Spline methods for a Birkhoff interpolation problem

In this paper, we study two new spline methods for the Birkhoff interpolation problem proposed in Costabile and Longo (Calcolo 47:49–63, 2010). Our methods are based on cubic spline and quintic spline respectively. The new methods are easy to be implemented. They are effective not only in approximating the original function but also in approximating its derivatives. The respective remainders and error bounds are given. Numerical tests are performed. Numerical results confirm the theoretical analysis and show that our methods are very practical.     

A time-varying Beurling-Lax theorem and a related interpolation problem

This article concerns linear time-varying interpretations of the Beurling-Lax-Ball-Helton theorem and of Sarason's interpolation problem. The former characterizes shift-invariantH ₂ (Krein) subspaces. Unilateral shift invariance reflects both causality and time invariance. Removing the stationarity requirement, a generalized theorem provides a characterization of certain causal subspace families M_t L₂(t, ), t . Sarason's interpolation problem is interpreted here as a search for a (close to) minimal induced norm system, given causal input-output specifications. The Beurling-Lax theorem helps in identifying admissible specification classes. The problem is then reduced to and solved in terms of a linear time-varying Nehari problem. Technically, developments are based on timedomain, state-space methods.This research was supported by the National Science Foundation and by the Army Research Office.     

On the scalar rational interpolation problem

A new technique is derived for determining a parametrization of all minimal complexity rational functionsa(x)/b(x) interpolating an arbitrary sequence of points. Complexity is measured in terms of max{deg(a), deg(b) +r } wherer is an arbitrary integer (so thatr=0 corresponds to the McMillan degree). Our construction uses Gröbner bases of submodules of the free module of rank 2 over the polynomial ring in one variable and extends previous work on the key equation of error control coding theory.     

A Java application for quality weighted 3-d interpolation

iw3d is a tool for 3-d interpolation of scattered data. The program is endowed with a user-friendly graphical interface. The interpolation-algorithm is a modified version of the inverse-distance weighting method. It has the capability for a quality-weighted interpolation. The interpolation can be carried out with a sector-search approach for reducing clustering effects. A semivariogram is calculated for the three cartesian main directions to give an estimate for a selectable search-distance. Tests of the program with a synthetic 3-d data set and a large set of measured subsurface temperatures with different qualities show good results.iw3d is programmed with Java (TM) and is a GPL licensed open source. It has been successfully tested with different Java versions on Microsoft Windows, Sun Solaris and on several Linux distributions.The current version is available at http://geomath.onlinehome.de/iw3d/.     

Nevanlinna-Pick interpolation problem for two frequency scalesystems

The Nevanlinna-Pick interpolation problem for a parametric set of data is considered. Sufficient conditions are obtained that guarantees a two frequency scale solution. Decomposition of the problem into two Nevanlinna-Pick interpolation problems with smaller data size is presented. An algorithm is developed which computes the solution as the result of combining the solutions for these two smaller problems     

General variational approach to the interpolation problem

The Talmi and Gilat variational approach to the interpolation problem in arbitrary dimension is presented together with the corresponding physical model. The connection of this approach to some known spline methods is demonstrated and new interpolation functions are derived for one-, two- and three-dimensional cases. They are designed to be flexible through the use of meaningful parameters and to give good approximations of both the function itself and its derivatives as well.     

Construction and application of fractal interpolation surfaces

Fractal geometry has unique advantages for a broad class of modeling problems, including natural objects and patterns. This paper presents an approach to the construction of fractal surfaces by triangulation. After introducing the notion of iterated function systems (IFSs), we prove theoretically that the attractors of this construction are continuous fractal interpolation surfaces (FISs). Two fast, parallel, and iterative algorithms are also provided. Several experiments in natural phenomena simulation verify that this method is suitable for generating complex 3D shapes with self-similar patterns.Project supported by the National Natural Science of China     

广义Birkhoff系统的梯度表示

研究广义 Birkhoff 系统的梯度表示. 给出广义 Birkhoff 系统可成为梯度系统的条件. 利用梯度系统的性质来研究广义 Birkhoff 系统的稳定性. 举例说明结果的应用.     

Efficient implementation of image interpolation as an inverse problem

This paper presents three computationally efficient solutions for the image interpolation problem which are developed in a general framework. This framework is based on dealing with the problem as an inverse problem. Based on the observation model, our objective is to obtain a high resolution image which is as close as possible to the original high resolution image subject to certain constraints. In the first solution, a linear minimum mean square error (LMMSE) approach is suggested. The necessary assumptions required to reduce the computational complexity of the LMMSE solution are presented. The sensitivity of the LMMSE solution to these assumptions is studied. In the second solution, the concept of entropy maximization of the required high resolution image a priori is used. The implementation of the suggested maximum entropy solution as a single sparse matrix inversion is presented. Finally, the well-known regularization technique used in iterative nature in image interpolation and image restoration is revisited. An efficient sectioned implementation of regularized image interpolation, which avoids the large number of iterations required in the interactive technique, is presented. In our suggested regularized solution, the computational time is linearly proportional to the dimensions of the image to be interpolated and a single matrix inversion of moderate dimensions is required. This property allows its implementation in interpolating images of any dimensions which is a great problem in iterative techniques. The effect of the choice of the regularization parameter on the suggested regularized image interpolation solution is studied. The performance of all the above-mentioned solutions is compared to traditional polynomial based interpolation techniques such as cubic O-MOMS and to iterative interpolation as well. The suitability of each solution to interpolating different images is also studied.     

