一种新的求非线性方程组的数值延拓法

针对迭代过程中的Jacobi奇异问题,本文提出了一种新的数值延拓法.通过构造双参数同伦算子,采用可控条件和适当选取参数的方式克服Jacobi奇异性,并分析了方法的收敛性.最后,通过数值实验对比,验证了方法的可行性和优越性.特别是具有可调控越过Jacobi奇异(点、线、面)的优势,从而也在某种程度上解决了数值延拓法严重依赖于初值的问题.     

广义线性多乘积问题的完全多项式时间近似算法

本文针对广义线性多乘积极小化问题,通过一系列的线性规划问题的解提出一种求其全局最优解的完全多项式时间近似算法,并给出该算法的计算复杂性,且数值算例验证该算法是可行的.     

Limit theorems for random walks

We consider a random walk $S_{\tau }$ which is obtained from the simple random walk $S$ by a discrete time version of Bochner’s subordination. We prove that under certain conditions on the subordinator $\tau$ appropriately scaled random walk $S_{\tau }$ converges in the Skorohod space to the symmetric $\alpha$-stable process $B^{\alpha }$. We also prove asymptotic formula for the transition function of $S_{\tau }$ similar to the Pólya’s asymptotic formula for $B^{\alpha }$.     

Continuous spin models on annealed generalized random graphs

We study Gibbs distributions of spins taking values in a general compact Polish space, interacting via a pair potential along the edges of a generalized random graph with a given asymptotic weight distribution $P$, obtained by annealing over the random graph distribution.First we prove a variational formula for the corresponding annealed pressure and provide criteria for absence of phase transitions in the general case.We furthermore study classes of models with second order phase transitions which include rotation-invariant models on spheres and models on intervals, and classify their critical exponents. We find critical exponents which are modified relative to the corresponding mean-field values when $P$ becomes too heavy-tailed, in which case they move continuously with the tail-exponent of $P$. For large classes of models they are the same as for the Ising model treated in Dommers et al. (2016). On the other hand, we provide conditions under which the model is in a different universality class, and construct an explicit example of such a model on the interval.     

Disjoint Paths in Decomposable Digraphs

The k‐linkage problem is as follows: given a digraph and a collection of k terminal pairs such that all these vertices are distinct; decide whether D has a collection of vertex disjoint paths such that is from to for . A digraph is k‐linked if it has a k‐linkage for every choice of 2k distinct vertices and every choice of k pairs as above. The k‐linkage problem is NP‐complete already for [11] and there exists no function such that every ‐strong digraph has a k‐linkage for every choice of 2k distinct vertices of D [17]. Recently, Chudnovsky et al. [9] gave a polynomial algorithm for the k‐linkage problem for any fixed k in (a generalization of) semicomplete multipartite digraphs. In this article, we use their result as well as the classical polynomial algorithm for the case of acyclic digraphs by Fortune et al. [11] to develop polynomial algorithms for the k‐linkage problem in locally semicomplete digraphs and several classes of decomposable digraphs, including quasi‐transitive digraphs and directed cographs. We also prove that the necessary condition of being ‐strong is also sufficient for round‐decomposable digraphs to be k‐linked, obtaining thus a best possible bound that improves a previous one of . Finally we settle a conjecture from [3] by proving that every 5‐strong locally semicomplete digraph is 2‐linked. This bound is also best possible (already for tournaments) [1].     

Local and parallel finite element algorithm for stationary incompressible magnetohydrodynamics

This article presents a local and parallel finite element method for the stationary incompressible magnetohydrodynamics problem. The key idea of this algorithm comes from the two‐grid discretization technique. Specifically, we solve the nonlinear system on a global coarse mesh, and then solve a series of linear problems on several subdomains in parallel. Furthermore, local a priori estimates are obtained on a general shape regular grid. The efficiency of the algorithm is also illustrated by some numerical experiments.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1513–1539, 2017     

Textile porosity determination based on a nonlinear heat and moisture transfer model

The inverse problems of textile materials design on heat and moisture transfer properties are important and indispensable in applications in the body-clothing-environment system. We present an inverse problem of textile porosity determination (IPTPD) based on a nonlinear heat and moisture transfer model. Adopting the idea of the least-squares, the mathematical formulation of IPTPD is deduced to a regularized optimization problem with collocation method applied. The continuity of the regularized minimization problem is proved. By means of genetic algorithm (GA), the approximate solution of the IPTPD is numerically obtained. To reduce the computational cost, an improved algorithm based on BP neural network with GA is proposed in the numerical simulation. Compared with the direct GA searching, the computational cost is greatly reduced, which presents a similar result.     

The limiting distributions of large heavy Wigner and arbitrary random matrices

A heavy Wigner matrix $X_N$ is defined similarly to a classical Wigner one. It is Hermitian, with independent sub-diagonal entries. The diagonal entries and the non-diagonal entries are identically distributed. Nevertheless, the moments of the entries of $\sqrt{N}{X}_N$ tend to infinity with N, as for matrices with truncated heavy tailed entries or adjacency matrices of sparse Erdös–Rényi graphs. Consider a family ${\mathbf{X}}_N$ of independent heavy Wigner matrices and an independent family ${\mathbf{Y}}_N$ of arbitrary random matrices with a bound condition and converging in ^?-distribution in the sense of free probability. We characterize the possible limiting joint ^?-distributions of $({\mathbf{X}}_N,{\mathbf{Y}}_N)$, giving explicit formulas for joint ^?-moments. We find that they depend on more than the ^?-distribution of ${\mathbf{Y}}_N$ and that in general ${\mathbf{X}}_N$ and ${\mathbf{Y}}_N$ are not asymptotically ^?-free. We use the traffic distributions and the associated notion of independence [21] to encode the information on ${\mathbf{Y}}_N$ and describe the limiting ^?-distribution of $({\mathbf{X}}_N,{\mathbf{Y}}_N)$. We develop this approach for related models and give recurrence relations for the limiting ^?-distribution of heavy Wigner and independent diagonal matrices.     

Distributed subgradient method for multi‐agent optimization with quantized communication

This paper focuses on a distributed optimization problem associated with a time‐varying multi‐agent network with quantized communication, where each agent has local access to its convex objective function, and cooperatively minimizes a sum of convex objective functions of the agents over the network. Based on subgradient methods, we propose a distributed algorithm to solve this problem under the additional constraint that agents can only communicate quantized information through the network. We consider two kinds of quantizers and analyze the quantization effects on the convergence of the algorithm. Furthermore, we provide explicit error bounds on the convergence rates that highlight the dependence on the quantization levels. Finally, some simulation results on a l₁‐regression problem are presented to demonstrate the performance of the algorithm. Copyright © 2016 John Wiley & Sons, Ltd.