Fully discrete dynamic mesh discontinuous Galerkin methods for the Cahn-Hilliard equation of phase transition

Fully discrete discontinuous Galerkin methods with variable mesh- es in time are developed for the fourth order Cahn-Hilliard equation arising from phase transition in materials science. The methods are formulated and analyzed in both two and three dimensions, and are proved to give optimal order error bounds. This coupled with the flexibility of the methods demonstrates that the proposed discontinuous Galerkin methods indeed provide an efficient and viable alternative to the mixed finite element methods and nonconforming (plate) finite element methods for solving fourth order partial differential equations.

     

多组变量典型相关分析SSQCOR准则的数值求解

本文研究多组变量相关分析SSQCOR.准则的数值方法.从KKT条件出发引入了Gauss-Seidel型方法,从SSQCOR出发引入了交替变量法.证明了前者是后者的非精确形式,都具有单调上升性.为了提高得到全局解的可能性,引入了初始点策略.用实际数据和模拟数据进行了数值试验以说明算法的有效性.     

Constructing higher-order methods for obtaining the multiple roots of nonlinear equations

This paper concentrates on iterative methods for obtaining the multiple roots of nonlinear equations. Using the computer algebra system Mathematica, we construct an iterative scheme and discuss the conditions to obtain fourth-order methods from it. All the presented fourth-order methods require one-function and two-derivative evaluation per iteration, and are optimal higher-order iterative methods for obtaining multiple roots. We present some special methods from the iterative scheme, including some known already. Numerical examples are also given to show their performance.     

Linearly implicit time discretization for free surface problems

Deeper investigation of time discretization for free surface problems is a widely neglected problem. Many existing approaches use an explicit decoupling which is only conditionally stable. Only few unconditionally stable methods are known, and known methods may suffer from too strong numerical dissipativity. They are also usually of first rder only [1, 9]. We are therefore looking for unconditionally stable, minimally dissipative methods of higher order. Linearly implicit Runge-Kutta (LIRK) methods are a class of one-step methods that require the solution of linear systems in each time step of a nonlinear system. They are well suited for discretized PDEs, e.g. parabolic problems [7]. They have been used successfully to solve the incompressible Navier-Stokes equations [5]. We suggest an adaption of these methods for free surface problems and compare different approximations to the Jacobian matrix needed for such methods. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Teacher time out as a site for studying mathematical knowledge for teaching

The special mathematical knowledge that is needed for teaching has been studied for decades but the methods for studying it have challenges. Some methods, such as measurement and cognitive interviews, are removed from the dynamics of teaching. Other methods, such as observation, are closer to practice but mostly involve an outsider perspective. Moreover, few methods tap into the tacit and often invisible demands that teachers encounter in teaching. This article develops an argument that teacher time outs in rehearsals and enactments might be a productive site for studying mathematical knowledge for teaching. Teacher time outs constitute a site for professional deliberation, which 1) preserves the complexity and gets inside the dynamics of teaching, where 2) tacit and implicit challenges and demands are made explicit, and where 3) insider and outsider perspectives are combined.     

Non-stiff integrators for differential–algebraic systems of index 2

Non-stiff differential-algebraic equations (DAEs) can be solved efficiently by partitioned methods that combine well-known non-stiff integrators from ODE theory with an implicit method to handle the algebraic part of the system. In the present paper we consider partitioned one-step and partitioned multi-step methods for index-2 DAEs in Hessenberg form and the application of these methods to constrained mechanical systems. The methods are presented from a unified point of view. The comparison of various classes of methods is completed by numerical tests for benchmark problems from the literature. This revised version was published online in June 2006 with corrections to the Cover Date.     

