OPTIMIZATION OF ADAPTIVE DIRECT METHOD FOR APPROXIMATE SOLUTION OF INTEGRAL EQUATIONS OF SEVERAL VARIABLES

This paper determines the exact error order on optimization of adaptive directmethods of approximate solution of the class of Fredholm integral equations of the secondkind with kernel belonging to the anisotropic Sobolev classes, and also gives an optimalalgorithm.     

多元积分方程自适应解法的最优化

In this paper it was considered that problem of optimization of adaptive direct algorithm of approximate solution of integral equations. For the Fredholm integral equations of second kind with kernels belonging to Besov classes there is determined the exact order of the error of an optimal adaptive direct algorithm and a algorithm for realizing it is indicated.     

多元积分方程自适应解法的优化

考虑核属于各向异性的Sobolev类的积分方程自适应直接方法的优化,得到误差阶的精确估计及相应的最优方法.     

多维平稳时间序列积分谱图的不变原理

用积分周期图估计平稳时间序列的谱函数，不论是高斯序列还是非高斯序列，其误差过程的不变原理都已被证明，许重光采用自迥归谱估计的积分估计谱函数也证明了误差过程的不变原理成立。本文讨论多维平稳时间序列，引进积分谱图来估计谱函数，证明了误差过程的不变原理在较弱的条件下成立。积分周期图是积分谱图的特例，因而［１］［３］［６］的一些结果是本文有关定理的特例。     

Note on the integral mean value method for Fredholm integral equations of the second kind

We study the paper of Avazzadeh et al. [Z. Avazzadeh, M. Heydari, G.B., Loghmani, Numerical solution of Fedholm integral equations of the second kind by using integral mean value theorem, Appl. Math. Model. 35 (2011) 2374–2383] with the integral mean value method for Fredholm integral equations of the second kind. The objective of the note is threefold. First, we point out a basic error in the paper. Second, we find that the given numerical examples are only related to the special cases of Fredholm integral equations of the second kind with the degenerate kernels, which can be solved simply. Third, due to the basic error, our observations reveal that generally the suggested method should not be considered for a Fredholm integral equation of the second kind.     

Optimization of Approximate Solution of Integral Equations of Several Variables

In this paper, we consider the problem of optimization of adaptive direct methods of operator equations. Adaptivity of a direct method is understood in the sense that the subspace on the basis of which it is constructed is chosen depending on the operator of the concrete equation (otherwise, nonadaptive direct method is then concerned), which would essentially let us increase the precision. For some classes of the second kind of Fredhlom integral equations with anisotropic smooth kernels we determine the exact order of the error of adaptive direct methods, and we also give an optimal algorithm. Received April 13, 2001, Accepted September 13, 2001     

Diagonal forms of the translation operators in the fast multipole algorithm for scattering problems

The integral equations of acoustic and electromagnetic scattering generate large dense systems of linear equations. These systems are efficiently solved with iterative methods where the matrix-vector multiplication is computed using a special fast method, such as the fast Fourier transform or the fast multipole method (FMM). In this paper, the so called diagonal forms of the translation operators for the fast multipole method are derived starting from integral representations of certain special functions. Error analysis of the FMM is given, considering both the truncation error of potential expansions and the errors from the use of numerical integration in the diagonal translation theorem. The implications of the error bounds on the FMM algorithm are discussed.This work has been financially supported by the Jenny and Antti Wihuri Foundation and by the Cultural Foundation of Finland.     

积分凸性及其应用

该文在Banach空间中通过向量值函数的Bochner积分引进集合与泛函的积分凸性以及集合的积分端点等概念. 文章主要证明有限维凸集、开凸集和闭凸集均是积分凸集，下半连续凸泛函与开凸集上的上半连续凸泛函均是积分凸的, 非空紧集具有积分端点, 对紧凸集来说其积分端点集与端点集一致, 最后给出积分凸性在最优化理论方面的两个应用.     

q-Hermite Polynomials and a q-Beta Integral

The main object of the present paper is to derive a formula for the q-Hahn.polynomials by using the q-Mehler formula about the q-Hermite polynomials.This result, together with other properties of the q-Hermite polynomials, provides an evalution of a q-beta integral.Both the celebrated Askey-Wilson integral and the Ismail-Stanton-Viennot integral are special cases of this integral.     

