NURBS曲线曲面拟合数据点的迭代算法

本文推广了文献[1]的结果,将文献[1]中关于B样条曲线曲面拟合数据点的迭代算法推广至有理形式,给出了无需求解方程组反求控制点及权因子即可得到拟合NURBS曲线曲面的迭代方法．该算法和文献[1]的算法本质上是统一的,而后者恰是前者的一种退化形式．文章还给出了收敛性证明以及一些定性分析．文末的数值实例说明该算法简单实用．     

逆向工程中基于NURBS的曲面重构

曲面重构是逆向工程中的关键技术.根据非均匀有理B样条曲面矩阵表达式,对于造型曲面上的(2m+1)×(2n+1)个型值数据点,推导了一个仅利用型值点数据反算二次非均匀有理B样条曲面控制顶点的算法.数值算例表明了该算法的有效性.     

Chebyshev rational interpolation

The Lanczos method and its variants can be used to solve efficiently the rational interpolation problem. In this paper we present a suitable fast modification of a general look-ahead version of the Lanczos process in order to deal with polynomials expressed in the Chebyshev orthogonal basis. The proposed approach is particularly suited for approximating analytic functions by means of rational interpolation at certain nodes located on the boundary of an elliptical region of the complex plane. In fact, in this case it overcomes some of the numerical difficulties which limited the applicability of the look-ahead Lanczos process for determining the coefficients both of the numerators and of the denominators with respect to the standard power basis. This revised version was published online in August 2006 with corrections to the Cover Date.     

Algorithms for G1 connection of multiple parametric bicubic NURBS surfaces

The objective of this paper is to introduce an innovative approach for constructing effective algorithms for removing gaps between parametric NURBS surfaces in three-space, while maintaining geometrical smoothness for the combined (or compound) surface. Similar to the degenerate case of tensor-product B-spline surfaces, if the underlying knot sequences along the connecting boundaries of two NURBS surfaces are proportional, then the parametric surfaces can be connected in a G¹ fashion. This approach can be easily extended to connecting three or four parametric NURBS surfaces. We will demonstrate the feasibility of our approach by focusing on the C¹ bicubic setting with knot sequences being equally-spaced and having double interior knots. This revised version was published online in June 2006 with corrections to the Cover Date.     

Shape recovery for sparse‐data tomography

A two‐dimensional sparse‐data tomographic problem is studied. The target is assumed to be a homogeneous object bounded by a smooth curve. A nonuniform rational basis splines (NURBS) curve is used as a computational representation of the boundary. This approach conveniently provides the result in a format readily compatible with computer‐aided design software. However, the linear tomography task becomes a nonlinear inverse problem because of the NURBS‐based parameterization. Therefore, Bayesian inversion with Markov chain Monte Carlo sampling is used for calculating an estimate of the NURBS control points. The reconstruction method is tested with both simulated data and measured X‐ray projection data. The proposed method recovers the shape and the attenuation coefficient significantly better than the baseline algorithm (optimally thresholded total variation regularization), but at the cost of heavier computation.     

A look-ahead Bareiss algorithm for general Toeplitz matrices

Summary. The Bareiss algorithm is one of the classical fast solvers for systems of linear equations with Toeplitz coefficient matrices. The method takes advantage of the special structure, and it computes the solution of a Toeplitz system of order~ with only~ arithmetic operations, instead of~ operations. However, the original Bareiss algorithm requires that all leading principal submatrices be nonsingular, and the algorithm is numerically unstable if singular or ill-conditioned submatrices occur. In this paper, an extension of the Bareiss algorithm to general Toeplitz systems is presented. Using look-ahead techniques, the proposed algorithm can skip over arbitrary blocks of singular or ill-conditioned submatrices, and at the same time, it still fully exploits the Toeplitz structure. Implementation details and operations counts are given, and numerical experiments are reported. We also discuss special versions of the proposed look-ahead Bareiss algorithm for Hermitian indefinite Toeplitz systems and banded Toeplitz systems. Received August 27, 1993 / Revised version received March 1994     

基于FETI的非协调等几何分析

基于非均匀有理B样条的等几何分析方法是一种无需网格划分的新的计算方法,旨在实现直接利用CAD模型进行分析,有望取代目前传统有限元技术．等几何分析已被成功应用在固体力学,流固耦合及拓扑优化等诸多领域．等几何分析方法要求CAD曲面或者实体高阶连续,而绝大多数CAD模型内多个曲面不但无法保持高阶连续,而且在公共界面处是几何非协调的．这一缺陷严重制约了等几何分析技术的进一步发展和应用．另外,由于采用高阶单元,等几何分析计算量较等自由度传统有限元要耗时．为解决这些难题,笔者在先前工作基础之上,提出了基于FETI方法的非协调等几何分析．新方法较以往的零空间解法更加快捷,适用于大规模数据的并行计算．数值算例表明本方法无需修改CAD模型,实施简单,精度满足要求,可处理复杂CAD模型．     

基于改进自适应混沌控制的逆可靠度分析方法

自适应混沌控制方法是一种高效、稳健的逆可靠度分析方法,但在求解强非线性凹功能函数时,计算效率仍然有待提高,且可能会陷入局部最优.通过对混沌控制因子更新策略进行改进,提出了基于改进自适应混沌控制的逆可靠度分析方法.数值算例分析表明:该方法能够有效地改善混沌控制因子自适应选取时的合理性,具有更好的收敛性和更高的计算效率,为结构可靠度分析和可靠度优化问题提供了更加高效、稳健的求解途径.     

