of which characterizes the operator. The basic tool in our analysis is a family of enlargements, recently introduced by Svaiter. This family of convex functions is in a one-to-one relation with a subfamily of these enlargements. We study the family of convex functions, and determine its extremal elements. An operator closely related to the Legendre–Fenchel conjugacy is introduced and we prove that this family of convex functions is invariant under this operator. The particular case in which the operator is a subdifferential of a convex function is discussed.     

On the Existence of Anti-periodic Solutions for Implicit Differential Equations

This paper is devoted to anti-periodic solutions for a class of implicit differential equations with nonmonotone perturbations. The main tools in our study will be the maximal monotone property of the derivative operator with anti-periodic conditions and a convergent approximation procedure.     

On the Existence of Fixed Points for Approximate Value Iteration and Temporal-Difference Learning

Approximate value iteration is a simple algorithm that combats the curse of dimensionality in dynamic programs by approximating iterates of the classical value iteration algorithm in a spirit reminiscent of statistical regression. Each iteration of this algorithm can be viewed as an application of a modified dynamic programming operator to the current iterate. The hope is that the iterates converge to a fixed point of this operator, which will then serve as a useful approximation of the optimal value function. In this paper, we show that, in general, the modified dynamic programming operator need not possess a fixed point; therefore, approximate value iteration should not be expected to converge. We then propose a variant of approximate value iteration for which the associated operator is guaranteed to possess at least one fixed point. This variant is motivated by studies of temporal-difference (TD) learning, and existence of fixed points implies here existence of stationary points for the ordinary differential equation approximated by a version of TD that incorporates exploration.     

Persistence Approximation Property for Maximal Roe Algebras

Persistence approximation property was introduced by Hervé Oyono-Oyono and Guoliang Yu. This property provides a geometric obstruction to Baum-Connes conjecture. In this paper, the authors mainly discuss the persistence approximation property for maximal Roe algebras. They show that persistence approximation property of maximal Roe algebras follows from maximal coarse Baum-Connes conjecture. In particular, let X be a discrete metric space with bounded geometry, assume that X admits a fibred coarse embedding into Hilbert space and X is coarsely uniformly contractible, then C_(max)~*(X) has persistence approximation property. The authors also give an application of the quantitative K-theory to the maximal coarse Baum-Connes conjecture.     

Lipschitz condition for the operator of best uniform approximation at points of uniqueness

The operator of best uniform approximation of real, continuous functions by elements of a finite space is considered. It is shown that the Lipschitz condition for the operator of best approximation by generalized polynomials is satisfied for each function having a characteristic set of maximal dimension.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 17–19, 1987.     

Convergence analysis of non-quadratic proximal methods for variational inequalities in Hilbert spaces

We consider a general approach for the convergence analysis of proximal-like methods for solving variational inequalities with maximal monotone operators in a Hilbert space. It proves to be that the conditions on the choice of a non-quadratic distance functional depend on the geometrical properties of the operator in the variational inequality, and –- in particular –- a standard assumption on the strict convexity of the kernel of the distance functional can be weakened if this operator possesses a certain `reserve of monotonicity'. A successive approximation of the `feasible set' is performed, and the arising auxiliary problems are solved approximately. Weak convergence of the proximal iterates to a solution of the original problem is proved.     

The prox-Tikhonov regularization method for the proximal point algorithm in Banach spaces

It is known, by Rockafellar (SIAM J Control Optim 14:877–898, 1976), that the proximal point algorithm (PPA) converges weakly to a zero of a maximal monotone operator in a Hilbert space, but it fails to converge strongly. Lehdili and Moudafi (Optimization 37:239–252, 1996) introduced the new prox-Tikhonov regularization method for PPA to generate a strongly convergent sequence and established a convergence property for it by using the technique of variational distance in the same space setting. In this paper, the prox-Tikhonov regularization method for the proximal point algorithm of finding a zero for an accretive operator in the framework of Banach space is proposed. Conditions which guarantee the strong convergence of this algorithm to a particular element of the solution set is provided. An inexact variant of this method with error sequence is also discussed.     

