退化正则半群

引入了退化正则半群的定义，给出退化正则半群的一些基本性质，并证明了用多值线性算子刻划的指数有界退化正则半群的生成定理．     

Erdős-Ko-Rado theorems in certain semilattices

Suda (2012) extended the Erds-Ko-Rado theorem to designs in strongly regularized semilattices.In this paper we generalize Suda’s results in regularized semilattices and partition regularized semilattices,give many examples for these semilattices and obtain their intersection theorems.     

Semi-discrete stabilized finite element methods for Navier–Stokes equations with nonlinear slip boundary conditions based on regularization procedure

Based on the pressure projection stabilized methods, the semi-discrete finite element approximation to the time-dependent Navier–Stokes equations with nonlinear slip boundary conditions is considered in this paper. Because this class of boundary condition includes the subdifferential property, then the variational formulation is the Navier–Stokes type variational inequality problem. Using the regularization procedure, we obtain a regularized problem and give the error estimate between the solutions of the variational inequality problem and the regularized problem with respect to the regularized parameter \({\varepsilon}\), which means that the solution of the regularized problem converges to the solution of the Navier–Stokes type variational inequality problem as the parameter \({\varepsilon\longrightarrow 0}\). Moreover, some regularized estimates about the solution of the regularized problem are also derived under some assumptions about the physical data. The pressure projection stabilized finite element methods are used to the regularized problem and some optimal error estimates of the finite element approximation solutions are derived.     

A class of semigroups regularized in space and time

We consider α-times integrated C-regularized semigroups, which are a hybrid between semigroups regularized in space (C-semigroups) and integrated semigroups regularized in time. We study the basic properties of these objects, also in absence of exponential boundedness. We discuss their generators and establish an equivalence theorem between existence of integrated regularized semigroups and well-posedness of certain Cauchy problems. We investigate the effect of smoothing regularized semigroups by fractional integration.     

A new regularized quasi-Newton method for unconstrained optimization

In this paper, we propose a new regularized quasi-Newton method for unconstrained optimization. At each iteration, a regularized quasi-Newton equation is solved to obtain the search direction. The step size is determined by a non-monotone Armijo backtracking line search. An adaptive regularized parameter, which is updated according to the step size of the line search, is employed to compute the next search direction. The presented method is proved to be globally convergent. Numerical experiments show that the proposed method is effective for unconstrained optimizations and outperforms the existing regularized Newton method.     

Projection Method Approach for General Regularized Non-convex Variational Inequalities

In this paper, we investigate or analyze non-convex variational inequalities and general non-convex variational inequalities. Two new classes of non-convex variational inequalities, named regularized non-convex variational inequalities and general regularized non-convex variational inequalities, are introduced, and the equivalence between these two classes of non-convex variational inequalities and the fixed point problems are established. A projection iterative method to approximate the solutions of general regularized non-convex variational inequalities is suggested. Meanwhile, the existence and uniqueness of solution for general regularized non-convex variational inequalities is proved, and the convergence analysis of the proposed iterative algorithm under certain conditions is studied.     

双连续C正则预解算子族的生成定理

基于C正则预解算子族和双连续C_0半群引入了双连续C正则预解算子族的概念,考察了双连续C正则预解算子族生成元与预解式之间的关系,给出了双连续C正则预解算子族Hille-Yosida型生成定理,从而对Bananch空间强连续半群的生成定理进行了推广.     

A minimum effort optimal control problem for the wave equation

A minimum effort optimal control problem for the undamped wave equation is considered which involves L ^∞-control costs. Since the problem is non-differentiable a regularized problem is introduced. Uniqueness of the solution of the regularized problem is proven and the convergence of the regularized solutions is analyzed. Further, a semi-smooth Newton method is formulated to solve the regularized problems and its superlinear convergence is shown. Thereby special attention has to be paid to the well-posedness of the Newton iteration. Numerical examples confirm the theoretical results.     

Generalization performance of least-square regularized regression algorithm with Markov chain samples

The previously known works describing the generalization of least-square regularized regression algorithm are usually based on the assumption of independent and identically distributed (i.i.d.) samples. In this paper we go far beyond this classical framework by studying the generalization of least-square regularized regression algorithm with Markov chain samples. We first establish a novel concentration inequality for uniformly ergodic Markov chains, then we establish the bounds on the generalization of least-square regularized regression algorithm with uniformly ergodic Markov chain samples, and show that least-square regularized regression algorithm with uniformly ergodic Markov chains is consistent.     