一种二次保形插值参数曲面

1．引言保形插值是工业设计和制造中经常遇到的问题，有关这方面的研究已有许多文献【‘－u1．设n二Fx；，yi，人，川7一0，1，…；n；j＝0；l，…，。；x；＜x；＋1．l＝0，1，…．n—1；yi＜的十；，J二O，L…，。一卫｝是一给定的数据集，Cadson和［ltsch、Beatson和zejerJ－1985年分别提出的方法只保持被插数据集的轴向单调性；Dodd和Roulier等人于1983和1987年提出的方法只保持被插点集网格线上的轴向凸凹性和单调性；Constantini和FOntanella于1990年提出的方法可保持被插点集在所有于区域的边界及共内部的轴向凸凹性和单调性；…     

Interpolation-based nonlinear parametric MOR for gas pipelines

This work deals with the model order reduction (MOR) of nonlinear, parametric systems of partial differential equations as they arise in gas pipeline modeling. We present an approach that is based on a linearization around parametric working points, linear modal reduction and interpolation. The choice of the working points as well as the interpolation strategy crucially determine the approximation quality. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

High-Dimensional Adaptive Sparse Polynomial Interpolation and Applications to Parametric PDEs

We consider the problem of Lagrange polynomial interpolation in high or countably infinite dimension, motivated by the fast computation of solutions to partial differential equations (PDEs) depending on a possibly large number of parameters which result from the application of generalised polynomial chaos discretisations to random and stochastic PDEs. In such applications there is a substantial advantage in considering polynomial spaces that are sparse and anisotropic with respect to the different parametric variables. In an adaptive context, the polynomial space is enriched at different stages of the computation. In this paper, we study an interpolation technique in which the sample set is incremented as the polynomial dimension increases, leading therefore to a minimal amount of PDE solving. This construction is based on the standard principle of tensorisation of a one-dimensional interpolation scheme and sparsification. We derive bounds on the Lebesgue constants for this interpolation process in terms of their univariate counterpart. For a class of model elliptic parametric PDE’s, we have shown in Chkifa et al. (Modél. Math. Anal. Numér. 47(1):253–280, 2013) that certain polynomial approximations based on Taylor expansions converge in terms of the polynomial dimension with an algebraic rate that is robust with respect to the parametric dimension. We show that this rate is preserved when using our interpolation algorithm. We also propose a greedy algorithm for the adaptive selection of the polynomial spaces based on our interpolation scheme, and illustrate its performance both on scalar valued functions and on parametric elliptic PDE’s.     

Computing parallel curves on parametric surfaces

Generating parallel curves on parametric surfaces is an important issue in many industrial settings. Given an initial curve (called the base curve or generator) on a parametric surface, the goal is to obtain curves on the surface that are parallel to the generator. By parallel curves we mean curves that are at a given distance from the generator, where distance is measured point-wise along certain characteristic curves (on the surface) orthogonal to the generator. Except for a few particular cases, computing these parallel curves is a very difficult and challenging problem. In fact, only partial, incomplete solutions have been reported so far in the literature. In this paper we introduce a simple yet efficient method to fill this gap. In clear contrast with other existing techniques, the most important feature of our method is its generality: it can be successfully applied to any differentiable parametric surface and to any kind of characteristic curves on surfaces. To evaluate our proposal, some illustrative examples (not addressed with previous methods) for the cases of section, vector-field, and geodesic parallels are discussed. Our experimental results show the excellent performance of the method even for the complex case of NURBS surfaces.     

广义Bézier曲线与曲面在连接中的应用

通常的贝齐尔（Ｂｅｚｉｅｒ）曲线、曲面，在其端点或边界只具有ＧＣ１阶插值性．本文在保持通常贝齐尔曲线、曲面性质的基础上，定义了一种广义的贝齐尔曲线、曲面，使其在曲线段的端点和曲面片的边界具有高阶光滑插值性，它可方便地光滑连接两条参数型的曲线段和两张以上参数型曲面片，并且连接方式是ＧＣｒ（ｒ≥１）的．所以广义贝齐尔曲线、曲面在计算机辅助设计应用中更具有独特的意义．     

NURBS曲面的形状修改的一种方法

NURBS曲面是计算机辅助几何设计和计算机图形中最常用的参数曲面。本文采用NURBS曲面的齐次坐标表示，给出了通过控制顶点和权因子同时改变来修改NURBS曲面形状的一种方法。     