F格上的逼近算子

粗集理论是波兰学者Pawlak提出的知识表示新理论.Pawlak代数是粗集理论中粗集系统的抽象,其公理系统包含了知识粗表示所必须的全部性质.本文深入研究了F格上的逼近算子,建立了F格上弱逼近算子之间的某些代数运算,从而从理论上建立了各种知识粗表示之间的联系.我们还定义了逼近算子的闭包,进而用逼近算子导出拓扑,为信息系统的近似提供必要的数学基础.最后,作为特例,我们研究了粗集理论中由相似关系导出逼近算子的某些性质.     

Integral operators of Sturm-Liouville type

This paper extends the class of integral equations whose solutions can be generated from a finite number of particular cases to include those of Sturm-Liouville type, including the case where the associated operators are not self-adjoint. An explicit expression for the resolvent operator is generated from two particular solutions, in a form amenable to the use of approximation techniques. A usable estimate of the norm of the inverse operator is obtained even in cases where approximate solutions have to be used.     

一类奇异积分算子在加权空间的界的估计

设T是由Grubb和Moore引入的一类奇异积分算子,它的核满足一种新型利普希茨正则性.T^*是由T确定的极大奇异积分算子.本文通过建立与T和T^*相关的grand极大算子的弱型端点估计,得到了算子T和T^*在加权空间的由A_p权常数表示的界的估计和弱型端点估计.     

Yosida approximations for multivalued stochastic partial differential equations driven by Lévy noise on a Gelfand triple

We prove the existence and uniqueness of solutions for a class of multivalued stochastic partial differential equations with maximal monotone drift on Banach space driven by multiplicative Lévy noise. We also establish the strong convergence result for solutions of the approximating equations where the maximal monotone drift operator is replaced by its Yosida approximation. As an application, the existence and uniqueness of solutions for multivalued stochastic porous medium equations is obtained.     

模糊形式背景下的类近似算子

分别在形式背景和模糊形式背景下定义类下近似算子和类模糊下近似算子,并研究它们的性质.证明这两种算子分剐等价于形式背景和模糊形式背景下的*算子和模糊*算子.进一步给出类下近似算子与类模糊下近似算子的公理刺画.最后,对偶地讨论类上近似算子和类模糊上近似算子的定义和性质.     

非周期神经网络及平移网络在L_w~p中的逼近

设s≥d≥1为整数, 1≤p≤+∞,借助于正交多元代数多项式系而构造了一类s维网络算子,并用于逼近Lpw[-1,1]s中的函数,给出了逼近的上界以及当此算子为平移网络算子及神经网络算子时的导数型估计.     

Statistical approximation properties of q-Bleimann,Butzer and Hahn operators

The main aim of this study is to introduce a new generalization of q-Bleimann, Butzer and Hahn operators and obtain statistical approximation properties of these operators with the help of the Korovkin type statistical approximation theorem. Rates of statistical convergence by means of the modulus of continuity and the Lipschitz type maximal function are also established. Our results show that rates of convergence of our operators are at least as fast as classical BBH operators. The second aim of this study is to construct a bivariate generalization of the operator and also obtain the statistical approximation properties.     

New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory

A multi(sub)linear maximal operator that acts on the product of m Lebesgue spaces and is smaller than the m-fold product of the Hardy-Littlewood maximal function is studied. The operator is used to obtain a precise control on multilinear singular integral operators of Calderón-Zygmund type and to build a theory of weights adapted to the multilinear setting. A natural variant of the operator which is useful to control certain commutators of multilinear Calderón-Zygmund operators with BMO functions is then considered. The optimal range of strong type estimates, a sharp end-point estimate, and weighted norm inequalities involving both the classical Muckenhoupt weights and the new multilinear ones are also obtained for the commutators.