We describe the burgeoning field of numerical relativity, which aims to solve Einstein's equations of general relativity numerically. The field presents many questions that may interest numerical analysts, especially problems related to nonlinear partial differential equations: elliptic systems, hyperbolic systems, and mixed systems. There are many novel features, such as dealing with boundaries when black holes are excised from the computational domain, or how even to pose the problem computationally when the coordinates must be determined during the evolution from initial data. The most important unsolved problem is that there is no known general 3-dimensional algorithm that can evolve Einstein's equations with black holes that is stable. This review is meant to be an introduction that will enable numerical analysts and other computational scientists to enter the field. No previous knowledge of special or general relativity is assumed.

We consider the efficient evaluation of accurate radiation boundary conditions for time domain simulations of wave propagation on unbounded spatial domains. This issue has long been a primary stumbling block for the reliable solution of this important class of problems. In recent years, a number of new approaches have been introduced which have radically changed the situation. These include methods for the fast evaluation of the exact nonlocal operators in special geometries, novel sponge layers with reflectionless interfaces, and improved techniques for applying sequences of approximate conditions to higher order. For the primary isotropic, constant coefficient equations of wave theory, these new developments provide an essentially complete solution of the numerical radiation condition problem.

In this article we review the image reconstruction algorithms used in tomography. We restrict ourselves to the standard problems in the reconstruction of function from line or plane integrals as they occur in X-ray tomography, nuclear medicine, magnetic resonance imaging, and electron microscopy. Nonstandard situations, such as incomplete data, unknown orientations, local tomography, and discrete tomography are not dealt with. Nor do we treat nonlinear tomographic techniques such as impedance, ultrasound, and near-infrared imaging.

In this survey we discuss various approximation-theoretic problems that arise in the multilayer feedforward perceptron (MLP) model in neural networks. The MLP model is one of the more popular and practical of the many neural network models. Mathematically it is also one of the simpler models. Nonetheless the mathematics of this model is not well understood, and many of these problems are approximation-theoretic in character. Most of the research we will discuss is of very recent vintage. We will report on what has been done and on various unanswered questions. We will not be presenting practical (algorithmic) methods. We will, however, be exploring the capabilities and limitations of this model.

This paper aims to give an overview and summary of numerical methods for the solution of stochastic differential equations. It covers discrete time strong and weak approximation methods that are suitable for different applications. A range of approaches and results is discussed within a unified framework. On the one hand, these methods can be interpreted as generalizing the well-developed theory on numerical analysis for deterministic ordinary differential equations. On the other hand they highlight the specific stochastic nature of the equations. In some cases these methods lead to completely new and challenging problems.

There is more to the computation of pseudospectra than the obvious algorithm of computing singular value decompositions on a grid and sending the results to a contour plotter. Other methods may be hundreds of times faster. The state of the art is reviewed, with emphasis on methods for dense matrices, and a Matlab code is given.