Optimized look-ahead recurrences for adjacent rows in the Padé table

A new look-ahead algorithm for recursively computing Padé approximants is introduced. It generates a subsequence of the Padé approximants on two adjacent rows (defined by fixed numerator degree) of the Padé table. Its two basic versions reduce to the classical Levinson and Schur algorithms if no look-ahead is required. The new algorithm can be viewed as a combination of the look-ahead sawtooth and the look-ahead Levinson and Schur algorithms that we proposed before, but now the look-ahead step size is minimal (as in the sawtooth version) and the computational costs are as low as in the least expensive competing algorithms (including our look-ahead Levinson and Schur algorithms). The underlying recurrences link well-conditioned basic pairs,i.e., pairs of sufficiently different neighboring Padé forms.The algorithm can be used to solve Toeplitz systems of equationsTx = b. In this application it comes in several versions: anO(N ²) Levinson-type form, anO(N ²) Schur-type form, and a superfastO(N log² N) Schur-type version. As an option of the first two versions, the corresponding block LDU decompositions ofT ^–1 orT, respectively, can be found.     

Interaction between intelligent agent strategies for real-time transportation planning

In this paper we study the real-time scheduling of time-sensitive full truckload pickup-and-delivery jobs. The problem involves the allocation of jobs to a fixed set of vehicles which might belong to different collaborating transportation agencies. A recently proposed solution methodology for this problem is the use of a multi-agent system where shipper agents offer jobs through sequential auctions and vehicle agents bid on these jobs. In this paper we consider such a system where both the vehicle agents and the shipper agents are using profit maximizing look-ahead strategies. Our main contribution is that we study the interrelation of these strategies and their impact on the system-wide logistical costs. From our simulation results, we conclude that the system-wide logistical costs (i) are always reduced by using the look-ahead strategies instead of a myopic strategy (10–20%) and (ii) the joint effect of two look-ahead strategies is larger than the effect of an individual strategy. To provide an indication of the savings that might be realized under centralized decision making, we benchmark our results against an integer programming approach.     

A General Hybrid Optimization Strategy for Curve Fitting in the Non-uniform Rational Basis Spline Framework

In this paper, a general methodology to approximate sets of data points through Non-uniform Rational Basis Spline (NURBS) curves is provided. The proposed approach aims at integrating and optimizing the full set of design variables (both integer and continuous) defining the shape of the NURBS curve. To this purpose, a new formulation of the curve fitting problem is required: it is stated in the form of a constrained nonlinear programming problem by introducing a suitable constraint on the curvature of the curve. In addition, the resulting optimization problem is defined over a domain having variable dimension, wherein both the number and the value of the design variables are optimized. To deal with this class of constrained nonlinear programming problems, a global optimization hybrid tool has been employed. The optimization procedure is split in two steps: firstly, an improved genetic algorithm optimizes both the value and the number of design variables by means of a two-level Darwinian strategy allowing the simultaneous evolution of individuals and species; secondly, the optimum solution provided by the genetic algorithm constitutes the initial guess for the subsequent gradient-based optimization, which aims at improving the accuracy of the fitting curve. The effectiveness of the proposed methodology is proven through some mathematical benchmarks as well as a real-world engineering problem.     

Precomputing strategy for Hamiltonian Monte Carlo method based on regularity in parameter space

Markov Chain Monte Carlo (MCMC) algorithms play an important role in statistical inference problems dealing with intractable probability distributions. Recently, many MCMC algorithms such as Hamiltonian Monte Carlo (HMC) and Riemannian Manifold HMC have been proposed to provide distant proposals with high acceptance rate. These algorithms, however, tend to be computationally intensive which could limit their usefulness, especially for big data problems due to repetitive evaluations of functions and statistical quantities that depend on the data. This issue occurs in many statistic computing problems. In this paper, we propose a novel strategy that exploits smoothness (regularity) in parameter space to improve computational efficiency of MCMC algorithms. When evaluation of functions or statistical quantities are needed at a point in parameter space, interpolation from precomputed values or previous computed values is used. More specifically, we focus on HMC algorithms that use geometric information for faster exploration of probability distributions. Our proposed method is based on precomputing the required geometric information on a set of grids before running sampling algorithm and approximating the geometric information for the current location of the sampler using the precomputed information at nearby grids at each iteration of HMC. Sparse grid interpolation method is used for high dimensional problems. Tests on computational examples are shown to illustrate the advantages of our method.     

Comparison Between the Mean-Variance Optimal and the Mean-Quadratic-Variation Optimal Trading Strategies

ABSTRACT

We compare optimal liquidation policies in continuous time in the presence of trading impact using numerical solutions of Hamilton–Jacobi–Bellman (HJB) partial differential equations (PDEs). In particular, we compare the time-consistent mean-quadratic-variation strategy with the time-inconsistent (pre-commitment) mean-variance strategy. We show that the two different risk measures lead to very different strategies and liquidation profiles. In terms of the optimal trading velocities, the mean-quadratic-variation strategy is much less sensitive to changes in asset price and varies more smoothly. In terms of the liquidation profiles, the mean-variance strategy is much more variable, although the mean liquidation profiles for the two strategies are surprisingly similar. On a numerical note, we show that using an interpolation scheme along a parametric curve in conjunction with the semi-Lagrangian method results in significantly better accuracy than standard axis-aligned linear interpolation. We also demonstrate how a scaled computational grid can improve solution accuracy.