NONLINEAR STABILITY OF RAREFACTION WAVES FOR A RATE-TYPE VISCOELASTIC SYSTEM

51.IntroductionInthispaper,westudythefollowingrate-typeviscoelasticsystem,i.e.,wherevand(--p)denotestrainandstress,uisrelatedtotheparticlevelocity,EisapositiveconstantcalledthedynamicYOung'smodulus,T>0isarelaxationtime.Thissystemwasproposedinfi6]tointroducearelaxationapproximationtothefollowingsystemSincethesystem(1.2)canbeobtainedfrom(1.1)byanexpansionprocedureasthefirstorder,itisnaturaltoexpectthatthesolutionof(1.1)convergestothatof(1.2)asT-0.However,thezerolimitconvergencehasnotbeenestabl…     

ON MONOTONE CONVERGENCE OF NONLINEARMULTISPLITTING RELAXATION METHODS

ＯＮＭＯＮＯＴＯＮＥＣＯＮＶＥＲＧＥＮＣＥＯＦＮＯＮＬＩＮＥＡＲＭＵＬＴＩＳＰＬＩＴＴＩＮＧＲＥＬＡＸＡＴＩＯＮＭＥＴＨＯＤＳ￥ＷＡＮＧＤＥＲＥＮ；ＢＡＩＺＨＯＮＧＺＨＩ（ＤｅｐａｚａｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＳｈｓｌｌｇｈａｉＵｎｉｖｅ...     

Electroelastic waves in polarizing media : Report to the fifth all-union congress on theoretical and applied mechanics. Alma-Ata, 27 May 1981

A system of equations is considered that describes a certain class of polarizing media in an electromagnetic field with both the spatial inhomogeneities and the relaxation processes of electrical polarization of the media taken into account. The nonlinear and the linearized theories are elucidated within the framework of the electrostatic approximation. Solutions are given for the equations considered in an electrostatic approximation, in the form of electroelastic waves. Without taking the relaxation of the electrical polarization into account, electorelastic waves were considered in piezoceramic media in /1/ and in ferroelectrics in /2/.     

带有非紧条件的拟线性Schrodinger-Poisson系统非平凡解的存在性

本文研究了如下带有非紧条件的拟线性Schrodinger-Poisson系统{-△u+V(x)u+Фu+k/2u△u2=λ|u|^p-2u+f(u),x ∈R^3,-ΔФ=u^2,x∈R^3, 其中κ<0,λ>0,p≥12,f∈C(R,R),V∈C(R3,R).文中首先构造截断函数,利用集中紧性原理和逼近的方法,得到了截断后系统非平凡解的存在性;然后利用Moser迭代技巧,讨论上述系统非平凡解的存在性.     

THE NONLINEAR STABILITY OF TRAVELLING WAVE SOLUTIONS FOR A REACTING FLOW MODEL WITH SOURCE TERM

1IntroductionConsidertilefollowingsystemtlwiwasproposedin[91todescribereactinggasinwhichthereealsttwo"lodes.Where,pcistiledensityofthemajormodeandpsisOftheminormode,f s=1.uisthevelocity,andp=PC'(r as)ispressurewhichcanbederivedbyAvogadro'sLaw.Here,cisthesoundspeedofthemajormode.TheParameterPprovidessometenuouslinkwithrealphysics.Sisthesollrcetermwhere,risareactiontime,rE(p)alldSE(P)areequilibriumdistributions.Thereaderisrefere(lto[9]formorephysicsandnumericalbackgroulld.TheLagrangianform…     

The relaxation schemes for systems of conservation laws in arbitrary space dimensions

We present a class of numerical schemes (called the relaxation schemes) for systems of conservation laws in several space dimensions. The idea is to use a local relaxation approximation. We construct a linear hyperbolic system with a stiff lower order term that approximates the original system with a small dissipative correction. The new system can be solved by underresolved stable numerical discretizations without using either Riemann solvers spatially or a nonlinear system of algebraic equations solvers temporally. Numerical results for 1-D and 2-D problems are presented. The second-order schemes are shown to be total variation diminishing (TVD) in the zero relaxation limit for scalar equations. ©1995 John Wiley & Sons, Inc.     

