求解无限维非线性互补问题的光滑化牛顿法

研究一类无限维非线性互补问题的光滑化牛顿法.借助于非线性互补函数,将无限维非线性互补问题转化为一个非光滑算子方程.构造光滑算子逼近非光滑算子,在光滑逼近算子满足方向可微相容性的条件下,证明了光滑化牛顿法具有超线性收敛性.     

含各阶导数的非线性弹性梁方程的一个存在定理

通过选择适当的Banach空间并利用Leray-Schauder非线性抉择对于含各阶导数的非线性弹性梁方程{u(4)(t)=f(t,u(t),u′(t),u″(t),u′″(t)),0≤t≤1, u(0)=u′(1)=u″(0)=u′″(1)=0.建立了一个解的存在定理．在材料力学中，该方程描述了一端简单支撑，另一端被滑动夹子夹住的弹性梁的形变．这个存在定理说明只要非线性项满足某种线性增长条件该方程至少有一个解．     

Rigorous Derivation of a Nonlinear Diffusion Equation as Fast-Reaction Limit of a Continuous Coagulation-Fragmentation Model with Diffusion

Weak solutions of the spatially inhomogeneous (diffusive) Aizenmann–Bak model of coagulation-breakup within a bounded domain with homogeneous Neumann boundary conditions are shown to converge, in the fast reaction limit, towards local equilibria determined by their mass. Moreover, this mass is the solution of a nonlinear diffusion equation whose nonlinearity depends on the (size-dependent) diffusion coefficient. Initial data are assumed to have integrable zero order moment and square integrable first order moment in size, and finite entropy. In contrast to our previous result [5 Carrillo , J. A. , Desvillettes , L. , Fellner , K. ( 2008 ). Fast-reaction limit for the inhomogeneous aizenman-bak model . Kinetic and Related Models 1 : 127 – 137 . [Google Scholar]], we are able to show the convergence without assuming uniform bounds from above and below on the number density of clusters.     

An Isotonic Regression Problem for Infinite-Dimensional Parameters

We consider an infinite-dimensional isotonic regression problem which is an extension of the suitably revised classical isotonic regression problem. Given p-summable data, for p finite and at least one, there exists an optimal estimator to our problem. For p greater than one, this estimator is unique and is the limit in the p-norm of the sequence of unique estimators in canonical finite-dimensional truncations of our problem. However, for p equal to one, our problem, as well as the finite-dimensional truncations, admit multiple optimal estimators in general. In this case, the sequence of optimal estimator sets to the truncations converges to the optimal estimator set of the infinite problem in the sense of Kuratowski. Moreover, the selection of natural best optimal estimators to the truncations converges in the 1-norm to an optimal estimator of the infinite problem.     

Reduction Methods for FE Analysis in Nonlinear Structural Dynamics

The computation of the nonlinear motion of large structures with implicit time integration schemes is costly. In each time step a large system of linear equations needs to be solved several times. In finite element models often a fine discretization is necessary to represent the geometry and to yield accurate results for the stress field. But from experience it is known that only a small number of degrees of freedom is sufficient to account for the dominant parts of a dynamic motion. Similar to modal decomposition, methods were developed to reduce the number of degrees of freedoms in nonlinear problems. Even though it is not possible to decompose the motion into decoupled modes, a reduction of the number of degrees of freedom yields less computational effort in many cases. The choice of appropriate basis vectors is important. Often used are load-dependent ‘Ritz’ vectors, which should be updated during the computation to yield sufficient accuracy. Dominant modes, computed by a proper orthogonal decomposition of a previous calculation can be used for repeated analyses of the same system with different loads. Significant time savings can be achieved with reduction methods. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Approximate Analysis of a Nonlinear Dissipative Model

From the approximate symmetry point of view, a perturbed system of partial differential equations (PDEs) for viscoelastic media is investigated. The corresponding complete approximate symmetry classification is derived and a fundamental task of the paper is the proof of a theorem that explores the relationship between the symmetries of our model and other models. In some physical cases, approximate solutions are computed by means of the approximate generator of the first order approximate group of transformations.     

