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 develop and test a 6th-order Richardson extrapolation scheme as an error estimator for reaction-diffusion problems. Supporting numerical experiments are presented.

A note on Richardson extrapolation as an error estimator for non‐linear reaction–diffusion problems

Transport of a chemical species through a viscous fluid with reaction at an interface is modelled by a quasi-static process in which the transient behaviour is dominated by motion of the reactive surface. The reaction is accompanied by a molar volume change which leads to motion of the free surface. A decoupled stream function-vorticity formulation is introduced in conjunction with a moving finite element method. This also permits convenient treatment of the free and reactive surface boundary conditions. The model is applied to the growth of oxide films on silicon surfaces and reveals the effect of surface curvature on film growth.

Viscous flow and transport with moving free and reactive surfaces

Mapped discretization strategies for curvilinear adaptively redistributed grids in semiconductor device modeling

The PCG package is a software system for solving systems of linear equations by means of preconditioned conjugate gradient (PCG)-type iterative methods on a variety of computer architectures. The software is designed to give high performance with a nearly identical user interface across different scalar, vector and parallel platforms, as well as across different programming models, such as shared-memory, data-parallel and message-passing programming interfaces. The basic features of the package are discussed, as well as techniques used to attain portability and high performance. Representative results for several computers are given. >

PCG: A Software Package for the Iterative Solution of Linear Systems on Scalar, Vector and Parallel Computers.

A new approach is proposed for the a posteriori error estimation of both global spatial and parameter error in parameterized nonlinear reaction–diffusion problems. The technique is based on linear equations relating the linearized spatial and parameter error to the weak residual. Computable local element error indicators are derived for local contributions to the global spatial and parameter error, along with corresponding global error indicators. The effectiveness of the error indicators is demonstrated using model problems for the case of regular points and simple turning points. In addition, a new turning point predictor and adaptive algorithm for accurately computing turning points are introduced. Copyright © 2006 John Wiley & Sons, Ltd.

https://www.researchgate.net/profile/Brian_Carnes/publication/228684559_Estimating_spatial_and_parameter_error_in_parameterized_nonlinear_reaction-diffusion_equations/links/0deec52ed8c436af3a000000.pdf

Graham F. Carey

Papers

A note on Richardson extrapolation as an error estimator for non‐linear reaction–diffusion problems

Viscous flow and transport with moving free and reactive surfaces

Mapped discretization strategies for curvilinear adaptively redistributed grids in semiconductor device modeling

PCG: A Software Package for the Iterative Solution of Linear Systems on Scalar, Vector and Parallel Computers.

Estimating spatial and parameter error in parameterized nonlinear reaction–diffusion equations