Strong tractability of multivariate integration using quasi-Monte Carlo algorithms

We study quasi-Monte Carlo algorithms based on low discrepancy sequences for multivariate integration. We consider the problem of how the minimal number of function evaluations needed to reduce the worst-case error from its initial error by a factor of depends on and the dimension . Strong tractability means that it does not depend on and is bounded by a polynomial in . The least possible value of the power of is called the -exponent of strong tractability. Sloan and Wozniakowski established a necessary and sufficient condition of strong tractability in weighted Sobolev spaces, and showed that the -exponent of strong tractability is between 1 and 2. However, their proof is not constructive.

In this paper we prove in a constructive way that multivariate integration in some weighted Sobolev spaces is strongly tractable with -exponent equal to 1, which is the best possible value under a stronger assumption than Sloan and Wozniakowski's assumption. We show that quasi-Monte Carlo algorithms using Niederreiter's -sequences and Sobol sequences achieve the optimal convergence order for any 0$"> independent of the dimension with a worst case deterministic guarantee (where is the number of function evaluations). This implies that strong tractability with the best -exponent can be achieved in appropriate weighted Sobolev spaces by using Niederreiter's -sequences and Sobol sequences.

     

Asymptotic Behavior of the Density in a Parabolic SPDE

Consider the density of the solution X(t, x) of a stochastic heat equation with small noise at a fixed t[0, T], x[0, 1]. In this paper we study the asymptotics of this density as the noise vanishes. A kind of Taylor expansion in powers of the noise parameter is obtained. The coefficients and the residue of the expansion are explicitly calculated. In order to obtain this result some type of exponential estimates of tail probabilities of the difference between the approximating process and the limit one is proved. Also a suitable iterative local integration by parts formula is developed.     

Remarks on Herz-Type Hardy Spaces

Basic properties of the Herz-type Hardy spaces , such as the boundedness of singular integral operators and the fractional integration operators, atomic decomposition, dense subspaces, etc., are established in the full range 0 < q < ∞. Partly supported by the Grants-in-Aid for Scientific Research (A)(1)11304009, (B)(1)10440046, Japan Society for the Promotion of Science.     

Effect of the stagnation temperature on supersonic flow parameters: Application for air in nozzles

When the stagnation temperature of a perfect gas increases, the specific heats and their ratio do not remain constant any more and start to vary with this temperature. The gas remains perfect and its state equation always remains valid, except it will be called a calorically imperfect gas. The aim of this research is to develop the relations of the necessary thermodynamics and geometrical ratios and to study the supersonic flow at high temperature (lower than the threshold of dissociation). The results are found by the resolution of nonlinear algebraic equations and integration of complex analytical functions where the exact calculation is impossible. The dichotomy method is used to solve the nonlinear equation, and the Simpson algorithm is used for the numerical integration of the found integrals. A condensation of the nodes is used, since the functions to be integrated have a high gradient at the extremity of the interval of integration. A comparison is made with a calorifcally perfect gas to determine the error of this. An application is made for air in a supersonic nozzle. The text was submitted by the authors in English.     

On the integrated behaviour of non-stationary volatility in stock markets

This paper analyses the behaviour of volatility for several international stock market indexes, namely the SP 500 (USA), the Nikkei (Japan), the PSI 20 (Portugal), the CAC 40 (France), the DAX 30 (Germany), the FTSE 100 (UK), the IBEX 35 (Spain) and the MIB 30 (Italy), in the context of non-stationarity. Our empirical results point to the evidence of the existence of integrated behaviour among several of those stock market indexes of different dimensions. It seems, therefore, that the behaviour of these markets tends to some uniformity, which can be interpreted as the existence of a similar behaviour facing to shocks that may affect the worldwide economy. Whether this is a cause or a consequence of market globalization is an issue that may be stressed in future work.     

