Periodic solutions of parabolic inclusions and the averaging method

We obtain an analog of the second Bogolyubov theorem for differential inclusions with multimappings acting in Sobolev spaces and satisfying some monotonicity and compactness conditions. As a consequence, we obtain criteria for the existence of periodic solutions of secondorder parabolic inclusions.     

Method of generalized integral guiding functions in the problem of the existence of periodic solutions for functional-differential inclusions

We suggest new methods for the solution of a periodic problem for a nonlinear object described by the differential inclusion x′(t) ∈ F(t, x_t) under the assumption that the multimapping F has convex compact values and satisfies the upper Carathéodory conditions. We also study the case in which this multimapping is not convex-valued but is normal. The class of normal multimappings includes, for example, bounded almost lower semicontinuous multimappings with compact values and mappings satisfying the Carathéodory conditions. In both cases, a generalized integral guiding function is used to study the problem.     

On the Existence of Optimal Solutions to an Optimal Control Problem

In this paper, some results concerning the existence of optimal solutions to an optimal control problem are derived. The problem involves a quasilinear hyperbolic differential equation with boundary condition and a nonlinear integral functional of action. The assumption of convexity, under which the main theorem is proved, is not connected directly with the convexity of the functional of action. In the proof, the implicit function theorem for multimappings is used.Communicated by L. D. Berkovitz     

Averaging of differential inclusions

We prove a version of the Krasnosel’skii-Krein theorem for differential inclusions with multimappings satisfying certain one-sided constraints. As a corollary, we obtain an analog of the first Bogolyubov theorem for the inclusion 0 ? x′ + ??(t, x).     

Sequential fixed accuracy estimation for nonstationary autoregressive processes

For an autoregressive process of order p, the paper proposes new sequential estimates for the unknown parameters based on the least squares (LS) method. The sequential estimates use p stopping rules for collecting the data and presumes a special modification the sample Fisher information matrix in the LS estimates. In case of Gaussian disturbances, the proposed estimates have non-asymptotic normal joint distribution for any values of unknown autoregressive parameters. It is shown that in the i.i.d. case with unspecified error distributions, the new estimates have the property of uniform asymptotic normality for unstable autoregressive processes under some general condition on the parameters. Examples of unstable autoregressive models satisfying this condition are considered.

     

Window estimates of location and scale with applications to the cauchy distribution

Location and scale parameters are estimated via “window estimates”. The consistency and asymptotic normality of the estimates are established. The special case of the Cauchy distribution is considered, where the estimates are shown to have the same asymptotic distribution as the maximum-likelihood estimates. Additional applications are given for the Pearson type-VII distributions. The estimates have the advantages of ease of computation and high asymptotic efficiencies for certain heavy-tailed distributions.     

Estimates of MM type for the multivariate linear model

We propose a class of robust estimates for multivariate linear models. Based on the approach of MM-estimation (Yohai 1987, [24]), we estimate the regression coefficients and the covariance matrix of the errors simultaneously. These estimates have both a high breakdown point and high asymptotic efficiency under Gaussian errors. We prove consistency and asymptotic normality assuming errors with an elliptical distribution. We describe an iterative algorithm for the numerical calculation of these estimates. The advantages of the proposed estimates over their competitors are demonstrated through both simulated and real data.     

指数分布参数的最短区间估计

本文研究了指数分布参数的区间估计方法。给出了指数分布参数的最短区间估计方法;通过一个实例介绍了最短区间估计方法的使用;最后指出最短区间估计方法比传统区间估计方法所具有的优越性。     

Linear estimates of accuracy for approximate solutions of inverse problems

In this paper, we explore the question of which non-linear inverse problems, which are solved by a selected regularization method, may have so-called linear a priori accuracy estimates – that is, the accuracy of corresponding approximate solutions linearly depends on the error level of the data. In particular, we prove that if such a linear estimate exists, then the inverse problem under consideration is well posed, according to Tikhonov. For linear inverse problems, we find that the existence of linear estimates lead to, under some assumptions, the well-posedness (according to Tikhonov) on the whole space of solutions. Moreover, we consider a method for solving inverse problems with guaranteed linear estimates, called the residual method on the correctness set (RMCS). The linear a priori estimates of absolute and relative accuracy for the RMCS are presented, as well as analogous a posteriori estimates. A numerical illustration of obtaining linear a priori estimates for appropriate parametric sets of solutions using RMCS is given in comparison with Tikhonov regularization. The a posteriori estimates are calculated on these parametric sets as well.     

A geometric theory for preconditioned inverse iteration IV: On the fastest convergence cases

In order to compute the smallest eigenvalue together with an eigenfunction of a self-adjoint elliptic partial differential operator one can use the preconditioned inverse iteration scheme, also called the preconditioned gradient iteration. For this iterative eigensolver estimates on the poorest convergence have been published by several authors. In this paper estimates on the fastest possible convergence are derived. To this end the convergence problem is reformulated as a two-level constrained optimization problem for the Rayleigh quotient. The new convergence estimates reveal a wide range between the fastest possible and the slowest convergence.     

