The Wavelet Element Method Part II. Realization and Additional Features in 2D and 3D

The Wavelet Element Method (WEM) provides a construction of multiresolution systems and biorthogonal wavelets on fairly general domains. These are split into subdomains that are mapped to a single reference hypercube. Tensor products of scaling functions and wavelets defined on the unit interval are used on the reference domain. By introducing appropriate matching conditions across the interelement boundaries, a globally continuous biorthogonal wavelet basis on the general domain is obtained. This construction does not uniquely define the basis functions but rather leaves some freedom for fulfilling additional features. In this paper we detail the general construction principle of the WEM to the 1D, 2D, and 3D cases. We address additional features such as symmetry, vanishing moments, and minimal support of the wavelet functions in each particular dimension. The construction is illustrated by using biorthogonal spline wavelets on the interval.     

一类基于小波基函数插值的有限元方法

在分析具有大的梯度问题中,将具有紧支集的小波基函数引入到传统的有限元插值函数的构造中,对传统的插值方法进行修正。对新的插值模式进行了数值稳定性（解的唯一存在性）分析并通过分片分析讨论了解的收敛性,新的插值模式所引入的附加自由度通过静力凝聚法来消除,最后得到了基于变分原理的小波有限元列式。     

A Conservative Space-time Mesh Refinement Method for the 1-D Wave Equation. Part I: Construction

Summary. We propose a new method for space-time refinement for the 1-D wave equation. This method is based on the conservation of a discrete energy through two different discretization grids which guarantees the stability of the scheme. Our approach results in a non-interpolatory scheme whose stability condition is not affected by the transition between the two grids. Mathematics Subject Classification (1991):65M12     

Multivalued Exponentiation Analysis. Part I: Maclaurin Exponentials

The exponentiation theory of linear continuous operators on Banach spaces can be extended in manifold ways to a multivalued context. In this paper we explore the Maclaurin exponentiation technique which is based on the use of a suitable power series. More precisely, we discuss about the existence and characterization of the Painlevé–Kuratowski limit

The Fourier Singular Complement Method for the Poisson problem. Part I: prismatic domains

This is the first part of a threefold article, aimed at solving numerically the Poisson problem in three-dimensional prismatic or axisymmetric domains. In this first part, the Fourier Singular Complement Method is introduced and analysed, in prismatic domains. In the second part, the FSCM is studied in axisymmetric domains with conical vertices, whereas, in the third part, implementation issues, numerical tests and comparisons with other methods are carried out. The method is based on a Fourier expansion in the direction parallel to the reentrant edges of the domain, and on an improved variant of the Singular Complement Method in the 2D section perpendicular to those edges. Neither refinements near the reentrant edges of the domain nor cut-off functions are required in the computations to achieve an optimal convergence order in terms of the mesh size and the number of Fourier modes used. This author was supported in part by the France/Hong Kong Joint Research Scheme. This author was supported by DGA/DSP-ENSTA 00.60.075.00.470.75.01 Research Programme This author was fully supported by Hong Kong RGC grants (Project CUHK4048/02P and project 403403).     

Nonsmooth Analysis of Singular Values. Part I: Theory

The singular values of a rectangular matrix are nonsmooth functions of its entries. In this work we study the nonsmooth analysis of functions of singular values. In particular we give simple formulae for the regular subdifferential, the limiting subdifferential, and the horizon subdifferential, of such functions. Along the way to the main result we give several applications and in particular derive von Neumann’s trace inequality for singular values. Mathematics Subject Classifications (2000) Primary 90C31, 15A18; secondary 49K40, 26B05.Research supported by NSERC.     

有限元法中大单元的构造

在通常的有限元法中,单元内的插值多项式的阶数固定不变,通过加密剖分网格来提高精度.大单元法则剖分的网格固定不变而通过增加单元内逼近级数的项数来提高精度. 本文提出采用两套变量的办法来构造大单元,即单元内采用一套变量,单元的边界上采用另一套变量,然后用杂交罚函数法把两者联系起来.这种方法能适用于任何椭圆型方程,任意几何形状区域以及任何复杂的边界条件.本文用严密的数学方法证明了:在一般情况下,这种方法的精度比通常的有限元法和文[7]的大单元法高得多.即在达到相同的精度时,本文方法所需要的自由度(即未知数数目)比上述两种方法少得多.     