柔性梁内平衡模型降阶与主动控制的实验研究

内平衡降阶方法能够有效地解决柔性结构的模型降阶问题,尤其是对于频率密集结构．然而由于存在如何从物理传感器测量中提取内平衡模态坐标的问题,目前关于内平衡降阶方法的研究大多是在理论上进行探索,少有实验研究和工程应用报道．该文以柔性梁为对象,开展内平衡降阶方法的理论与实验研究,并且基于降阶模型进行主动控制的设计．文中介绍了一个基于DSP TMS320F2812芯片的实验系统,提出了一个从物理传感器测量中提取内平衡模态坐标的近似方法,并且通过仿真与实验验证了该方法的可行性和有效性,基于降阶模型的控制设计能够有效地抑制梁的弹性振动．     

Algorithm for Solving the Cauchy Problem for One Infinite-Dimensional System of Nonlinear Differential Equations

Computational Mathematics and Mathematical Physics - The Cauchy problem for an infinite-dimensional system of nonlinear evolution equations, which is a generalization of the Langmuir chain, is...     

An Infinite-Dimensional Semismooth Newton Method for Elasto-Plastic Contact Problems

A Fenchel dualization scheme for the one-step time-discretized elasto-plastic contact problem with kinematic or isotropic hardening is considered. The associated path is induced by a combined Moreau-Yosida / Tichonov regularization of the dual problem. The sequence of solutions to the regularized problems is shown to converge strongly to the solution of the original problem. This property relies on the density of the intersection of certain convex sets. The corresponding conditions are worked out and customary regularization approaches are shown to be valid in this context. It is also argued that without higher regularity assumptions on the data the resulting problems possess Newton differentiable optimality systems in infinite dimensions [2]. Consequently, each regularized subsystem can be solved mesh-independently at a local superlinear rate of convergence [6]. Numerically the problems are solved using conforming finite elements. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear Model Order Reduction in Nanoelectronics: Combination of POD and TPWL

In this paper we demonstrate model order reduction of a nonlinear academic model of an inverter chain. Two reduction methods, which are suitable for nonlinear differential algebraic equation systems are combined, the trajectory piecewise linear approach and the proper orthogonal decomposition. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sliced Coordinate Analysis for Effective Dimension Reduction and Nonlinear Extensions

Sliced inverse regression (SIR) is an important method for reducing the dimensionality of input variables. Its goal is to estimate the effective dimension reduction directions. In classification settings, SIR is closely related to Fisher discriminant analysis. Motivated by reproducing kernel theory, we propose a notion of nonlinear effective dimension reduction and develop a nonlinear extension of SIR called kernel SIR (KSIR). Both SIR and KSIR are based on principal component analysis. Alternatively, based on principal coordinate analysis, we propose the dual versions of SIR and KSIR, which we refer to as sliced coordinate analysis (SCA) and kernel sliced coordinate analysis (KSCA), respectively. In the classification setting, we also call them discriminant coordinate analysis and kernel discriminant coordinate analysis. The computational complexities of SIR and KSIR rely on the dimensionality of the input vector and the number of input vectors, respectively, while those of SCA and KSCA both rely on the number of slices in the output. Thus, SCA and KSCA are very efficient dimension reduction methods.     

An Estimation of Exact Penalty for Infinite-Dimensional Inequality-Constrained Minimization Problems

We use the penalty approach in order to study inequality-constrained minimization problems in infinite dimensional spaces. A penalty function is said to have the exact penalty property if there is a penalty coefficient for which a solution of an unconstrained penalized problem is a solution of the corresponding constrained problem. In this paper we consider a large class of inequality-constrained minimization problems for which a constraint is a mapping with values in a normed ordered space. For this class of problems we introduce a new type of penalty functions, establish the exact penalty property and obtain an estimation of the exact penalty. Using this exact penalty property we obtain necessary and sufficient optimality conditions for the constrained minimization problems.     

非线性弹性梁方程的一个孪生正解的存在定理

对于非线性四阶两点边值问题建立了一个孪生正解的存在定理．该边值问题通常描述了具有固定两端点的弹性梁的形变．     

Stability and Bifurcation Analysis of a Nonlinear DDE Model for Drilling

We discuss several examples of synchronous dynamical phenomena in coupled cell networks that are unexpected from symmetry considerations, but are natural using a theory developed by Stewart, Golubitsky, and Pivato. In particular we demonstrate patterns of synchrony in networks with small numbers of cells and in lattices (and periodic arrays) of cells that cannot readily be explained by conventional symmetry considerations. We also show that different types of dynamics can coexist robustly in single solutions of systems of coupled identical cells. The examples include a three-cell system exhibiting equilibria, periodic, and quasiperiodic states in different cells; periodic $2n\times 2n$ arrays of cells that generate $2^n$ different patterns of synchrony from one symmetry-generated solution; and systems exhibiting multirhythms (periodic solutions with rationally related periods in different cells). Our theoretical results include the observation that reduced equations on a center manifold of a skew product system inherit a skew product form.     