Exploratory Data Analysis for Complex Models

“Exploratory” and “confirmatory” data analysis can both be viewed as methods for comparing observed data to what would be obtained under an implicit or explicit statistical model. For example, many of Tukey's methods can be interpreted as checks against hypothetical linear models and Poisson distributions. In more complex situations, Bayesian methods can be useful for constructing reference distributions for various plots that are useful in exploratory data analysis. This article proposes an approach to unify exploratory data analysis with more formal statistical methods based on probability models. These ideas are developed in the context of examples from fields including psychology, medicine, and social science.     

Exactness and optimality of methods for recovering functions from their spectrum

Optimal methods are constructed for recovering functions and their derivatives in a Sobolev class of functions on the line from exactly or approximately defined Fourier transforms of these functions on an arbitrary measurable set. The methods are exact on certain subspaces of entire functions. Optimal recovery methods are also constructed for wider function classes obtained as the sum of the original Sobolev class and a subspace of entire functions.     

Mixing OR methods in practice: Past, present and future directions

Although mixing of OR methods is an area of increasing interest to the OR community, there has been little discussion regarding generic lessons that can be learnt from mixing methods in practice. The aim of this paper is to carry out such an analysis through considering generic lessons that may be associated with mixing methods, regardless of the methods chosen. To identify these lessons, published case studies on how OR methods have been mixed are analysed giving rise to a number of themes revealing the lessons. These themes include; the implications from the use of different facilitators/modellers, how methods have been mixed together, the nature of the modelling interventions, the client value and the rationale given for mixing methods. The paper discusses the lessons learnt in each of these themes and presents opportunities for future work.     

Exponentially fitted variants of Euler's method for ODEs

A new class of Euler's method for the numerical solution of ordinary differential equations is presented in this article. The methods are iterative in nature and admit their geometric derivation from an exponentially fitted osculating straight line. They are single-step methods and do not require evaluation of any derivatives. The accuracy and stability of the proposed methods are considered and their applicability to stiff problems is also discussed.     

On the Application of the Auxiliary Problem Principle

The auxiliary problem principle (APP) derives from a general theory on decomposition-coordination methods establishing a comprehensive framework for both one-level and two-level methods. In this paper, the results of the two-level methods of APP are specialized for an efficient application to some engineering problems.     

Analysis of two Chebyshev-like third order methods free from second derivatives for solving systems of nonlinear equations

In this paper, two Chebyshev-like third order methods free from second derivatives are considered and analyzed for systems of nonlinear equations. The methods can be obtained by having different approximations to the second derivatives present in the Chebyshev method. We study the local and third order convergence of the methods using the point of attraction theory. The computational aspects of the methods are also studied using some numerical experiments including an application to the Chandrasekhar integral equations in Radiative Transfer.     

The Mixturegram: A Visualization Tool for Assessing the Number of Components in Finite Mixture Models

Certain practical and theoretical challenges surround the estimation of finite mixture models. One such challenge is how to determine the number of components when this is not assumed a priori. Available methods in the literature are primarily numerical and lack any substantial visualization component. Traditional numerical methods include the calculation of information criteria and bootstrapping approaches; however, such methods have known technical issues regarding the necessary regularity conditions for testing the number of components. The ability to visualize an appropriate number of components for a finite mixture model could serve to supplement the results from traditional methods or provide visual evidence when results from such methods are inconclusive. Our research fills this gap through development of a visualization tool, which we call a mixturegram. This tool is easy to implement and provides a quick way for researchers to assess the number of components for their hypothesized mixture model. Mixtures of univariate or multivariate data can be assessed. We validate our visualization assessments by comparing with results from information criteria and an ad hoc selection criterion based on calculations used for the mixturegram. We also construct the mixturegram for two datasets.     

Nonlinear Stability of General Linear Methods

This paper considers the nonlinear stability of general linear methods. The diagonal matrix of an algebraically stable method is shown to be fixed. Readily testable necessary and sufficient conditions are obtained for algebraic stability and, more generally, for nonlinear stability in closed disk regions of the complex plane. It is shown that the latter criteria are satisfied by some explicit methods. It is also shown that certain methods, including some that are L-stable, suffer from nonautonomous instability along the negative real line near zero. A loose classification of methods is given according to nonlinear stability properties.     