Regularized Lagrangian duality for linearly constrained quadratic optimization and trust-region problems

In this paper we first establish a Lagrange multiplier condition characterizing a regularized Lagrangian duality for quadratic minimization problems with finitely many linear equality and quadratic inequality constraints, where the linear constraints are not relaxed in the regularized Lagrangian dual. In particular, in the case of a quadratic optimization problem with a single quadratic inequality constraint such as the linearly constrained trust-region problems, we show that the Slater constraint qualification (SCQ) is necessary and sufficient for the regularized Lagrangian duality in the sense that the regularized duality holds for each quadratic objective function over the constraints if and only if (SCQ) holds. A new theorem of the alternative for systems involving both equality constraints and two quadratic inequality constraints plays a key role. We also provide classes of quadratic programs, including a class of CDT-subproblems with linear equality constraints, where (SCQ) ensures regularized Lagrangian duality.     

Semi-smooth Newton methods for the Signorini problem

Semi-smooth Newton methods are analyzed for the Signorini problem. A proper regularization is introduced which guarantees that the semi-smooth Newton method is superlinearly convergent for each regularized problem. Utilizing a shift motivated by an augmented Lagrangian framework, to the regularization term, the solution to each regularized problem is feasible. Convergence of the regularized problems is shown and a report on numerical experiments is given.     

Multivalued Regularized Equilibrium Problems

In this paper, we introduce a new class of equilibrium problems known as the multivalued regularized equilibrium problems. We use the auxiliary principle technique to suggest some iterative methods for solving multivalued regularized equilibrium problems. The convergence of the proposed methods is studied under some mild conditions. As special cases, we obtain a number of known and new results for solving various classes of regularized equilibrium problems and related optimization problems.     

On dynamics of quantum states generated by the Cauchy problem for the Schrödinger equation with degeneration on the half-line

The paper considers the Cauchy problem for the Schrödinger equation with operator degenerate on the semiaxis and the family of regularized Cauchy problems with uniformly elliptic operators whose solutions approximate the solution of the degenerate problem. The author studies the strong and weak convergences of the regularized problems and the convergence of values of quadratic forms of bounded operators on solutions of the regularized problems when the regularization parameter tends to zero.     

Regularized modified α-processes for nonlinear equations with monotone operators

For a stable approximation of the solution to a nonlinear irregular equation with a monotone operator, a two-step method based on Lavrent’ev scheme and nonlinear regularized α-processes is constructed. These processes are shown to have a linear convergence rate when used to approximate the solution of a regularized equation. The error of the regularized solution is estimated, and the two-step method is shown to be order optimal in the well-posedness class of sourcewise representable solutions.     

Tikhonov regularization methods for inverse variational inequalities

The purpose of this paper is to study Tikhonov regularization methods for inverse variational inequalities. A rather weak coercivity condition is given which guarantees that the solution set of regularized inverse variational inequality is nonempty and bounded. Moreover, the perturbation analysis for the solution set of regularized inverse variational inequality is established. As an application, we show that solutions of regularized inverse variational inequalities form a minimizing sequence of the D-gap function under a mild condition.     

Automatic extensions of local regularized semigroups and local regularized cosine functions

This paper establishes automatic extensions for local regularized semigroups and local regularized cosine functions in a certain sense and applies the results to abstract Cauchy problems.

     

正则预解算子族的乘积扰动

设K∈C（R＋）和B是一个有界线性算子．作者证明如果犃生成一个指数有界的A正则预解算子族，那么BA，AB或A（I＋B），（I＋B）A也生成一个指数有界的k-正则预解算子族．此外，作者也给出了k正则预解算子族的加法扰动的相应结果．     

Lagrange multipliers, (exact) regularization and error bounds for monotone variational inequalities

We examine two central regularization strategies for monotone variational inequalities, the first a direct regularization of the operative monotone mapping, and the second via regularization of the associated dual gap function. A key link in the relationship between the solution sets to these various regularized problems is the idea of exact regularization, which, in turn, is fundamentally associated with the existence of Lagrange multipliers for the regularized variational inequality. A regularization is said to be exact if a solution to the regularized problem is a solution to the unregularized problem for all parameters beyond a certain value. The Lagrange multipliers corresponding to a particular regularization of a variational inequality, on the other hand, are defined via the dual gap function. Our analysis suggests various conceptual, iteratively regularized numerical schemes, for which we provide error bounds, and hence stopping criteria, under the additional assumption that the solution set to the unregularized problem is what we call weakly sharp of order greater than one.     

正则解析半群与正则Cosine函数

在本文中，我们的主要结果是：若Ａ是一个正则解析半群的生成元，则Ａ也一定是某正则Ｃｏｓｉｎｅ函数的生成元．     

Polylogarithms and multiple zeta values from free Rota-Baxter algebras

We show that the shuffle algebras for polylogarithms and regularized MZVs in the sense of Ihara, Kaneko and Zagier are both free commutative nonunitary Rota-Baxter algebras with one generator. We apply these results to show that the full sets of shuffle relations of polylogarithms and regularized MZVs are derived by a single series. We also take this approach to study the extended double shuffle relations of MZVs by comparing these shuffle relations with the quasi-shuffle relations of the regularized MZVs in our previous approach of MZVs by renormalization.