Shape reconstruction of an inverse boundary value problem of two‐dimensional Navier–Stokes equations

This paper is concerned with the problem of the shape reconstruction of two‐dimensional flows governed by the Navier–Stokes equations. Our objective is to derive a regularized Gauss–Newton method using the corresponding operator equation in which the unknown is the geometric domain. The theoretical foundation for the Gauss–Newton method is given by establishing the differentiability of the initial boundary value problem with respect to the boundary curve in the sense of a domain derivative. The numerical examples show that our theory is useful for practical purpose and the proposed algorithm is feasible. Copyright © 2009 John Wiley & Sons, Ltd.     

X-ray photoelectron diffraction study of discommensurate Cu/Ge(111)

We have studied the atomic structure in the interior of discommensurate domains of the Cu/Ge(111) surface by using scanned-angle X-ray photoelectron diffraction (XPED). XPED patterns of Cu 2p_3/2 intensity provided direct information on the local structure in the vicinity of photoelectron emitters. It has been found that a certain number of Cu atoms are embedded within the surface layer, so that the surface has some structural similarity with the discommensurate Cu/Si(111)-‘5×5’.     

基于解码技术的全息显微术研究

提出了一种利用解码技术对全息显微畸变图像进行校正的方法。把全息记录中的显微镜预放大过程作为物光波产生畸变的编码过程;在波前再现过程中,用全息记录中的预放大显微镜对再现的赝实像再次放大进行解码,从而获得清晰的全息显微图像。实验验证了该方法的有效性。     

Building nonredundant adaptive wavelets by update lifting

In this paper, we propose a technique for building adaptive wavelets by means of an extension of the lifting scheme. Our scheme comprises an adaptive update lifting step and a fixed prediction lifting step. The adaptivity consists hereof that the system can choose between two different update filters, and that this choice is triggered by the local gradient of the original signal. If the gradient is large (in some seminorm sense) it chooses one filter, if it is small the other. We derive necessary and sufficient conditions for the invertibility of such an adaptive system for various scenarios. Furthermore, we present some examples to illustrate our theoretical results.     

Numerical control of kohonen neural network for scattered data approximation

Surface reconstruction from scattered data using Kohonen neural network is presented in this paper. The network produces a topologically predefined grid from the unordered data which can be applied as a rough approximation of the input set or as a base surface for further process. The quality and computing time of the approximation can be controlled by numerical parameters. As a further application, ruled surface is produced from a set of unordered lines by the network. AMS subject classification 68U07, 65D17, 68T20     

Shannon sampling II: Connections to learning theory

We continue our study [S. Smale, D.X. Zhou, Shannon sampling and function reconstruction from point values, Bull. Amer. Math. Soc. 41 (2004) 279–305] of Shannon sampling and function reconstruction. In this paper, the error analysis is improved. Then we show how our approach can be applied to learning theory: a functional analysis framework is presented; dimension independent probability estimates are given not only for the error in the L² spaces, but also for the error in the reproducing kernel Hilbert space where the learning algorithm is performed. Covering number arguments are replaced by estimates of integral operators.     

A new volume of fluid method in three dimensions—Part II: Piecewise‐planar interface reconstruction with cubic‐Bézier fit

A new interface reconstruction method in 3D is presented. The method involves a conservative level‐contour reconstruction coupled to a cubic‐Bézier interpolation. The use of the proposed piecewise linear interface calculation (PLIC) reconstruction scheme coupled to a multidimensional time integration provides solutions of second‐order spatial and temporal accuracy. The accuracy and efficiency of the proposed reconstruction algorithm are demonstrated through several tests, whose results are compared with those obtained with other recently proposed methods. An overall improvement in accuracy with respect to other recent methods has been achieved, along with a substantial reduction in the central processing unit time required. Copyright © 2008 John Wiley & Sons, Ltd.     

The discontinuous profile method for simulating two‐phase flow in pipes using the single component approximation

Godunov‐type algorithms are very attractive for the numerical solution of discontinuous flows. The reconstruction of the profile inside the cells is crucial to scheme performance. The non‐linear generalization of the discontinuous profile method (DPM) presented here for the modelling of two‐phase flow in pipes uses a discontinuous reconstruction in order to capture shocks more efficiently than schemes using continuous functions. The reconstructed profile is used to define the Riemann problem at cell interfaces by averaging of the components of the variable in the base of eigenvectors over their domain of dependence. Intercell fluxes are computed by solving the Riemann problem with an approximate‐state solver. The adapted treatment of boundary conditions is essential to ensure the quality of the computational results and a specific procedure using virtual cells at both extremities of the computational domain is required. Internal boundary conditions can be treated in the same way as external ones. Application of the DPM to test cases is shown to improve the quality of computational results significantly. Copyright © 2001 John Wiley & Sons, Ltd.     

Scanning heterodyne reconstruction of coded aperture images

The reconstruction of coded aperture images using an optical scanning heterodyne processor is described for a class of apertures which can be synthesized by two beam interference. The analysis of Fresnel zone apertures indicates that a lateral and longitudinal (tomographic) resolution close to the theoretical limit is achievable. Experimental results of optical simulations confirm this assertion. Advantages of such technique include real-time operation and large multiplexing capability.     

Shape reconstruction of an inverse Stokes problem

This paper deals with the shape reconstruction of a viscous incompressible fluid driven by the Stokes flow. For the approximate solution of the ill-posed and nonlinear problem we propose a regularized Newton method. A theoretical foundation for the Newton method is given by establishing the differentiability of the initial boundary value problem with respect to the interior boundary curve in the sense of a domain derivative. The numerical examples show that our theory is useful for practical purpose and the proposed algorithm is feasible.