Machine Learning is the study of methods for programming computers to learn. Computers are applied to a wide range of tasks, and for most of these it is relatively easy for programmers to design and implement the necessary software. However, there are many tasks for which this is difficult or impossible. These can be divided into four general categories. First, there are problems for which there exist no human experts. For example, in modern automated manufacturing facilities, there is a need to predict machine failures before they occur by analyzing sensor readings. Because the machines are new, there are no human experts who can be interviewed by a programmer to provide the knowledge necessary to build a computer system. A machine learning system can study recorded data and subsequent machine failures and learn prediction rules. Second, there are problems where human experts exist, but where they are unable to explain their expertise. This is the case in many perceptual tasks, such as speech recognition, hand-writing recognition, and natural language understanding. Virtually all humans exhibit expert-level abilities on these tasks, but none of them can describe the detailed steps that they follow as they perform them. Fortunately, humans can provide machines with examples of the inputs and correct outputs for these tasks, so machine learning algorithms can learn to map the inputs to the outputs. Third, there are problems where phenomena are changing rapidly. In finance, for example, people would like to predict the future behavior of the stock market, of consumer purchases, or of exchange rates. These behaviors change frequently, so that even if a programmer could construct a good predictive computer program, it would need to be rewritten frequently. A learning program can relieve the programmer of this burden by constantly modifying and tuning a set of learned prediction rules. Fourth, there are applications that need to be customized for each computer user separately. Consider, for example, a program to filter unwanted electronic mail messages. Different users will need different filters. It is unreasonable to expect each user to program his or her own rules, and it is infeasible to provide every user with a software engineer to keep the rules up-to-date. A machine learning system can learn which mail messages the user rejects and maintain the filtering rules automatically. Machine learning addresses many of the same research questions as the fields of statistics, data mining, and psychology, but with differences of emphasis. Statistics focuses on understanding the phenomena that have generated the data, often with the goal of testing different hypotheses about those phenomena. Data mining seeks to find patterns in the data that are understandable by people. Psychological studies of human learning aspire to understand the mechanisms underlying the various learning behaviors exhibited by people (concept learning, skill acquisition, strategy change, etc.).

Machine learning

The organization of behavior: A neuropsychological theory

Introduction to linear and nonlinear programming

This is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation. In this paper, a general recovery technique is developed for determining the derivatives (stresses) of the finite element solutions at nodes. The implementation of the recovery technique is simple and cost effective. The technique has been tested for a group of widely used linear, quadratic and cubic elements for both one and two dimensional problems. Numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent. One order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements. In particular, an O(h4) convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time. The performance of the proposed technique is compared with the widely used smoothing procedure of global L2 projection and other methods. It is found that the derivatives recovered at interelement nodes, by using L2 projection, are also superconvergent for linear elements but not for quadratic elements. Numerical experiments on the convergence of the recovered solutions in the energy norm are also presented. Higher rates of convergence are again observed. The results presented in this part of the paper indicate clearly that a new, powerful and economical process is now available which should supersede the currently used post-processing procedures applied in most codes.

The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique

The authors develop finite difference methods for elliptic equations of the form \[ \nabla \cdot (\beta (x)\nabla u(x)) + \kappa (x)u(x) = f(x)\] in a region $\Omega $ in one or two space dimension...

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources

A class of vector-parallel schemes for solution of steady compressible or incompressible viscous flow is developed and performance studies carried out The algorithms employ an artificial transient treatment that permits rapid integration to a steady state In the present work a four-stage explicit Runge-Kutta scheme employing variable local step size is utilized for the ODE system integration The RK-4 scheme is restructured to allow vectorization and enhance concurrency in the calculation for a streamfunction-vorticity formulation of the flow problem The parameters of the resulting RK scheme can be selected to accelerate convergence of the RK recursion Four main procedures are considered which permit vector-parallel solution: a Jacobi update, a hybrid of the Jacobi and Gauss-Seidel method, red-black ordering and domain decomposition Numerical performance studies are conducted with a representative viscous incompressible flow calculation Results indicate that a scheme involving domain decomposition with a Gauss-Seidel type of update for the RK four-stage scheme is most effective and provides performance in excess of 8 Gflops on the Cray C-90

