On the locking phenomenon for a class of elliptic problems

Summary. In this paper we study the numerical behaviour of elliptic problems in which a small parameter is involved and an example concerning the computation of elastic arches is analyzed using this mathematical framework. At first, the statements of the problem and its Galerkin approximations are defined and an asymptotic analysis is performed. Then we give general conditions ensuring that a numerical scheme will converge uniformly with respect to the small parameter. Finally we study an example in computation of arches working in linear elasticity conditions. We build one finite element scheme giving a locking behaviour, and another one which does not. Revised version received October 25, 1993     

Extension of the hierarchy for k-OBDDs of small width

In this paper we explore the well-known k-OBDD model of branching programs. We develop a method of representation of the k-OBDD computation process as an “automata-communication protocol” computation process. Our method allows us to extend the hierarchy proved by Bolling-Sauerhoff-Sieling-Wegener in 1996 for k-OBDDs. Moreover, using the PJM function (a modification of well-known PJ and ISA functions), we prove a new hierarchy.     

On the dimensions of secant varieties of Segre-Veronese varieties

This paper explores the dimensions of higher secant varieties to Segre-Veronese varieties. The main goal of this paper is to introduce two different inductive techniques. These techniques enable one to reduce the computation of the dimension of the secant variety in a high-dimensional case to the computation of the dimensions of secant varieties in low-dimensional cases. As an application of these inductive approaches, we will prove non-defectivity of secant varieties of certain two-factor Segre-Veronese varieties. We also use these methods to give a complete classification of defective sth Segre–Veronese varieties for small s. In the final section, we propose a conjecture about defective two-factor Segre–Veronese varieties.     

A Refined Model of Computation for Continuous Problems

The real-number model of computation is used in computational geometry, in the approach suggested by Blum, Shub, and Smale and in information based complexity. In this paper we propose a refinement of this model, the TTE-model of computation. In contrast to the real-number model, which is unrealistic or at least too optimistic, the TTE-model is very realistic; i.e., for TTE-algorithms there are digital computers, which behave exactly the same way as predicted by the theoretical model. We start with a detailed discussion of some objections to the real-number model. We introduce the refined model by adding the condition “every partial input or output information of an algorithm is finite” to the assumptions of the IBC-model of computation. First, we explain computability and computational complexity in TTE for the simple case of real functions. Then we apply the refined model to two typical IBC-problems, integration and zero-finding on the spaceC[0; 1] of continuous real functions. We study computability and computational complexity and compare TTE-results with IBC-results. Finally, we briefly discuss the computation of discontinuous problems. This paper does not present new technical results but should be considered as a fresh impetus to reflect on models of computation for numerical analysis and as an invitation to try out the TTE-model of computation in information based complexity.     

On the Poset of Computation Rules for Nonassociative Calculus

The symmetric maximum, denoted by ?, is an extension of the usual maximum ∨ operation so that 0 is the neutral element, and ??x is the symmetric (or inverse) of x, i.e., x???(???x)?=?0. However, such an extension does not preserve the associativity of ∨. This fact asks for systematic ways of parenthesing (or bracketing) terms of a sequence (with more than two arguments) when using such an extended maximum. We refer to such systematic (predefined) ways of parenthesing as computation rules. As it turns out there are infinitely many computation rules, each of which corresponds to a systematic way of bracketing arguments of sequences. Essentially, computation rules reduce to deleting terms of sequences based on the condition x???(???x)?=?0. This observation gives raise to a quasi-order on the set of such computation rules: say that rule 1 is below rule 2 if for all sequences of numbers, rule 1 deletes more terms of the sequence than rule 2. In this paper we present a study of this quasi-ordering of computation rules. In particular, we show that the induced poset of all equivalence classes of computation rules is uncountably infinite, has infinitely many maximal elements, has infinitely many atoms, and it embeds the powerset of natural numbers ordered by inclusion.     

