Implementation and investigation of iterative solvers in the Discrete Sources Method

The implementation of iterative methods as solvers for the Discrete Sources Method (DSM) is presented. In this method, light scattering computation linear systems with dense and relative small matrices are generated. The linear systems are traditionally solved using the QR-decomposition method. For large particles or particles with extreme geometries even this commonly stable solver can fail. In these cases, we expect that iterative methods can provide a satisfying solution nevertheless.We will present our investigation in two consecutive papers. Here, we study four different iterative solvers (RGMRES, BiCGStab, BiCGStab(l), and MinRes) considering the performance and the accuracy for typical light scattering problems. Using these iterative methods we increased the quality of a solution, especially for oblate spheroids with a higher aspect ratio. Preconditioning technique is considered in the following paper.     

Lie-Mei symmetry and conserved quantities of the Rosenberg problem

The Rosenberg problem is a typical but not too complicated problem of nonholonomic mechanical systems.The Lie-Mei symmetry and the conserved quantities of the Rosenberg problem are studied.For the Rosenberg problem,the Lie and the Mei symmetries for the equation are obtained,the conserved quantities are deduced from them and then the definition and the criterion for the Lie-Mei symmetry of the Rosenberg problem are derived.Finally,the Hojman conserved quantity and the Mei conserved quantity are deduced from the Lie-Mei symmetry.     

资料同化中的数字滤波弱约束试验及分析

气象数值预报中,由于分析过程引入初始非平衡,从而引起虚假快波振荡,重力波控制弱约束把资料分析过程和初始化过程结合在一起,通过数字滤波弱约束在极小化过程中实现对分析场的平衡约束,克服非平衡问题. 以2008年初的一次南方雨雪天气为研究个例,进行了数字滤波弱约束的同化试验和预报试验,结果表明,数字滤波弱约束4D-Var能充分控制快波振荡的出现和初始调整现象,使得到的分析场不仅能更好的逼近观测,而且能更好地与模式动力相协调. 预报检验的结果表明,在同化过程中施加数字滤波弱约束,能有效滤除由于地形或观测资料等因素 关键词： 变分同化 初始非平衡 数字滤波 弱约束     

Defocus and binocular vision based stereo particle pairing method for 3D particle tracking velocimetry

We propose a new method, based on the depth-from-defocus technique and binocular vision, for solving the stereo particle pairing problem in 3D particle tracking velocimetry (PTV). Firstly, the apparent particle depth is measured with a single camera, using the depth-from-defocus technique. Secondly, a strict mathematical model of the particle-to-particle correspondence relationship between the left and right images, taking into account the refractions at the interfaces in the optical path, is presented, with the assumption that the apparent particle depth is measured. Thirdly, based on the apparent particle depth and particle-to-particle correspondence relationship, the epipoplar line is truncated into a short line segment by cutting off, where the apparent particle depth extends beyond its estimated range. For the first time, the range of the blur circle radius is employed as an additional stereo particle pairing constraint. Finally, the optimal pairing particle is selected by applying the epipolar line segment and blur circle radius constraints. The experimental results show that the rate of correct pairing is significantly improved compared with the epipolar line nearest neighbor analysis, especially when the particle density is increasing.     

The solution of the global relation for the derivative nonlinear Schrödinger equation on the half-line

We consider initial-boundary value problems for the derivative nonlinear Schrödinger (DNLS) equation on the half-line x>0. In a previous work, we showed that the solution q(x,t) can be expressed in terms of the solution of a Riemann-Hilbert problem with jump condition specified by the initial and boundary values of q(x,t). However, for a well-posed problem, only part of the boundary values can be prescribed; the remaining boundary data cannot be independently specified, but are determined by the so-called global relation. In general, an effective solution of the problem therefore requires solving the global relation. Here, we present the solution of the global relation in terms of the solution of a system of nonlinear integral equations. This also provides a construction of the Dirichlet-to-Neumann map for the DNLS equation on the half-line.     

Positive solution of singularly perturbed Dirichlet problem with singularities

A class of singularly perturbed boundary value problem with singularities is considered. Introducing the stretched variables, the boundary layer corrective terms near x = 0 and x = 1 are constructed. Under suitable conditions, by using the theory of differential inequalities the existence and asymptotic behavior of solution for boundary value problem are proved, uniformly valid asymptotic expansion of solution with boundary layers are obtained,     

A Remark on the Cauchy Problem of 1D Compressible Navier-Stokes Equations with Density-dependent Viscosity Coefcients

This paper is concerned with the Cauchy problems of one-dimensional compressible Navier-Stokes equations with density-dependent viscosity coefcients.By assumingρ0∈L1(R),we will prove the existence of weak solutions to the Cauchy problems forθ0.This will improve results in Jiu and Xin’s paper(Kinet.Relat.Models,1(2):313–330(2008))in whichθ12is required.In addition,We will study the large time asymptotic behavior of such weak solutions.     

Spatial‐skin effect for eigenvibrations of a thick cascade junction with ‘heavy’ concentrated masses

A spectral problem for the Laplace operator in a thick cascade junction with concentrated masses is considered. This cascade junction consists of the junction's body and a great number of ?‐alternating thin rods belonging to two classes. One class consists of rods of finite length, and the second one consists of rods of small length of order . The density of the junction is of order on the rods from the second class and outside of them. The asymptotic behavior of eigenvalues and eigenfunctions of this problem is studied as ? → 0. There exist five qualitatively different cases in the asymptotic behavior of eigenmagnitudes as ? → 0, namely the case of ‘light’ concentrated (α ∈ (0,1)), ‘middle’ concentrated (α = 1), and ‘heavy’ concentrated masses (α ∈ (1, + ∞ )) that we divide into ‘slightly heavy’ concentrated (α ∈ (1,2)), ‘intermediate heavy’ concentrated (α = 2), and ‘very heavy’ concentrated masses (α > 2). In the paper, we study in detail the influence of the concentrated masses on the asymptotic behavior if α ∈ (1,2). We construct the leading terms of asymptotic expansions both for the eigenvalues and eigenfunctions and prove the corresponding asymptotic estimates. Copyright © 2013 John Wiley & Sons, Ltd.