Efficient Convex Optimization Approaches to Variational Image Fusion

Image fusion is an imaging technique to visualize information from multiple imaging sources in one single image, which is widely used in remote sensing, medical imaging etc. In this work, we study two variational approaches to image fusion which are closely related to the standard TV-$L_2$ and TV-$L_1$ image approximation methods. We investigate their convex optimization formulations, under the perspective of primal and dual, and propose their associated new image decomposition models. In addition, we consider the TV-$L_1$ based image fusion approach and study the specified problem of fusing two discrete-constrained images $f_1(x) ∈\mathcal{L}_1$ and $f_2(x) ∈\mathcal{L}_2$, where$\mathcal{L}_1$ and$\mathcal{L}_2$ are the sets of linearly-ordered discrete values. We prove that the TV-$L_1$ based image fusion actually gives rise to the exact convex relaxation to the corresponding nonconvex image fusion constrained by the discrete-valued set $u(x) ∈\mathcal{L}_1 ∪\mathcal{L}_2$. This extends the results for the global optimization of the discrete-constrained TV-$L_1$ image approximation [8, 36] to the case of image fusion. As a big numerical advantage of the two proposed dual models, we show both of them directly lead to new fast and reliable algorithms, based on modern convex optimization techniques. Experiments with medical images, remote sensing images and multi-focus images visibly show the qualitative differences between the two studied variational models of image fusion. We also apply the new variational approaches to fusing 3D medical images.     

Relaxation schemes for curvature‐dependent front propagation

In this paper we study analytically and numerically a novel relaxation approximation for front evolution according to a curvature‐dependent local law. In the Chapman‐Enskog expansion, this relaxation approximation leads to the level‐set equation for transport‐dominated front propagation, which includes the mean curvature as the next‐order term. This approach yields a new and possibly attractive way of calculating numerically the propagation of curvature‐dependent fronts. Since the relaxation system is a symmetrizable, semilinear, and linearly convective hyperbolic system without singularities, the relaxation scheme captures the curvature‐dependent front propagation without discretizing directly the complicated yet singular mean curvature term. © 1999 John Wiley & Sons, Inc.     

求解非定常对流扩散方程的流函数松弛方法

提出了一种求解线性和非线性对流扩散方程的流函数松弛方法.方法的主要思想是利用流函数松弛近似将原始的方程转化成等价的松弛方程组,新的松弛方程组是带源项的双曲系统.通过稳定性分析可以知道新系统的耗散系数可由松弛系数调整.数值实现亦证明这个方法可以快速有效地描述对流扩散方程的解.     

A system of conservation laws including a stiff relaxation term; the 2D case

We analyze a system of conservation laws in two space dimensions with a stiff relaxation term. A semi-implicit finite difference method approximating the system is studied and an error bound of order measured inL ¹ is derived. This error bound is independent of the relaxation time > 0. Furthermore, it is proved that the solutions of the system converge towards the solution of an equilibrium model as the relaxation time tends to zero, and that the rate of convergence measured inL ¹ is of order . Finally, we present some numerical illustrations.This research has been supported by the Norwegian Research Council (NFR), program no. STP 110673/420, at the Department of Applied Mathematics, SINTEF, Oslo, Norway     

Approximation of Dense-n/2-Subgraph and the Complement of Min-Bisection

We consider the DENSE-n/2-SUBGRAPH problem, i.e., determine a block of half number nodes from a weighted graph such that the sum of the edge weights, within the subgraph induced by the block, is maximized. We prove that a strengthened semidefinite relaxation with a mixed rounding technique yields a 0.586 approximations of the problem. The previous best-known results for approximating this problem are 0.25 using a simple coin-toss randomization, 0.48 using a semidefinite relaxation, 0.5 using a linear programming relaxation or another semidefinite relaxation. In fact, an un-strengthened SDP relaxation provably yields no more than 0.5 approximation. We also consider the complement of the graph MIN-BISECTION problem, i.e., partitioning the nodes into two blocks of equal cardinality so as to maximize the weights of non-crossing edges. We present a 0.602 approximation of the complement of MIN-BISECTION.     

Improved approximation of Max-Cut on graphs of bounded degree

Let α0.87856 denote the best approximation ratio currently known for the Max-Cut problem on general graphs. We consider a semidefinite relaxation of the Max-Cut problem, round it using the random hyperplane rounding technique of Goemans and Williamson [J. ACM 42 (1995) 1115–1145], and then add a local improvement step. We show that for graphs of degree at most Δ, our algorithm achieves an approximation ratio of at least α+ε, where ε>0 is a constant that depends only on Δ. Using computer assisted analysis, we show that for graphs of maximal degree 3 our algorithm obtains an approximation ratio of at least 0.921, and for 3-regular graphs the approximation ratio is at least 0.924. We note that for the semidefinite relaxation of Max-Cut used by Goemans and Williamson the integrality gap is at least 1/0.885, even for 2-regular graphs.     

