Two-level periodic marketing networks without market news

A model is constructed for a two-level periodic marketing network wherein traders buy goods from farmers in a number of spatially separated markets and transport them to urban centers for sale to consumers. It is assumed that there is no market news dissemination. As a result, market disturbances are transmitted throughout the network at a rate limited by the trading activity. Moreover, distant markets may even be isolated from local disturbances. This model possesses one and only one equilibrium state. That equilibrium state is asymptotically stable if the slopes of the various supply and demand functions are sufficiently restricted in the vicinity of the equilibrium state.     

Multilevel preconditioning

Summary This paper is concerned with multilevel techniques for preconditioning linear systems arising from Galerkin methods for elliptic boundary value problems. A general estimate is derived which is based on the characterization of Besov spaces in terms of weighted sequence norms related to corresponding multilevel expansions. The result brings out clearly how the various ingredients of a typical multilevel setting affect the growth rate of the condition numbers. In particular, our analysis indicates how to realize even uniformly bounded condition numbers. For example, the general results are used to show that the Bramble-Pasciak-Xu preconditioner for piecewise linear finite elements gives rise to uniformly bounded condition numbers even when the refinements of the underlying triangulations are highly nonuniform. Furthermore, they are applied to a general multivariate setting of refinable shift-invariant spaces, in particular, covering those induced by various types of wavelets.The work of this author was partially supported by the Air Force Office of Scientific Research (Contract No. 89/0455) and by the Office of Naval Research (Contract No. N00014/90/1343) during her stay at the Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA.     

Multilevel Schwarz methods

Summary We consider the solution of the algebraic system of equations which result from the discretization of second order elliptic equations. A class of multilevel algorithms are studied using the additive Schwarz framework. We establish that the condition number of the iteration operators are bounded independent of mesh sizes and the number of levels. This is an improvement on Dryja and Widlund's result on a multilevel additive Schwarz algorithm, as well as Bramble, Pasciak and Xu's result on the BPX algorithm. Some multiplicative variants of the multilevel methods are also considered. We establish that the energy norms of the corresponding iteration operators are bounded by a constant less than one, which is independent of the number of levels. For a proper ordering, the iteration operators correspond to the error propagation operators of certain V-cycle multigrid methods, using Gauss-Seidel and damped Jacobi methods as smoothers, respectively.This work was supported in part by the National Science Foundation under Grants NSF-CCR-8903003 at Courant Institute of Mathematical Sciences, New York University and NSF-ASC-8958544 at Department of Computer Science, University of Maryland.     

Multilevel ILU decomposition

Summary. In this paper, the multilevel ILU (MLILU) decomposition is introduced. During an incomplete Gaussian elimination process new matrix entries are generated such that a special ordering strategy yields distinct levels. On these levels, some smoothing steps are computed. The MLILU decomposition exists and the corresponding iterative scheme converges for all symmetric and positive definite matrices. Convergence rates independent of the number of unknowns are shown numerically for several examples. Many numerical experiments including unsymmetric and anisotropic problems, problems with jumping coefficients as well as realistic problems are presented. They indicate a very robust convergence behavior of the MLILU method. Received June 13, 1997 / Revised version received March 17, 1998     

Multilevel Stackelberg strategies in linear-quadratic systems

Open-loop multilevel Stackelberg strategies in deterministic, sequential decision-making problems for continuous linear systems and quadratic criteria are developed. Characterization of the Stackelberg controls via the solution of a higher-order square-matrix-Riccati differential equation is established; also, the basic structural properties of the coefficient matrices of this differential equation are established, and the basic structural properties of its solution are inferred.This work was supported in part by the Energy Research and Development Administration, Contract No. ERDA E(49-18)-2088.on leave from the Mihailo Pupin Institute, Belgrade, Yugoslavia.     

Topographical Multilevel Single Linkage

An iterative topographical Multilevel Single Linkage (TMSL) method has been introduced. The approach uses topographical information on the objective function, in particular theg-nearest-neighbour graph. The algorithm uses evenly distributed points from a Halten sequence of uniform limiting density. We discuss the implementation of the algorithm and compare its performance with other well-known algorithms. The new algorithm performs much better (in some cases several times) than the Multilevel Single Linkage method in terms of number of function evaluations but is not quite so competitive with respect to CPU time.     

多级概率群试

本文主要分为两个部分,第一部分是将群试的方法推广到多级群式。第二部分是在多级群式的基础上添加一个值得考虑的因素。这样做不仅使得问题更加切合实际并且在最后可使最优方案变得十分的简洁。     

Multilevel Interpolation and Approximation

Interpolation by translates of a given radial basis function (RBF) has become a well-recognized means of fitting functions sampled at scattered sites in ^d. A major drawback of these methods is their inability to interpolate very large data sets in a numerically stable way while maintaining a good fit. To circumvent this problem, a multilevel interpolation (ML) method for scattered data was presented by Floater and Iske. Their approach involves m levels of interpolation where at the jth level, the residual of the previous level is interpolated. On each level, the RBF is scaled to match the data density. In this paper, we provide some theoretical underpinnings to the ML method by establishing rates of approximation for a technique that deviates somewhat from the Floater–Iske setting. The final goal of the ML method will be to provide a numerically stable method for interpolating several thousand points rapidly.     