LQR based improved discrete PID controller design via optimum selection of weighting matrices using fractional order integral performance index

The continuous and discrete time Linear Quadratic Regulator (LQR) theory has been used in this paper for the design of optimal analog and discrete PID controllers respectively. The PID controller gains are formulated as the optimal state-feedback gains, corresponding to the standard quadratic cost function involving the state variables and the controller effort. A real coded Genetic Algorithm (GA) has been used next to optimally find out the weighting matrices, associated with the respective optimal state-feedback regulator design while minimizing another time domain integral performance index, comprising of a weighted sum of Integral of Time multiplied Squared Error (ITSE) and the controller effort. The proposed methodology is extended for a new kind of fractional order (FO) integral performance indices. The impact of fractional order (as any arbitrary real order) cost function on the LQR tuned PID control loops is highlighted in the present work, along with the achievable cost of control. Guidelines for the choice of integral order of the performance index are given depending on the characteristics of the process, to be controlled.     

Imbedding methods for integral equations with applications

During the last decade or two, significant progress has been made in the development of imbedding methods for the analytical and computational treatment of integral equations. These methods are now well known in radiative transfer, neutron transport, optimal filtering, and other fields. In this review paper, we describe the current status of imbedding methods for integral equations. The paper emphasizes new analytical and computational developments in control and filtering, multiple scattering, inverse problems of wave propagation, and solid and fluid mechanics. Efficient computer programs for the determination of complex eigenvalues of integral operators, analytical investigations of stability for significant underlying Riccati integrodifferential equations, and comparisons against other methods are described.     

Optimization of adaptive direct algorithms for approximate solution of integral equation

The concern of this paper is to study the optimization of adaptive direct algorithms for approximate solution of the second kind Fredholm integral equations. We derive the exact order of the problem with integral kernels belonging to the Besov classes. Furthermore, we also present an almost optimal adaptive direct algorithm for above equation class.     

Lebesgue-Stieltjes形式的Choquet积分Ⅱ

研究了Lebesgue-Stieltjes形式的Choquet积分的收敛性定理,如单调收敛定理、法都引理、控制收敛定理等.     

空间平面内整点n边形上的整点个数

本文给出了计算空间平面内整点n边形上的整点个数的一个公式。     

Dirichlet Boundary Control of Hyperbolic Equations in the Presence of State Constraints

We study optimal control problems for hyperbolic equations (focusing on the multidimensional wave equation) with control functions in the Dirichlet boundary conditions under hard/pointwise control and state constraints. Imposing appropriate convexity assumptions on the cost integral functional, we establish the existence of optimal control and derive new necessary optimality conditions in the integral form of the Pontryagin Maximum Principle for hyperbolic state-constrained systems.     

Choquet积分的收敛定理

Murofushi、Sugeno等学者已对关于模糊测度的Choquet积分进行了详细的研究,但关于积分的收敛理论是不够的。本文讨论这一问题,给出Choquet积分的一些收敛定理,包括广义单调收敛定理、Fatou引理等。从中可以看出,Choquet积分与Sugeno模糊积分具有相同的收敛定理。     

The integral convexity of sets and functionals in Banach spaces

By way of the Bochner integral of vector-valued functions, the integral convexity of sets and functionals and the concept of integral extreme points of sets are introduced in Banach spaces. The relations between integral convexity and convexity are mainly discussed, two integral extreme points theorems and their applications are obtained at last.     

A new approach to young measure theory, relaxation and convergence in energy

The main idea of this paper is to reduce analysis of behavior of integral functionals along weakly convergent sequences to operations with Young measures generated by these sequences. We show that Young measures can be characterized as measurable functions with values in a special compact metric space and that these functions have a spectrum of properties sufficiently broad to realize this idea.These new observations allow us to give simplified proofs of the results of gradient Young measure theory and to use them for deriving the results on relaxation and convergence in energy under optimal assumptions on integrands.We think that this work helps to clarify role of Young measures.     

Integral refinable operators exact on polynomials

We study integral refinable operators of integral type exact on polynomials of even degree constructed by using refinable B-bases of GP type. We prove a general theorem of existence and uniqueness. Then we study the L^p$L^p$-norm of these operators and we give error bounds in approximating functions and their derivatives belonging to suitable classes. Numerical results and comparisons with other quasi-interpolatory operators having the same order of exactness on polynomial reproduction are presented.     

Adaptive-Additive Multilevel Methods for Hypersingular Integral Equation

In this article we present the combined adaptive-additive multilevel methods for the Galerkin approximation of hypersingular integral equation on the interval o = ( m 1,1). We also derive an efficient and reliable a posteriori error estimate for the error between the exact solution u and the approximated multilevel solution $ \tilde u_ {\cal M} $ , measuring locally the quality of $ \tilde u_ {\cal M} $ . The algorithm is carefully designed to obtain minimal complexity. A limitation of our analysis approach is that the meshes must be assumed to be quasi-uniform.