Continuity Corrections for Discrete Distributions Under the Edgeworth Expansion

The approximation of discrete distributions by Edgeworth expansion series for continuity points of a discrete distribution F _n implies that if t is a support point of F _n, then the expansion should be performed at a continuity point . When a value is selected to improve the approximation of , and especially when a single term of the expansion is used, the selected is defined to be a continuity correction. This paper investigates the properties of the approximations based on several terms of the expansion, when is the value at which the infimum of a residual term is attained. Methods of selecting the estimation and the residual terms are investigated and the results are compared empirically for several discrete distributions. The results are also compared with the commonly used approximation based on the normal distribution with . Some numerical comparisons show that the developed procedure gives better approximations than those obtained under the standard continuity correction technique, whenever is close to 0 and 1. Thus, it is especially useful for p-value computations and for the evaluation of probabilities of rare events.     

Global solutions of nonconcave hyperbolic conservation laws with relaxation arising from traffic flow

We establish global solutions of nonconcave hyperbolic equations with relaxation arising from traffic flow. One of the characteristic fields of the system is neither linearly degenerate nor genuinely nonlinear. Furthermore, there is no dissipative mechanism in the relaxation system. Characteristics travel no faster than traffic. The global existence and uniqueness of the solution to the Cauchy problem are established by means of a finite difference approximation. To deal with the nonconcavity, we use a modified argument of Oleinik (Amer. Math. Soc. Translations 26 (1963) 95). It is also shown that the zero relaxation limit of the solutions exists and is the unique entropy solution of the equilibrium equation.     

A relaxation approximation for transport equations in the diffusive limit

A stable relaxation approximation for a transport equation with the diffusive scaling is developed. The relaxation approximation leads in the small mean free path limit to the higher-order diffusion equation obtained from the asymptotic analysis of the transport equation.     

Convergence of the Equations for a Maxwell Fluid

The equations for the flow of a viscoelastic fluid of the Maxwell type are analyzed in a linear approximation. First, we establish that the solution depends continuously on changes in the relaxation time. Next, we investigate how the solution to the linearized Maxwell system converges to the solution to Stokes flow as the relaxation time tends to zero. Convergence in different measures is examined and specific a priori bounds are derived.     

Improved Approximation Algorithms for MAX \frac{n}{{\text{2}}}-DIRECTED-BISECTION and MAX \frac{n}{{\text{2}}}-DENSE-SUBGRAPH

We consider the MAX $\frac{n}{{\text{2}}}$ -DIRECTED-BISECTION problem, i.e., partitioning the vertices of a directed graph into two blocks of equal cardinality so as to maximize the total weight of the edges in the directed cut. A polynomial approximation algorithm using a semidefinite relaxation with 0.6458 performance guarantee is presented for the problem. The previous best-known results for approximating this problem are 0.5 using a linear programming relaxation, 0.6440 using a semidefinite relaxation. We also consider the MAX $\frac{n}{{\text{2}}}$ -DENSE-SUBGRAPH problem, i.e., determine a block of half the number of vertices from a weighted undirected graph such that the sum of the edge weights, within the subgraph induced by the block, is maximized. We present an 0.6236 approximation of the problem as opposed to 0.6221 of Halperin and Zwick.     

Approximation Bounds for Trilinear and Biquadratic Optimization Problems Over Nonconvex Constraints

This paper presents new approximation bounds for trilinear and biquadratic optimization problems over nonconvex constraints. We first consider the partial semidefinite relaxation of the original problem, and show that there is a bounded approximation solution to it. This will be achieved by determining the diameters of certain convex bodies. We then show that there is also a bounded approximation solution to the original problem via extracting the approximation solution of its semidefinite relaxation. Under some conditions, the approximation bounds obtained in this paper improve those in the literature.     

Convex approximations for complete integer recourse models

We consider convex approximations of the expected value function of a two-stage integer recourse problem. The convex approximations are obtained by perturbing the distribution of the random right-hand side vector. It is shown that the approximation is optimal for the class of problems with totally unimodular recourse matrices. For problems not in this class, the result is a convex lower bound that is strictly better than the one obtained from the LP relaxation.This research has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences.Key words.integer recourse – convex approximationMathematics Subject Classification (1991):90C15, 90C11     

An improved upper bound for the TSP in cubic 3-edge-connected graphs

We consider the travelling salesman problem (TSP) problem on (the metric completion of) 3-edge-connected cubic graphs. These graphs are interesting because of the connection between their optimal solutions and the subtour elimination LP relaxation. Our main result is an approximation algorithm better than the 3/2-approximation algorithm for TSP in general.