低能N+注入植物种子射程分布的模拟计算

伴随着离子束生物技术的广泛应用, 国内许多单位开展了低能离子注入植物种子的实验研究. 其中关于低能离子注入植物种子诱变的物理机理, 集中在离子注入的深度-浓度分布上. 一些单位直接使用纵向非静态(LSS)理论和TRIM程序来计算低能离子注入植物种子的深度-浓度分布, 却发现计算结果与实验测量结果相差甚远. 所以在对植物种子靶材料进行处理和对LSS理论进行修正的基础上, 在二维近似情况下, 用蒙特卡罗方法分别模拟计算了200keV V⁺和20keV Ti⁺注入花生和棉花种子的射程分布, 得到了与实验结果较符合的曲线. 在此模型基础上, 计算了同样初始条件和理论计算模型下无法从实验上测量的N⁺注入植物种子的射程分布, 初步地为低能N⁺注入植物种子射程分布提供了一种理论计算方法.     

普物实验教学应科学引导学生进行知能整合

以“电池电动势和内阻的测定”为例,简述了在普物实验教学中,科学引导学生进行知能整合的过程,使科学知识与相关知识的应用实现了最佳的整合,提高了学生设计实验、实验测量、科学评定实验过程及测量结果的能力。     

Temporal reflectance from a light pulse irradiated medium embedded with highly scattering cores

This paper presents a new approach to utilize ultrashort pulsed laser for optical diagnostics with numerical simulations. The method is based on the use of ultrafast pulses with a pulsewidth selected according to the probed medium's radiative property and/or size. Our previous work in nonhomogeneous media has shown that the resulting time-resolved reflectance signal will have a unique characteristic: it will show a direct correlation of ballistic photon travel time and interface location, which is in between different layers or nonhomogeneous regions. The premise is based on utilizing the medium's structural information carried by the ballistic and snake photons without being masked by the diffuse photons. In this study, the space-time correlation is further explored in the case of minimally scattered photons from a large scattering coefficient core region embedded within a less-scattering medium. Time-resolved reflectance signals of the single scattering core and multiple scattering cores within a three-dimensional medium demonstrate the concept and illustrate the additional effect due to the scattered photons from the core region. A unique temporal signal profile's correlation at various detector positions with respect to the scattering core is explained in detail. The result has important implications. This approach will lead to a much simpler and more precise determination of the probed medium's composition or structure. Due to the large computational requirement to obtain the physical details of the light pulse propagation inside highly scattering multi-dimensional media, the reverse Monte-Carlo method is used. The potential applications of the method include non-destructive diagnostics, optical imaging, and remote sensing of underwater objects.     

Extending the applicability of the discrete dipole approximation for multi-scale features on surface

Many problems in science and engineering involve significant physical entities and processes that span a substantial range of dimensions. In the case of characterization of bacteria on growth media using light scattering the length scales of interest can be classified as micro-scale (single bacterium), macro-scale (bacterial colonies of more than 10¹² bacterium that have passed through the exponential growth phase and reached mm size), and the intermediate or meso-scale of several tens of hundreds of bacteria. Light scattering approaches, to be effective in determining physical properties such as morphology and material composition, must comprehend this spectrum of length scales. The discrete dipole approximation (DDA), a powerful modeling tool for rigorous 3-D vector scattering, has shown its capability to predict the light scattering from micro-scale objects. To be able to accommodate meso-scale objects, we need to extend the computational limits of the DDA method such that it could compute object sizes of 10λ-30λ characteristic dimension (i.e. volumes of 10³-10⁴ cubic wavelengths). To accomplish this, an analysis of the DDA method was performed for meso-scale cases of interest especially in biological applications. Based on this study, we propose new Sommerfeld integration paths and a revised iterative algorithm that combine to provide substantial improvements in the size of the computational domain that can be modeled for a given convergence criterion.     

抽取信号的χ2分布拟合方法

BESⅡ上测量ψ(2S)辐射衰变时, 鉴于本底道的污染非常严重, 提出了一种抽取信号事例数的χ²拟合方法. 大量的蒙特卡罗模拟表明, 信号道与各个本底道的χ²分布形状各不相同. 这些特点提供了从数据的χ²分布中抽出信号道事例数的χ²拟合方法. 从蒙特卡罗模拟的结果来看, χ²拟合方法的误差小于2%.