New interpolants for asymptotically correct defect control of BVODEs

The defect of a continuous approximate solution to an ODE is the amount by which that approximation fails to satisfy the ODE. A number of studies have explored the use of asymptotically correct defect estimates in the numerical solution of initial value ODEs (IVODEs). By employing an appropriately constructed interpolant to an approximate discrete solution to the ODE, various researchers have shown that it is possible to obtain estimates of the local error and/or the maximum defect that are asymptotically correct on each step, as the stepsize h →0. In this paper, we investigate the usefulness of asymptotically correct defect estimates for defect control in boundary value ODE (BVODE) codes. In the BVODE context, for a sequence of meshes which partition the problem interval, one computes a discrete numerical solution, constructs an interpolant, and estimates the maximum defect. The estimates (typically obtained by sampling the defect at a small number of points on each subinterval of the mesh) are used in a redistribution process to determine the next mesh and thus the availability of these more reliable maximum defect estimates can lead to improved meshes. As well, when such estimates are available, the code can terminate with more confidence that the defect is bounded throughout the problem domain by the user-prescribed tolerance. In this paper we employ a boot-strapping approach to derive interpolants that allow asymptotically correct defect estimates. Numerical results are included to demonstrate the validity of this approach.     

The Construction and Properties of Boundary Kernels for Smoothing Sparse Multinomials

In recent years several authors have investigated the use of smoothing methods for sparse multinomial data. In particular, Hall and Titterington (1987) studied kernel smoothing in detail. It is pointed out here that the bias of kernel estimates of probabilities for cells near the boundaries of the multinomial vector can dominate the mean sum of squared error of the estimator for most true probability vectors. Fortunately, boundary kernels devised to correct boundary effects for kernel regression estimators can achieve the same result for these estimators. Properties of estimates based on boundary kernels are investigated and compared to unmodified kernel estimates and maximum penalized likelihood estimates. Monte Carlo evidence indicates that the boundary-corrected kernel estimates usually outperform uncorrected kernel estimates and are quite competitive with penalized likelihood estimates.     

Consistent estimation in a linear regression problem with random errors in coefficients

We consider the linear regression model in the case when the independent variables are measured with errors, while the variances of the main observations depend on an unknown parameter. In the case of normally distributed replicated regressors we propose and study new classes of two-step estimates for the main unknown parameter. We find consistency and asymptotic normality conditions for first-step estimates and an asymptotic normality condition for second-step estimates. We discuss conditions under which these estimates have the minimal asymptotic variance.     

Conditional Stability Estimates for Ill-Posed PDE Problems by Using Interpolation

The focus of this article is on conditional stability estimates for ill-posed inverse problems in partial differential equations. Conditional stability estimates have been obtained in related literature by a couple different methods. In this article, we propose a method called interpolation method, which is based on interpolation in variable Hilbert scales. We provide the theoretical background of this method and show that optimal conditional stability estimates are obtained. The capabilities of our method are illustrated by a comprehensive collection of different inverse and ill-posed PDE problems containing elliptic and parabolic problems, one source problem and the problem of analytic continuation.     

Perturbations of positive semigroups on $$L_p$$-spaces

We give a characterization of a variation of constants type estimate relating two positive semigroups on (possibly different) \(L_p\)-spaces to one another in terms of corresponding estimates for the respective generators and of estimates for the respective resolvents. The results have applications to kernel estimates for semigroups induced by accretive and non-local forms on \(\sigma \)-finite measure spaces.     

The Quantum Faedo-Galerkin Equation: Evolution Operator and Time Discretization

Time dependent quantum systems have become indispensable in science and nanotechnology. Disciplines including chemical physics and electrical engineering have used approximate evolution operators to solve these systems for targeted physical quantities. Here, we discuss the approximation of closed time dependent quantum systems on bounded domains via evolution operators. The work builds upon the use of weak solutions, which includes a framework for the evolution operator based upon dual spaces. We are able to derive the corresponding Faedo-Galerkin equation as well as its time discretization, yielding a fully discrete theory. We obtain corresponding approximation estimates. These estimates make no regularity assumptions on the weak solutions, other than their inherent properties. Of necessity, the estimates are in the dual norm, which is natural for weak solutions. This appears to be a novel aspect of this approach.     

Estimates of approximation numbers and applications

In this study, the estimates of approximation numbers of embedding operators in weighted spaces have been analyzed. These estimates depend on orders of differential operators, dimensions of function spaces and weighted functions. This fact implies that the associated embedding operators belong to Schatten class of compact operators. By using these estimates, the discreetness of spectrum and completion of root elements relating to principal nonselfedjoint degenerate differential operators is obtained.     

Reference Bayesian analysis of inverse Gaussian degradation process

In this paper, objective Bayesian method is applied to analyze degradation model based on the inverse Gaussian process. Noninformative priors (Jefferys prior and two reference priors) for model parameters are obtained and their properties are discussed. Moreover, we propose a class of modified reference priors to remedy weaknesses of the usual reference priors and show that the modified reference priors not only have proper posterior distributions but also have probability matching properties for model parameters. Gibbs sampling algorithms for Bayesian inference based on the Jefferys prior and the modified reference priors are studied. Simulations are conducted to compare the objective Bayesian estimates with the maximum likelihood estimates and subjective Bayesian estimates and shows better performance of the objective method than the other two estimates especially for the case of small sample size. Finally, two real data examples are analyzed for illustration.     

随机套分类模型中方差分量区间估计的改进

首先给出了随机套分类模型中方差分量基于由方差分析产生的平方和的区间估计.然后以此为基础进行了改进,推导出了同时依赖于均值与平方和的区间估计.二者的区间长度相同,但后者有较高的置信度.     

Efficient Estimation of Two Seemingly Unrelated Regression Equations

We derive simpler expressions under a certain structure of design matrices for the two-stage Aitken estimates of the regression coefficients of two seemingly unrelated regression equations. The estimates are shown to have smaller variance than the ordinary least squares estimates for sufficiently large samples.