A Multilevel AINV Preconditioner

In this paper we describe an algebraic multilevel extension of the approximate inverse AINV preconditioner for solving symmetric positive definite linear systems Ax=b with the preconditioned conjugate gradient method. The smoother is the approximate inverse M and the coarse grids and the interpolation operator are constructed by looking at the entries of M. Numerical examples are given for problems arising from discretization of partial differential equations.     

Multilevel hybrid split-step implicit tau-leap

In biochemically reactive systems with small copy numbers of one or more reactant molecules, the dynamics is dominated by stochastic effects. To approximate those systems, discrete state-space and stochastic simulation approaches have been shown to be more relevant than continuous state-space and deterministic ones. In systems characterized by having simultaneously fast and slow timescales, existing discrete space-state stochastic path simulation methods, such as the stochastic simulation algorithm (SSA) and the explicit tau-leap (explicit-TL) method, can be very slow. Implicit approximations have been developed to improve numerical stability and provide efficient simulation algorithms for those systems. Here, we propose an efficient Multilevel Monte Carlo (MLMC) method in the spirit of the work by Anderson and Higham (SIAM Multiscal Model. Simul. 10(1), 2012). This method uses split-step implicit tau-leap (SSI-TL) at levels where the explicit-TL method is not applicable due to numerical stability issues. We present numerical examples that illustrate the performance of the proposed method.     

Multilevel decompositions of functional spaces

A unified abstract framework for the multilevel decomposition of both Banach and quasi-Banach spaces is presented. The characterization of intermediate spaces and their duals is derived from general Bernstein and Jackson inequalities. Applications to compactly supported biorthogonal wavelet decompositions of families of Besov spaces are also given. The first author was partially supported by grants from MURST (40% Analisi Numerica) and ASI (Contract ASI-92-RS-89), whereas the second author was partially supported by grants from MURST (40% Analisi Funzionale) and CNR (Progetto Strategico “Applicazioni della Matematica per la Tecnologia e la Società”).     

Multilevel algorithms for ill-posed problems

Summary In this paper new multilevel algorithms are proposed for the numerical solution of first kind operator equations. Convergence estimates are established for multilevel algorithms applied to Tikhonov type regularization methods. Our theory relates the convergence rate of these algorithms to the minimal eigenvalue of the discrete version of the operator and the regularization parameter. The algorithms and analysis are presented in an abstract setting that can be applied to first kind integral equations.Dedicated to Jim Bramble on the occasion of his sixtieth birthday     

Multilevel particle-partition of unity method

This paper is concerned with the particle-partition of unity method, a meshfree generalization of the finite element method. We present the fundamental construction principles and abstract approximation properties of the resulting function spaces V ^PU. Moreover, we discuss the construction of optimal approximation spaces for a reference application in linear elastic fracture mechanics in particular. The presented construction not only yields optimal convergence rates globally independently of the regularity of the solution, our method shows a super-convergence near the singular points of the solution.     

Multilevel and multidimensional Hadamard matrices

Multilevel Hadamard matrices (MHMs), whose entries are integers as opposed to the traditional restriction to {±1}, were introduced by Trinh, Fan, and Gabidulin in 2006 as a way to construct multilevel zero-correlation zone sequences, which have been studied for use in approximately synchronized code division multiple access systems. We answer the open question concerning the maximum number of distinct elements permissible in an order n MHM by proving the existence of an order n MHM with n elements of distinct absolute value for all n. We also define multidimensional MHMs and prove an analogous existence result.      

Artificial Multilevel Boundary Element Preconditioners

A hierarchical multilevel preconditioner is constructed for an efficient solution of a first kind boundary integral equation with the single layer potential operator discretized by a boundary element method. This technique is based on a hierarchical clustering of all boundary elements as used in fast boundary element methods. This hierarchy is applied to define a sequence of nested boundary element spaces of piecewise constant basis functions as used in the definition of the preconditioning multilevel operator.     

多水平模型及其在经济分析中的应用

针对具有层次或聚类数据的多水平模型能准确地反映变量间基于层次框架下的关系,并给出不同层次数据的差异性估计及跨级相关估计,为具有层次结构数据的统计建模提供了重要的研究工具,在社会学、心理学、生物医学及经济学领域具有广泛的应用价值。本文简要介绍常用的多水平线性模型和多水平Logistic模型的构建过程,重点介绍其在经济领域中应用。同时对多水平模型的估计理论,应用软件以和发展展望进行了讨论。     

多水平模型及其在经济分析中的应用

针对具有层次或聚类数据的多水平模型能准确地反映变量间基于层次框架下的关系,并给出不同层次数据的差异性估计及跨级相关估计,为具有层次结构数据的统计建模提供了重要的研究工具,在社会学、心理学、生物医学及经济学领域具有广泛的应用价值。本文简要介绍常用的多水平线性模型和多水平Logistic模型的构建过程,重点介绍其在经济领域中的应用。同时对多水平模型的估计理论、应用软件以及发展展望进行了讨论。