多分散Au-Ag合金纳米球的光谱消光法反演

贵金属纳米颗粒具有局域表面等离子体共振特性而引起了广泛的关注,其中Au-Ag合金纳米颗粒具有良好的结构稳定性、光热性能以及潜在的抗癌功效而得到普遍研究。在众多应用中的特性与其粒径和浓度密切相关,然而目前常用的电子显微镜观察法和动态光散射法不能同时获得粒径和浓度信息,因此采取有效手段测量颗粒粒径和浓度信息至关重要。基于光谱消光法,利用非负的Tikhonov正则化方法解决反演问题,并根据Mie理论计算消光矩阵。针对噪声问题,采取两种情况研究多分散Au-Ag合金纳米球粒径分布与浓度的反演问题。未添加噪声情况下,颗粒系Ⅰ的反演相对误差小于颗粒系Ⅱ,在波长范围300～500 nm之间的反演相对误差最小,对应平均粒径、粒径标准差和颗粒数浓度的反演相对误差分别为0%,-0.03%和0%。添加随机噪声情况下,将0.5%和1.0%的随机噪声添加进颗粒系Ⅰ中的消光谱,经过数据比较发现在波长范围200～600 nm之间的反演相对误差最小。当添加0.5%的随机噪声时,粒径分布、粒径标准差和颗粒数浓度的变化范围分别为79.76～80.15 nm, 5.60～6.61 nm和0.995 8×1010～1.005 9×1010个·cm-3;当添加1.0%的随机噪声时,粒径分布、粒径标准差和颗粒数浓度的变化范围分别为78.87～80.27 nm, 5.36～9.00 nm和0.992 4×1010～1.027 7×1010个·cm-3。反演结果随着随机噪声的增大,变化范围也明显增大即反演相对误差增大,并且每次添加相同随机噪声后的反演结果不同。为了减少随机噪声导致的不稳定性,对100次反演结果进行平均得到平均粒径、粒径标准差和颗粒数浓度。当随机噪声从0.5%增大至1.0%时,其反演结果的相对误差均增大,但是反演得到的粒径分布、粒径标准差和颗粒数浓度相对误差均小于6%,这说明通过反演算法得到的反演结果具有较好的稳定性。研究表明,光谱消光法为反演多分散Au-Ag合金纳米球粒径分布与浓度提供了一种简单、快速的表征手段,也对研究非球形纳米颗粒有启示作用。     

土中爆炸成腔问题中确定炸点状态的一种计算反求方法

提出了一种通过给定的土中爆炸成腔毁伤效应确定炸点状态的计算反求方法。该方法将确定炸点状态的反问题转化为求解爆炸毁伤效应的计算值与给定值误差函数最小的优化问题。在反求过程中,采用基于误差减小比率技术的多项式近似模型代替土中爆炸数值分析模型,以便提高反求效率。采用Tikhonov正则化方法克服反求过程中出现的病态问题。在此基础上,引入信赖域管理策略判断当前近似模型与实际模型的逼近程度,以确定最优的反求向量。炸点状态反求结果与实验结果的对比分析表明,该方法能够有效且稳定地通过给定的毁伤效应实现炸点状态的反求,这可为炸点状态的设计提供参考。     

Kustaanheimo–Stiefel变量与二体问题的正规化

本文叙述Kustaanheimo–Stiefel变量及其从四元数发展形成的过程,以及Kustaanheimo–Stiefel变换在二体问题正规化中的应用。

     

Regularization of a two-dimensional strongly damped wave equation with statistical discrete data

In this paper, we consider an inverse problem for a strongly damped wave equation in two dimensional with statistical discrete data. Firstly, we give a representation for the solution and then present a discretization form of the Fourier coefficients. Secondly, we show that the solution does not depend continuously on the data by stating a concrete example, which makes the solution be not stable and thus the present problem is ill-posed in the sense of Hadamard. Next, we use the trigonometric least squares method associated with the Fourier truncation method to regularize the instable solution of the problem. Finally, the convergence rate of the error between the regularized solution and the sought solution is estimated and also investigated numerically.     

Approximation of Attainable Sets of an Evolution Inclusion of Subdifferential Type

In a separable Hilbert space we consider an evolution inclusion with a multivalued perturbation and evolution operators that are subdifferentials of a proper convex lower semicontinuous function depending on time. Along with the original inclusion, we consider a sequence of approximating evolution inclusions with the same perturbation and the evolution operators that are subdifferentials of the Moreau–Yosida regularizations of the original function. We show that the attainable set of the original inclusion, regarded as a multivalued function of time, is the uniform (in time) limit in the Hausdorff metric of the sequence of attainable sets of the approximating inclusions. As an application we consider an example of a control system with discontinuous nonlinearity.     

Multilevel regularization of wavelet based fitting of scattered data – some experiments

In [6], an adaptive method to approximate unorganized clouds of points by smooth surfaces based on wavelets has been described. The general fitting algorithm operates on a coarse-to-fine basis. It selects on each refinement level in a first step a reduced number of wavelets which are appropriate to represent the features of the data set. In a second step, the fitting surface is constructed as the linear combination of the wavelets which minimizes the distance to the data in a least squares sense. This is followed by a thresholding procedure on the wavelet coefficients to discard those which are too small to contribute much to the surface representation. In this paper, we firstly generalize this strategy to a classically regularized least squares functional by adding a Sobolev norm, taking advantage of the capability of wavelets to characterize Sobolev spaces of even fractional order. After recalling the usual cross-validation technique to determine the involved smoothing parameters, some examples of fitting severely irregularly distributed data, synthetically produced and of geophysical origin, are presented. In order to reduce computational costs, we then introduce a multilevel generalized cross-validation technique which goes beyond the Sobolev formulation and exploits the hierarchical setting based on wavelets. We illustrate the performance of the new strategy on some geophysical data. AMS subject classification 65T60, 62G09, 93E14, 93E24We gratefully acknowledge the support by the Deutsche Forschungsgemeinschaft (KU 1028/7 1 and SFB 611) and by the Basque Government.     

Subspace Preconditioned LSQR for Discrete Ill-Posed Problems

We present a novel implementation of a two-level iterative method for the solution of discrete linear ill-posed problems. The algorithm is algebraically equivalent to the two-level Schur complement CG algorithm of Hanke and Vogel, but involves less work per iteration. We review the algorithm, discuss our implementation, and show promising results from numerical experiments that give insight into the proper use of the algorithm.This revised version was published online in October 2005 with corrections to the Cover Date.     

Robust Adaptive Fuzzy Identification of Time-Varying Processes with Uncertain Data. Handling Uncertainties in the Physical Fitness Fuzzy Approximation with Real World Medical Data: An Application

This study considers the problem of Robust Fuzzy approximation of a time-varying nonlinear process in the presence of uncertainties in the identification data using a Sugeno Fuzzy System while maintaining the interpretability of the fuzzy model during identification. A recursive procedure for the estimation of fuzzy parameters is proposed based on solving local optimization problem that attempt to minimize the worst-case effect of data uncertainties on approximation performance. To illustrate the approach, several simulation studies on numerical examples are provided. The developed scheme was applied to handle the vagueness, ambiguity and uncertainty inherently present in the general notion of a Medical Expert about Physical Fitness based on a set of various Physiological parameters measurements.