Some computational methods for systems of nonlinear equations and systems of polynomial equations

This paper gives a brief survey and assessment of computational methods for finding solutions to systems of nonlinear equations and systems of polynomial equations. Starting from methods which converge locally and which find one solution, we progress to methods which are globally convergent and find an a priori determinable number of solutions. We will concentrate on simplicial algorithms and homotopy methods. Enhancements of published methods are included and further developments are discussed.     

Searching for knowledge in a world flooded with facts

The wide availability of computer technology and large electronic storage media has led to an enormous proliferation of databases in almost every area of human endeavour. This naturally creates an intense demand for powerful methods and tools for data analysis. Current methods and tools are primarily oriented toward extracting numerical and statistical data characteristics. While such characteristics are very important and useful, they are often insufficient. A decision maker typically needs an interpretation of these findings, and this has to be done by a data analyst. With the growth in the amount and complexity of the data, making such interpretations is an increasingly difficult problem. As a potential solution, this paper advocates the development of methods for conceptual data analysis. Such methods aim at semi-automating the processes of determining high-level data interpretations, and discovering qualitative patterns in data. It is argued that these methods could be built on the basis of algorithms developed in the area of machine learning. An exemplary system utilizing such algorithms, INLEN, is discussed. The system integrates machine learning and statistical analysis techniques with database and expert system technologies. Selected capabilities of the system are illustrated by examples from implemented modules.     

Nonadaptive methods for polyhedral approximation of the Edgeworth—Pareto hull using suboptimal coverings on the direction sphere

For multicriteria convex optimization problems, new nonadaptive methods are proposed for polyhedral approximation of the multidimensional Edgeworth-Pareto hull (EPH), which is a maximal set having the same Pareto frontier as the set of feasible criteria vectors. The methods are based on evaluating the support function of the EPH for a collection of directions generated by a suboptimal covering on the unit sphere. Such directions are constructed in advance by applying an asymptotically effective adaptive method for the polyhedral approximation of convex compact bodies, namely, by the estimate refinement method. Due to the a priori definition of the directions, the proposed EPH approximation procedure can easily be implemented with parallel computations. Moreover, the use of nonadaptive methods considerably simplifies the organization of EPH approximation on the Internet. Experiments with an applied problem (from 3 to 5 criteria) showed that the methods are fairly similar in characteristics to adaptive methods. Therefore, they can be used in parallel computations and on the Internet.     

Analysis of patient self-assessment visual analogue scales

The present paper describes an example of practical analysis of data from an application of visual analogue scales. Potentially useful and simple methods for analysing visual analogue scales are described for effective handling of complex data when the desire is to compare two or more groups of patients. Visual analogue scales were used to assess subjective tolerability in the International Prospective Primary Prevention Study in Hypertension, a large scale clinical trial. The methods are illustrated on the IPPPSH data.     

Combining Trust-Region Techniques and Rosenbrock Methods to Compute Stationary Points

Rosenbrock methods are popular for solving a stiff initial-value problem of ordinary differential equations. One advantage is that there is no need to solve a nonlinear equation at every iteration, as compared with other implicit methods such as backward difference formulas or implicit Runge–Kutta methods. In this article, we introduce a trust-region technique to select the time steps of a second-order Rosenbrock method for a special initial-value problem, namely, a gradient system obtained from an unconstrained optimization problem. The technique is different from the local error approach. Both local and global convergence properties of the new method for solving an equilibrium point of the gradient system are addressed. Finally, some promising numerical results are also presented. This research was supported in part by Grant 2007CB310604 from National Basic Research Program of China, and #DMS-0404537 from the United States National Science Foundation, and Grant #W911NF-05-1-0171 from the United States Army Research Office, and the Research Grant Council of Hong Kong.