Multi-scale method for the quasi-periodic structures of composite materials

In this paper, we shall present a new second-order and two-scale approximate solution of boundary value problems for the linear elasticity systems with quasi-periodic structures by multi-scale analysis. The computation of strain and stress is presented here. This idea can apply to the damage computation of composite materials. Because the computation of the local stress by this method having more precision, it can provide some inspiration for construct optimal design. Finally numerical results show that the method presented in this paper is effective and reliable.     

Computing the NTU-Shapley value of NTU-games defined by multiple objective linear programs

In this paper we study a class of cooperative games with non-transferable utility (NTU) derived from multiple objective linear programs (MOLP's). This is done in order to introduce the NTU-Shapley value as a solution to multiple objective linear programming.  We present an algorithm for the computation of the set of all NTU-Shapley values for the bicriterion cases, which is based on the simplex algorithm. Furthermore, we give a new optimal recursive procedure for the computation of the Shapley value of TU-games. Received: September 1997/Final version: May 1999     

A Riemannian submersion-based approach to the Wasserstein barycenter of positive definite matrices

In this paper, we introduce a novel geometrization on the space of positive definite matrices, derived from the Riemannian submersion from the general linear group to the space of positive definite matrices, resulting in easier computation of its geometric structure. The related metric is found to be the same as a particular Wasserstein metric. Based on this metric, the Wasserstein barycenter problem is studied. To solve this problem, some schemes of the numerical computation based on gradient descent algorithms are proposed and compared. As an application, we test the k-means clustering of positive definite matrices with different choices of matrix mean.     

Estimation of the workload correlation in a Markov fluid queue

This paper considers a Markov fluid queue, focusing on the correlation function of the stationary workload process. A simulation-based computation technique is proposed, which relies on a coupling idea. Then an upper bound on the variance of the resulting estimator is given, which reveals how the coupling time and the busy period of the Markov fluid queue affect the performance of the computation procedure. A numerical assessment, in which we compare the proposed technique with naive simulation, gives an indication of the achievable efficiency gain in various scenarios.     

A column generation approach to the discrete barycenter problem

The discrete Wasserstein barycenter problem is a minimum-cost mass transport problem for a set of discrete probability measures. Although an exact barycenter is computable through linear programming, the underlying linear program can be extremely large. For worst-case input, a best known linear programming formulation is exponential in the number of variables, but has a low number of constraints, making it an interesting candidate for column generation.In this paper, we devise and study two column generation strategies: a natural one based on a simplified computation of reduced costs, and one through a Dantzig–Wolfe decomposition. For the latter, we produce efficiently solvable subproblems, namely, a pricing problem in the form of a classical transportation problem. The two strategies begin with an efficient computation of an initial feasible solution. While the structure of the constraints leads to the computation of the reduced costs of all remaining variables for setup, both approaches may outperform a computation using the full program in speed, and dramatically so in memory requirement. In our computational experiments, we exhibit that, depending on the input, either strategy can become a best choice.     

Distributions of the location of maximuma and minimuma for diffusions with jumps

The paper deals with methods of computation of distributions of location for maxima and minima for diffusions with jumps. As an example, we obtain explicit formulas for distributions of location for the maximum of the process which is equal to the sum of a Brownian motion and the compound Poisson process. Bibliography: 8 titles.     

p-Adic Analogue of the Wave Equation

In this paper, a p-adic analogue of the wave equation with Lipschitz source is considered. Since it is hard to arrive the solution of the problem, we propose a regularized method to solve the problem from a modified p-adic integral equation. Moreover, we give an iterative scheme for numerical computation of the regularlized solution.

     

Solving time‐periodic fractional diffusion equations via diagonalization technique and multigrid

This paper addresses numerical computation of time‐periodic diffusion equations with fractional Laplacian. Time‐periodic differential equations present fundamental challenges for numerical computation because we have to consider all the discrete solutions once in all instead of one by one. An idea based on the diagonalization technique is proposed, which yields a direct parallel‐in‐time computation for all the discrete solutions. The major computation cost is therefore reduced to solve a series of independent linear algebraic systems with complex coefficients, for which we apply a multigrid method using the damped Richardson iteration as the smoother. Such a linear solver possesses mesh‐independent convergence factor, and we make an optimization for the damping parameter to minimize such a constant convergence factor. Numerical results are provided to support our theoretical analysis.     

