Logarithmic asymptotics of solutions of the large deviation problem for the boltzmann equation with small transfer of momentum

For the Boltzmann equation with small transfer of momentum, we derive a system of nonlinear integral-differential equations describing the logarithmic asymptotics of the solution to the Cauchy problem in the domain at the distanceO(1) from the support of the initial condition.Translated fromMatematicheskie Zametki, Vol. 64, No. 1, pp. 73–94, July, 1998.This research was partially supported by INTAS-RFBR under grant No. 95-91 in the case of the first author, and by INTAS under grant No. 96-0698 in the case of the second author.     

Uniform grid approximation of nonsmooth solutions to the mixed boundary value problem for a singularly perturbed reaction-diffusion equation in a rectangle

A mixed boundary value problem for a singularly perturbed reaction-diffusion equation in a square is considered. A Neumann condition is specified on one side of the square, and a Dirichlet condition is set on the other three. It is assumed that the coefficient of the equation, its right-hand side, and the boundary values of the desired solution or its normal derivative on the sides of the square are smooth enough to ensure the required smoothness of the solution in a closed domain outside the neighborhoods of the corner points. No compatibility conditions are assumed to hold at the corner points. Under these assumptions, the desired solution in the entire closed domain is of limited smoothness: it belongs only to the Hölder class C ^μ, where μ ∈ (0, 1) is arbitrary. In the domain, a nonuniform rectangular mesh is introduced that is refined in the boundary domain and depends on a small parameter. The numerical solution to the problem is based on the classical five-point approximation of the equation and a four-point approximation of the Neumann boundary condition. A mesh refinement rule is described under which the approximate solution converges to the exact one uniformly with respect to the small parameter in the L _∞ ^h norm. The convergence rate is O(N ^?2ln² N), where N is the number of mesh nodes in each coordinate direction. The parameter-uniform convergence of difference schemes for mixed problems without compatibility conditions at corner points was not previously analyzed.     

一种神经网络算子及其逼近阶估计

讨论了一种神经网络算子f_n(x)=sum from -n~2 to n~2 (f(k/n))/(n~α)b(n~(1-α)(x-k/n)),对f(x)的逼近误差|f_n(x)-f(x)|的上界在f(x)为连续和N阶连续可导两种情形下分别给出了该网络算子逼近的Jackson型估计.     

求解二维多区域声波散射问题的边界元方法

对于多散射区域的声波散射问题的外Neumann边值问题,用单层位势来逼近每个散射域上的散射波,再利用位势理论的跳跃关系将问题转换为第二类边界积分方程组的求解问题,然后用Nystrom方法进行了求解.对多个随机散射区域的声波散射问题,数值例子体现了该求解方法的可行性和准确性.     

THE NECESSARY AND SUFFICIENT CONDITIONS FOR THE SOLVABILITY OF A CLASS OF THE MATRIX INVERSE PROBLEM

Censider the solutions of the matrix inverse problem, which are symmetric positive semide finite on a subspace. Necessary and sufficient conditions for the solvability, as well as the general solution are obtained. The best approximate solution by the above solution set is given. Thus the open problem in [1] is solved.     

A note on approximate Bayesian credible sets based on modified loglikelihood ratios

Higher-order asymptotic arguments for a scalar parameter of interest have been widely investigated for Bayesian inference. In this paper the theory of asymptotic expansions is discussed for a vector parameter of interest. A modified loglikelihood ratio is suggested, which can be used to derive approximate Bayesian credible sets with accurate frequentist coverage. Three examples are illustrated.     

Due-window assignment with identical jobs on parallel uniform machines

A scheduling problem with a common due-window, earliness and tardiness costs, and identical processing time jobs is studied. We focus on the setting of both (i) job-dependent earliness/tardiness job weights and (ii) parallel uniform machines. The objective is to find the job allocation to the machines and the job schedule, such that the total weighted earliness and tardiness cost is minimized. We study both cases of a non-restrictive (i.e. sufficiently late), and a restrictive due-window. For a given number of machines, the solutions of the problems studied here are obtained in polynomial time in the number of jobs.     

奇异摄动问题内罚间断有限方法的最优阶一致收敛性分析

本文在Bakhvalov-Shishkin网格上分析了采用高次元的内罚间断有限元方法求解一维对流扩散型奇异摄动问题的最优阶一致收敛性. 取k(k≥1)次分片多项式和网格剖分单元数为N时,在能量范数度量下, Bakhvalov-Shishkin网格上可获得O(N^-k)的一致误差估计. 在数值算例部分对理论分析结果进行了验证.     

Robin型离散Schwarz波形松弛算法的收敛性分析

Schwarz波形松弛(Schwarz waveform relaxation,SWR)是一种新型区域分解算法,是当今并行计算研究领域的焦点之一,但针对该算法的收敛性分析基本上都停留在时空连续层面.从实际计算角度看,分析离散SWR算法的收敛性更重要.本文考虑SWR研究领域中非常流行的Robin型人工边界条件,分析时空离散参数t和x、模型参数等因素对算法收敛速度的影响.Robin型人工边界条件中含有一个自由参数p,可以用来优化算法的收敛速度,但最优参数的选取却需要求解一个非常复杂的极小-极大问题.本文对该极小-极大问题进行深入分析,给出最优参数的计算方法.本文给出的数值实验结果表明所获最优参数具有以下优点:(1)相比连续情形下所获最优参数,利用离散情形下获得的参数可以进一步提高Robin型SWR算法在实际计算中的收敛速度,当固定t或x而令另一个趋于零时,利用离散情形下所获参数可以使算法的收敛速度具有鲁棒性(即收敛速度不随离散参数的减小而持续变慢).(2)相比连续情形下所获收敛速度估计,离散情形下获得的收敛速度估计可以更加准确地预测算法的实际收敛速度.     

参数弱向量平衡问题解集的似Holder性和相依导数

本文研究了一类参数弱向量平衡问题解集的似Hölder性和相依导数. 首先, 讨论了该问题的一类实值间隙函数的Lipschitz连续性和Hadamard方向可微性. 然后, 借助这些性质, 建立了该问题解集的似Hölder性和Hölder连续性以及相依导数的具体表达式.