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

We present a parallel finite element solver for mesh-connected multicomputers (ie., message-passing, distributed-memory multiprocessors) and provide a theoretical proof tht the algorithms used are highly scalable and close to optimal. Further, the algorithms are written in an object-oriented language (ESP C++) to enhance reusability of code. Performance results are presented for a Motorola multicomputer and a network of Sun workstations.

A scalable, object-oriented finite element solver for partial differential equations on multicomputers

Irregular hybrid meshes may excessively distort the node-dual finite-volume discretization. A new scheme is formulated that uses a different type of polygonal control volume. Superior stability of the polygonal scheme over the conventional node-dual scheme is demonstrated on representative irregular hybrid meshes for incompressible viscous flow past a circular cylinder. Copyright © 2006 John Wiley & Sons, Ltd.

An enhanced polygonal finite-volume method for unstructured hybrid meshes

The present investigation considers element block preconditioners for discretizations employing elements of high degree. Several different element block preconditioners are constructed and compared in numerical studies for a model elliptic problem. For regular grids a 2-level preconditioning scheme is found to work quite well. This consists of a local element preconditioner based on transformation of polynomial basis functions, combined with a simple global preconditioner such as Jacobi diagonal type preconditioning. For grids with non-constant coefficients or highly skewed grids, the block preconditioners described here can be applied instead of Jacobi preconditioning.

Graham F. Carey

Papers

A scalable, object-oriented finite element solver for partial differential equations on multicomputers

An enhanced polygonal finite-volume method for unstructured hybrid meshes

Preconditioners for high degree elements

A distributed memory parallel element-by-element scheme for semiconductor device simulation

Maximizing Sparse Matrix-Vector Product Performance on RISC Based MIMD Computers