A polynomial expression for the Owen value in the maintenance cost game

The class of maintenance cost games was introduced in 2000 to deal with a cost allocation problem arising in the reorganization of the railway system in Europe. The main application of maintenance cost games regards the allocation of the maintenance costs of a facility among the agents using it. To that aim it was first proposed to utilize the Shapley value, whose computation for maintenance cost games can be made in polynomial time. In this paper, we propose to model this cost allocation problem as a maintenance cost game with a priori unions and to use the Owen value as a cost allocation rule. Although the computation of the Owen value has exponential complexity in general, we provide an expression for the Owen value of a maintenance cost game with cubic polynomial complexity. We finish the paper with an illustrative example using data taken from the literature of railways management.     

An improved sweeping domain decomposition preconditioner for the Helmholtz equation

In this paper we generalize and improve a recently developed domain decomposition preconditioner for the iterative solution of discretized Helmholtz equations. We introduce an improved method for transmission at the internal boundaries using perfectly matched layers. Simultaneous forward and backward sweeps are introduced, thereby improving the possibilities for parallellization. Finally, the method is combined with an outer two-grid iteration. The method is studied theoretically and with numerical examples. It is shown that the modifications lead to substantial decreases in computation time and memory use, so that computation times become comparable to that of the fastests methods currently in the literature for problems with up to 10⁸ degrees of freedom.     

Bounds for the reliability functions of coherent systems with heterogeneous components

The computation of the reliability function of a (complex) coherent system is a difficult task. Hence, sometimes, we should simply work with some bounds (approximations). The computation of these bounds has been widely studied in the case of coherent systems with independent and identically distributed (IID) components. However, few results have been obtained in the case of heterogeneous (non ID) components. In this paper, we derive explicit bounds for systems with heterogeneous (independent or dependent) components. Also some stochastic comparisons are obtained. Some illustrative examples are included where we compare the different bounds proposed in the paper.     

有缺失数据的META-MARKOV模型的局部计算

根据meta Markov模型的特征,本文提出了有缺失数据的meta Markov模型的局部计算方法．由于最小可压缩集与局部计算关系密切,本文给出了一种搜索包含预先给定的子集的最小可压缩集的算法．     

Performance enhancement and validation of the open-source software for modeling of ship infrared signatures (OSMOSIS)

Designing stealth technologies for modern warships requires methods for signatures reduction (radar, infrared, etc). First, we have to model these signatures. In Lapierre et al. (2006), Lapierre et al. (2007) [3] and [4], we proposed an infrared (IR) ship signature simulation software (OSMOSIS) that can manage parametric emissivity. OSMOSIS comprises a smart computation time and memory management. Evaluating the effectiveness of an IR signature reduction method implies simulating the evolution of the IR signature with time. This requires a huge computation time. In this paper, we propose a novel hierarchical meshing strategy leading to an important reduction of the computation time. The time complexity of the algorithm is shown to be , where N is the requested number of facets of the mesh of the ship’s surface. Then, we validate OSMOSIS on temperature measurements performed on an “L-shape” object.     

Filiform李超代数L_(1,2)上的Yang-Baxter方程

求解一般Filiform李超代数L_(n,m)的Yang-Baxter方程尚无一般方法.通过计算,刻画了特征零的代数闭域上四维Filiform李超代数L_(1,2)上的所有Yang-Baxter方程的解.     

A formula for the computation of the integral cohomology of the knot of an isolated singularity of a hypersurface

The present paper is devoted to the computation of the Jordan normal form of the monodromy operator of an isolated singularity of a hypersurface and also to the computation of the integral cohomologies of the knot of an isolated singularity of a hypersurface.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 100, pp. 106–112, 1980.