A vector-parallel scheme for Navier-Stokes computations at multi-gigaflop performance rates

The evolution of finite element programs and program systems for engineering analysis and the more recent pioneering development of expert systems are briefly described.1 From these we proceed to develop some general principles and guidelines using expert systems with finite element techniques and novel computer architectures.

Toward expert systems in finite element analysis

Finite element analysis of mass transport through a viscous fluid with reaction

DISCUSSION The reduced integration penalty finite element method has been frequently used to compute approximate velocity solutions to Navier- Stokes problems.’-4 The velocity approximation can be post-processed to obtain an approximate pressure solution. Numerical studies for Stokes have shown that for problems where the data (boundary conditions or the forcing function) are ‘rough’, these computed pressures may exhibit oscillations for certain elements and reduced integration schemes, whereas smoother pressures are obtained for other (‘stable’) elements. An example of such a problem is the familiar ‘driven cavity’ problem. For this problem, both in the case of Stokes flow and low Reynolds number Navier-Stokes flow, when the 4-node bilinear element with 1-point Gauss integration of the penalty term is used, the pressures computed from the velocity field are found to be oscillatory, whereas when the 9-node biquadratic element with 1point integration of the penalty term is used, smooth pressure profiles are obtained. Recently, we have been investigating the use of continuation techniques and iterative methods for computing approximate solutions at higher Reynolds numbers. On examining the computed pressures for the driven cavity we observed that as the Reynolds number increased, the local pressure oscillations for the bilinear element diminish and by Re = 2000 the pressure profiles at representative sections appear smooth. As an example, results are now given for the cavity problem and bilinear I-point element for calculations on a uniform mesh of size h = 1/32 and with penalty parameter E = The pressure profile at Re = 10 along the section y = 17/64 is given in Figure 1 (marked + ) and is seen to contain local pressure oscillations. The projected pressure obtained by averaging the four element pressures adjacent to a node is also given and is smooth (marked x ). Analogous results are also observed for Stokes flow (Re = 0). However, the computed pressure profile at J’ = 17/64 for Re = 2000 appears smooth (Figure 2).+ Similar behaviour is observed at other representative sections and is not shown here. Results with penalty parameter E = and E = Approximate solutions were also computed using the mixed method with Co biquadratic velocity and Co bilinear pressure on uniform meshes with h = 1/16 and h = 1/25. This element is known to be stable. The pressure solution is smooth as anticipated and qualitatively agrees with that obtained with the penalty method. However, quantitatively the pressure profiles agree were essentially the same.

On the computed pressures for navier-stokes problems at increasing reynolds numbers using the penalty finite element method

Preconditioning strategies based on incomplete LU factorization using thresholding with dual dropping (ILUT) are investigated for iterative solution of sparse linear systems arising in semiconductor dopant diffusion modelling. Of particular interest are questions associated with selection and adaption of threshold parameters with spatial resolution, timestep in the adaptive ODE integrator and the problem physics. We investigate these issues and carry out detailed numerical studies in one- and two-dimensions for a representative phosphorus diffusion model. Guidelines for optimally selecting the threshold parameters are deduced from the results, and strategies for adaptive parameter selection are presented. Copyright © 2002 John Wiley & Sons, Ltd.

Graham F. Carey

Papers

A vector-parallel scheme for Navier-Stokes computations at multi-gigaflop performance rates

Toward expert systems in finite element analysis

Finite element analysis of mass transport through a viscous fluid with reaction

On the computed pressures for navier-stokes problems at increasing reynolds numbers using the penalty finite element method

Performance of adaptive dual-dropping ILUT preconditioners in semiconductor dopant diffusion simulation