Sparsity in sums of squares of polynomials

Representation of a given nonnegative multivariate polynomial in terms of a sum of squares of polynomials has become an essential subject in recent developments of sums of squares optimization and semidefinite programming (SDP) relaxation of polynomial optimization problems. We discuss effective methods to obtain a simpler representation of a sparse polynomial as a sum of squares of sparse polynomials by eliminating redundancy.A considerable part of this work was conducted while this author was visiting Tokyo Institute of Technology. Research supported by Kosef R004-000-2001-00200Mathematics Subject Classification (1991): 90C22, 90C26, 90C30     

Polynomial approximation of analytic functions on a finite number of continua in the complex plane

The Dzjadyk-type theorem concerning the polynomial approximation of functions on a continuum in the complex plane is generalized to the case of polynomial approximation of functions on a compact set in which consists of a finite number of continua.     

基于多项式模型法的低端数码相机的颜色特性化

近年来随着数字技术,以及彩色数字图像输入输出设备的不断发展和广泛使用,颜色在不同的设备之间精确的传递或再现成为该领域的重要课题。本论文以彩色数码相机特性化为研究内容,采用多项式转换方法,研究了多项式回归模型不同系数矩阵、不同训练样本对数码相机特性化精度的影响,得出了系数矩阵为11×3多项式模型时误差较小(2.874ΔE*ab)的结论,达到了实用水平。     

A construction of real weight functions for certain orthogonal polynomials in two variables

H.L. Krall and I.M. Sheffer considered the problem of classifying certain second-order partial differential equations having an algebraically complete, weak orthogonal bivariate polynomial system of solutions. Two of the equations that they considered are

(x²+y)uxx+2xyuxy+y²uyy+gxux+g(y−1)uy=λu,     

Uncertainty quantification via random domain decomposition and probabilistic collocation on sparse grids

Quantitative predictions of the behavior of many deterministic systems are uncertain due to ubiquitous heterogeneity and insufficient characterization by data. We present a computational approach to quantify predictive uncertainty in complex phenomena, which is modeled by (partial) differential equations with uncertain parameters exhibiting multi-scale variability. The approach is motivated by flow in random composites whose internal architecture (spatial arrangement of constitutive materials) and spatial variability of properties of each material are both uncertain. The proposed two-scale framework combines a random domain decomposition (RDD) and a probabilistic collocation method (PCM) on sparse grids to quantify these two sources of uncertainty, respectively. The use of sparse grid points significantly reduces the overall computational cost, especially for random processes with small correlation lengths. A series of one-, two-, and three-dimensional computational examples demonstrate that the combined RDD–PCM approach yields efficient, robust and non-intrusive approximations for the statistics of diffusion in random composites.     

SUPERCONVERGENCE ANALYSIS OF THE POLYNOMIAL PRESERVING RECOVERY FOR ELLIPTIC PROBLEMS WITH ROBIN BOUNDARY CONDITIONS

We analyze the superconvergence property of the linear finite element method based on the polynomial preserving recovery(PPR)for Robin boundary elliptic problems on triangulartions.First,we improve the convergence rate between the finite element solution and the linear interpolation under the H1-norm by introducing a class of meshes satisfying the Condition(α,σ,μ).Then we prove the superconvergence of the recovered gradients post-processed by PPR and define an asymptotically exact a posteriori error estimator.Finally,numerical tests are provided to verify the theoretical findings.     

关于最小二乘QR分解算法(LSQR)的一个注记

本文从最小多项式出发,通过寻找包含奇异线性系统Ax=b最小范数解的一个解空间,获得了一个更简单的求解广义逆的计算公式.并从理论上对最小二乘QR分解算法（LSQR）收敛性进行了简单分析,分析表明LSQR的收敛性与矩阵A的非零奇异值密切相关,并用A的非零奇异值以及所寻找到的最小范数解空间将最小范数解线性表出.     

Intersection Poincaré Polynomial for Nagaraj–Seshadri moduli space

We compute Betti numbers of both the components of the moduli space of rank 2 semi‐stable torsion‐free sheaves with fixed determinant over a reducible nodal curve with two smooth components intersecting at a node. We also compute the intersection Betti numbers of the moduli space.