Approximate analytical solutions of systems of PDEs by homotopy analysis method

In this paper, the homotopy analysis method (HAM) is applied to obtain series solutions to linear and nonlinear systems of first- and second-order partial differential equations (PDEs). The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of series solutions. It is shown in particular that the solutions obtained by the variational iteration method (VIM) are only special cases of the HAM solutions.     

Risk-Sensitive and Robust Escape Control for Degenerate Diffusion Processes

This paper considers the problem of controlling a possibly degenerate small noise diffusion so as to prevent it from leaving a prescribed set. The criterion of interest is a risk-sensitive version of the mean escape time criterion. Using a general representation formula, this criterion is expressed as the upper value of a stochastic differential game. It is shown that in the small noise limit this upper value converges to the value of an associated deterministic differential game. Our approach differs from standard PDE approaches in a number of ways. For example, the upper game representation allows one to relate directly the prelimit and the limit controls and, in fact, strategies that are nearly maximizing for the robust problem can be used to define nearly minimizing controls for the risk-sensitive control problem for sufficiently small ε>0. The result provides a canonical example of the use of variational representations in connecting risk-sensitive and robust control. Date received: November 21, 1998. Date revised: June 20, 1999.     

Image thresholding by variational minimax optimization

In this paper we introduce an adaptive image thresholding technique via minimax optimization of a novel energy functional that consists of a non-linear convex combination of an edge sensitive data fidelity term and a regularization term. While the proposed data fidelity term requires the threshold surface to intersect the image surface only at places with large image gradient magnitude, the regularization term enforces smoothness in the threshold surface. To the best of our knowledge, all the previously proposed energy functional-based adaptive image thresholding algorithms rely on manually set weighting parameters to achieve a balance between the data fidelity and the regularization terms. In contrast, we use minimax principle to automatically find this weighting parameter value, as well as the threshold surface. Our conscious choice of the energy functional permits a variational formulation within the minimax principle leading to a globally optimum solution. The proposed variational minimax optimization is carried out by an iterative gradient descent with exact line search technique that we experimentally demonstrate to be computationally far more attractive than the Fibonacci search applied to find the minimax solution. Our method shows promising results to preserve edge/texture structures in different benchmark images over other competing methods. We also demonstrate the efficacy of the proposed method for delineating lung boundaries from magnetic resonance imagery (MRI).     

Variational Bayesian noise estimation of point sets

Scanning devices acquire geometric information from the surface of an object in the form of a 3D point set. Such point sets, as any data obtained by means of physical measurement, contain some noise. To create an accurate model of the scanned object, this noise should be resolved before or during the process of surface reconstruction. In this paper, we develop a statistical technique to estimate the noise in a scanned point set. The noise is represented as normal distributions with zero mean and their variances determine the amount of the noise. These distributions are estimated with a variational Bayesian method, which is known to provide more robust estimations than point estimate methods, such as maximum likelihood and maximum a posteriori. Validation experiments and further tests with real scan data show that the proposed technique can accurately estimate the noise in a 3D point set.     

A rate-dependent incremental variational formulation of ferroelectricity

This paper presents a variational-based modeling and computational implementation of the non-linear, rate-dependent response of piezoceramics under electro-mechanical loading. The point of departure is a general internal variable formulation that describes the hysteretic electro-mechanical response of the material as a standard dissipative solid. Consistent with this type of dissipative continua, we develop a variational formulation of the coupled electro-mechanical boundary-value-problem based on incremental potentials for the stresses and the electric displacement. We specify the variational formulation to a model that describes time-dependent, electric polarizations accompanied by remanent strains. It is governed by a dual dissipation function formulated in terms of the internal driving forces. The model reproduces experimentally observed dielectric and butterfly hystereses, which are characteristic for ferroelectric materials. It accounts for the rate-dependency of the hystereses and the macroscopically non-uniform distribution of the polarization in the solid. An important aspect of our treatment is the numerical implementation of the coupled problem. The monolithic discretization of the two-field problem appears, as a consequence of the proposed variational principle, in a symmetric format. The performance of the proposed methods is demonstrated by means of a spectrum of benchmark problems.     

Numerical solutions for system of third-order boundary value problems

We develop a numerical method for computing approximations for the solutions of a system of third order boundary value problems associated with odd order obstacle problems. Such a problem arise in physical oceanography and can be studied in the framework of variational inequality theory. We study the convergence analysis of the present method and we show that it gives numerical results which are better than the other available results. Numerical example is presented to illustrate the applicability of the new method.     

Formulation and performance of variational integrators for rotating bodies

Variational integrators are obtained for two mechanical systems whose configuration spaces are, respectively, the rotation group and the unit sphere. In the first case, an integration algorithm is presented for Euler’s equations of the free rigid body, following the ideas of Marsden et al. (Nonlinearity 12:1647–1662, 1999). In the second example, a variational time integrator is formulated for the rigid dumbbell. Both methods are formulated directly on their nonlinear configuration spaces, without using Lagrange multipliers. They are one-step, second order methods which show exact conservation of a discrete angular momentum which is identified in each case. Numerical examples illustrate their properties and compare them with existing integrators of the literature. Financial support for this work has been provided by grant DPI2006-14104 from the Spanish Ministry of Education and Science.     

用Liapunov直接方法估计非线性扰动常微分方程的解

用Liapunov函数，对解非线性扰动微分方程的变分方程的基本解矩阵，给出了一种指数估计的存在准则．