非线性梁方程的渐近吸引子

研究了一类非线性梁方程的渐近吸引子.即利用正交分解法构造一个有限维解序列.首先用数学归纳法证明了该解序列不会远离方程的整体吸引子,其次证明了它在长时间后无限趋于方程的整体吸引子,并给出了渐近吸引子的维数估计.     

Exact Solutions to a Finite-difference Model of a Nonlinear Reaction-advection Equation: Implications for Numerical Analysis

This article gives exact solutions to a finite-difference model of a nonlinear reaction-advection equation. We show that this partial difference equation and the corresponding stationary and spatially independent difference equations derived from this model give the best representation of the original partial differential equation. The relevance of this work to the elimination of chaotic behavior in numerical solutions of differential equations is discussed.     

An Adaptive Neural Network Model for Nonlinear Programming Problems

In this paper a canonical neural network with adaptively changing synaptic weights and activation function parameters is presented to solve general nonlinear programming problems. The basic part of the model is a sub-network used to find a solution of quadratic programming problems with simple upper and lower bounds. By sequentially activating the sub-network under the control of an external computer or a special analog or digital processor that adjusts the weights and parameters, one then solves general nonlinear programming problems. Convergence proof and numerical results are given.     

An Algorithm for Nonlinear,Nonparametric Model Choice and Prediction

We introduce an algorithm which, in the context of nonlinear regression on vector-valued explanatory variables, aims to choose those combinations of vector components that provide best prediction. The algorithm is constructed specifically so that it devotes attention to components that might be of relatively little predictive value by themselves, and so might be ignored by more conventional methodology for model choice, but which, in combination with other difficult-to-find components, can be particularly beneficial for prediction. The design of the algorithm is also motivated by a desire to choose vector components that become redundant once appropriate combinations of other, more relevant components are selected. Our theoretical arguments show these goals are met in the sense that, with probability converging to 1 as sample size increases, the algorithm correctly determines a small, fixed number of variables on which the regression mean, g say, depends, even if dimension diverges to infinity much faster than n. Moreover, the estimated regression mean based on those variables approximates g with an error that, to first order, equals the error which would arise if we were told in advance the correct variables. In this sense, the estimator achieves oracle performance. Our numerical work indicates that the algorithm is suitable for very high dimensional problems, where it keeps computational labor in check by using a novel sequential argument, and also for more conventional prediction problems, where dimension is relatively low.     

A Dynamical System for Plant Pattern Formation: A Rigorous Analysis

Summary. We present a rigorous mathematical analysis of a discrete dynamical system modeling plant pattern formation. In this model, based on the work of physicists Douady and Couder, fixed points are the spiral or helical lattices often occurring in plants. The frequent occurrence of the Fibonacci sequence in the number of visible spirals is explained by the stability of the fixed points in this system, as well as by the structure of their bifurcation diagram. We provide a detailed study of this diagram.     

A Finite Element Formulation for the Nonlinear Analysis of Piezoelectric Three-Dimensional Beam Structures

A finite element formulation for a three-dimensional piezoelectric beam which includes geometrical and material nonlinearities is presented. To account for the piezoelectric effect, the coupling between the mechanical stress and the electrical displacement is considered. Based on the Timoshenko theory, an eccentric beam formulation is introduced which provides an efficient model to analyze piezoelectric structures. The geometrically nonlinear assumption allows the calculation of large deformations including buckling analysis. A quadratic approximation of the electric potential through the cross section of the beam ensures the fulfilment of the charge conservation law exactly. This assumption leads to a finite element formulation with six mechanical and five electrical degrees of freedom per node. To take into account the typical ferroelectric hysteresis phenomena, a nonlinear material model is essential. For this purpose, the phenomenological Preisach model is implemented into the beam formulation which provides an efficient determination of the remanent part of the polarization. The applicability of the introduced beam formulation is discussed with respect to available data from literature. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)