The subspace Nevanlinna interpolation problem and the most powerful unfalsified model

A generalization of the tangential Nevanlinna interpolation problem will be studied from a behavioral point of view. Necessary and sufficient conditions for its solvability and a characterization of all its solutions are derived. These results are obtained by associating to the interpolation data a behavior that enjoys a special structure.     

A knowledge-based problem solving method in GIS application

Model design for theme analysis is one of the biggest challenges in GIS. Many real applications in GIS require functioning not only in data management and visualization, but also in analysis and decision-making. Confronted with an application of planning a new metro line in a city, a typical GIS is unable to accomplish the task in the absence of human experts or artificial intelligence technologies. Apart from being models for analyzing in different themes, some applications are also instances of problem solving in AI. Therefore, in order to strengthen its ability in automatic analysis, many theories and technologies from AI can be embedded in the GIS. In this paper, a state space is defined to formalize the metro line planning problem. By utilizing the defined state evaluation function, knowledge-based rules and strategies, a heuristic searching method is developed to optimize the solutions iteratively. Experiments are implemented to illuminate the validity of this AI-enhanced automatic analysis model of GIS.     

A quasilinearization algorithm and its application to a manipulator problem

A quasilinearization algorithm is proposed for the computation of optimal control of a class of constrained problem. The constraints are inequality constraints on functions of the state and control variables, and bounds on the values of the control variables. Necessary conditions for optimal control of the control problem are derived. In the iterative procedure, no prior information is required regarding the sequence of constrained and unconstrained arcs and the inequality constraints which are on their boundaries along a specific constrained arc of the optimal trajectory. All this information will be determined within the iterative procedure using some necessary conditions for optimal control. The ability of the proposed algorithm to solve practical problems is demonstrated by its application to several variations of two problems, one of which is a common manipulator problem in industry where transportation of open vessels of liquid is to be performed in a specified period of time. It is shown that the proposed quasilinearization algorithm is an effective tool in deriving optimal control policies for a common type of manipulator operation in industry.     

3D IFS在DEM重构中的应用研究

在基于DEM的地形表面重构中,传统的插值方法（B样条插值、双线形插值）获取的地形表面过于平滑,不能反映自然地形具有无限细节的事实。引入3维迭代函数系统（3D-IFS）插值方法来重构经随机简化的原始地形。在重构地形时,创造性地将垂直放缩因子作为分形维数的连续函数以简化计算。实验结果表明,在压缩率不大时,3D IFS能较好地保持地表地貌特征和统计特征。     

广义分数阶受迫Birkhoff方程

研究受迫Birkhoff系统的分数阶变分问题,建立具有这两种分数阶微分算子的广义分数阶受迫Birkhoff方程.〖JP〗然后,给出具有这两种分数阶微分算子的分数阶Hamilton方程和分数阶Lagrange方程.最后,讨论广义分数阶Lotka 生化振子模型和广义分数阶Hojman Urrutia模型.     

三层并行遗传算法及装箱问题中的应用

提出了一个新的基于线程构建模块(TBB)的三层并行遗传算法(TPGA)。与传统遗传算法相比,在保证了算法正确性的前提下提高了运行效率,并将遗传算法的数据编码、任务处理和数据解码分别进行并行化,提高了收敛速度。TBB是Intel提供的能够完整表现并行性的代码库。采用C++语言实现了基于TBB的TPGA和串行遗传算法(SGA),通过大量实验证明,TPGA同SGA相比,不但提高了收敛速度,而且能够取得一致的最优解。     

A combinatorial approximation algorithm for concurrent flow problem and its application

The multi-commodity flow problem involves simultaneous shipping several different commodities from sources to sinks in a directed network with total amount of flow going through an edge limited by its capacity. The optimization version of the multi-commodity flow problem is the maximum concurrent flow problem, which finds a flow with the minimum congestion. For any positive ε, the ε-optimal concurrent flow problem is to find a solution whose the congestion value is no more than (1+ε) times the minimum congestion. In recent years, a few fast combinatorial approximation algorithms for the ε-optimal concurrent flow problem have been presented. In this paper we propose a new variant of the combinatorial approximation algorithm: CACF with a tighter computation bound in decreasing the values of congestion and the potential function. Numerical comparisons are made between the results obtained by the combinatorial approximation algorithms and those did by the linear programming package CPLEX on large-scale test networks. The application of CACF to efficiently solving the system-optimal network flow problem is given where good results have been obtained. It has shown the capacity of the CACF in dealing with problems of concurrent flow associated.     

Symbolic computation of the Birkhoff normal form in the problem of stability of the triangular libration points

The problem of stability of the triangular libration points in the planar circular restricted three-body problem is considered. A software package, intended for normalization of autonomous Hamiltonian systems by means of computer algebra, is designed so that normalization problems of high analytical complexity could be solved. It is used to obtain the Birkhoff normal form of the Hamiltonian in the given problem. The normalization is carried out up to the 6th order of expansion of the Hamiltonian in the coordinates and momenta. Analytical expressions for the coefficients of the normal form of the 6th order are derived. Though intermediary expressions occupy gigabytes of the computer memory, the obtained coefficients of the normal form are compact enough for presentation in typographic format. The analogue of the Deprit formula for the stability criterion is derived in the 6th order of normalization. The obtained floating-point numerical values for the normal form coefficients and the stability criterion confirm the results by Markeev (1969) and Coppola and Rand (1989), while the obtained analytical and exact numeric expressions confirm the results by Meyer and Schmidt (1986) and Schmidt (1989). The given computational problem is solved without constructing a specialized algebraic processor, i.e., the designed computer algebra package has a broad field of applicability.