The pricing problem. Part I: Exact and approximate algorithms

In the article we consider the pricing problem and show it to be NP-hard in the strong sense. Some exact and approximate algorithms are developed based on decomposition, genetic local search, and tabu search. The results of some computational experiments are provided.     

Basic formulas and languages Part I. The theory

A general mathematical framework to deal with (the decidability status of) properties of derivations in E0L systems (forms) is developed. It is based on the theory of well-quasi-orders. This paper (the first of two parts) deals with the mathematical theory of the proposed approach.     

Mathematical Analysis of a Soccer Game. Part I: Individual and Collective Behaviors

Detailed time series analysis of a soccer match is given based on the detailed data of the 2D motions of all 22 players and of the ball for the match. The whole analysis includes two parts. In Part I, the individual and collective behaviors of the players of the two teams as well as the motion of the ball are presented as various time series. Geometrical centers, radii, expansion speeds, possession functions of the two teams are defined and calculated as functions of time. Major ranges of all players as well as of different groups of players (defenders, midfielders, forwards) of the two teams during the entire first half, the attacking phase of team A and the attacking phase of team B are calculated, respectively, showing the structures of the two teams during different phases. Distance coverage of each player and the mean distances covered by different groups of players (defenders, midfielders, forwards) during different phases are calculated. The time portions of possession of the ball by each team and the time portions of different phases are also calculated. In Part II, energy and spectral analysis and various correlations will be derived. The relation between various parameters and potential indicators will be discussed. The major purpose of the present study is to offer some general mathematical tools for the detailed analysis and to reveal some general features of soccer match when the detailed 2D data are available. The results would offer the raw materials for various potential indicators which may eventually be used in the coaching process to enhance the performance and in the prediction of the results of soccer matches.     

Constrained Extremum Problems,Regularity Conditions and Image Space Analysis. Part I: The Scalar Finite-Dimensional Case

Image space analysis has proved to be instrumental in unifying several theories, apparently disjoint from each other. With reference to constraint qualifications/regularity conditions in optimization, such an analysis has been recently introduced by Moldovan and Pellegrini. Based on this result, the present paper is a preliminary part of a work, which aims at exploiting the image space analysis to establish a general regularity condition for constrained extremum problems. The present part deals with scalar constrained extremum problems in a Euclidean space. The vector case as well as the case of infinite-dimensional image will be the subject of a subsequent part.     

Hybridized globally divergence-free LDG methods. Part I: The Stokes problem

We devise and analyze a new local discontinuous Galerkin (LDG) method for the Stokes equations of incompressible fluid flow. This optimally convergent method is obtained by using an LDG method to discretize a vorticity-velocity formulation of the Stokes equations and by applying a new hybridization to the resulting discretization. One of the main features of the hybridized method is that it provides a globally divergence-free approximate velocity without having to construct globally divergence-free finite-dimensional spaces; only elementwise divergence-free basis functions are used. Another important feature is that it has significantly less degrees of freedom than all other LDG methods in the current literature; in particular, the approximation to the pressure is only defined on the faces of the elements. On the other hand, we show that, as expected, the condition number of the Schur-complement matrix for this approximate pressure is of order in the mesh size . Finally, we present numerical experiments that confirm the sharpness of our theoretical a priori error estimates.

     

Problems, Models and Complexity. Part I: Theory

The meaning of the term problem in operations research (OR) deviates from the understanding in the theoretical computer sciences: While an OR problem is often conceived to be stated or represented by model formulations, a computer-science problem can be viewed as a mapping from encoded instances to solutions. Such a computer-science problem turns out to be rather similar to an OR model formulation. This ambiguity may cause difficulties if the computer-science theory of computational complexity is applied in the OR context. In OR, a specific model formulation is commonly used in the analysis of the underlying problem and the results obtained for this model are simply lifted to the problem level. But this may lead to erroneous results, if the model used is not appropriate for such an analysis of the problem.To resolve this issue, we first suggest a new precise formal definition of the term problem which is identified with an equivalence class of models describing it. Afterwards, a new definition is suggested for the size of a model which depends on the assumed encoding scheme. Only models which exhibit a minimal size with respect to a reasonable encoding scheme finally turn out to be suitable for the model-based complexity analysis of an OR problem. Therefore, the appropriate choice (or if necessary the construction) of a suitable representative of an OR problem becomes an important theoretical aspect of the modelling process.