In this article we shall present a unified and axiomatized view of several theories and algorithms of image multiscale analysis (and low level vision) which have been developed in the past twenty years. We shall show that under reasonable invariance and assumptions, all image (and shape) analyses can be reduced to a single partial differential equation. In the same way, movie analysis leads to a single parabolic differential equation. We discuss some applications to image segmentation and movie restoration. The experiments show how accurate and invariant the numerical schemes must be and we compare several (old and new) algorithms by discussing how well they match the axiomatic invariance requirements.

Domain decomposition refers to divide and conquer techniques for solving partial differential equations by iteratively solving subproblems defined on smaller subdomains. The principal advantages include enhancement of parallelism and localized treatment of complex and irregular geometries, singularities and anomalous regions. Additionally, domain decomposition can sometimes reduce the computational complexity of the underlying solution method.

In this article, we survey iterative domain decomposition techniques that have been developed in recent years for solving several kinds of partial differential equations, including elliptic, parabolic, and differential systems such as the Stokes problem and mixed formulations of elliptic problems. We focus on describing the salient features of the algorithms and describe them using easy to understand matrix notation. In the case of elliptic problems, we also provide an introduction to the convergence theory, which requires some knowledge of finite element spaces and elementary functional analysis.

This article starts with a brief introduction to neural networks for those unfamiliar with the basic concepts, together with a very brief overview of mathematical approaches to the subject. This is followed by a more detailed look at three areas of research which are of particular interest to numerical analysts.

The first area is approximation theory. If K is a compact set in ℝn, for some n, then it is proved that a semilinear feedforward network with one hidden layer can uniformly approximate any continuous function in C(K) to any required accuracy. A discussion of known results and open questions on the degree of approximation is included. We also consider the relevance of radial basis functions to neural networks.

The second area considered is that of learning algorithms. A detailed analysis of one popular algorithm (the delta rule) will be given, indicating why one implementation leads to a stable numerical process, whereas an initially attractive variant (essentially a form of steepest descent) does not. Similar considerations apply to the backpropagation algorithm. The effect of filtering and other preprocessing of the input data will also be discussed systematically.

Finally some applications of neural networks to numerical computation are considered.

Finite Difference (FD) methods approximate derivatives of a function by local arguments (such as du(x) / dx ≈ (u(x + h) − u(x − h))/2h, where h is a small grid spacing) – these methods are typically designed to be exact for polynomials of low orders. This approach is very reasonable: since the derivative is a local property of a function, it makes little sense (and is costly) to invoke many function values far away from the point of interest.

We consider a system whose state is given by the solution y to a Partial Differential Equation (PDE) of evolution, and which contains control functions, denoted by v.

A classical problem in electrostatics is the determination of the effective electrical conductivity in a composite material consisting of a collection of piecewise homogeneous inclusions embedded in a uniform background. We discuss recently developed fast algorithms for the evaluation of the potential and electrostatic fields induced in multiphase composites by an applied potential, from which the desired effective properties may be easily obtained. The schemes are based on combining a suitable boundary integral equation with the Fast Multipole Method and the GMRES iterative method; the CPU time required grows linearly with the number of points in the discretization of the interface between the inclusions and the background material.

A variety of other questions in electrostatics, magnetostatics and diffusion can be formulated in terms of interface problems. These include the evaluation of electrostatic fields in the presence of dielectric inclusions, the determination of magnetostatic fields in media with variable magnetic permeability, and the calculation of the effective thermal conductivity of a composite material. The methods presented here apply with minor modification to these other situations as well.

The mathematical techniques used within Computer Aided Design software for the representation and calculation of surfaces of objects are described. First the main techniques for dealing with surfaces as computational objects are described, and then the methods for enquiring of such surfaces the properties required for their assessment and manufacture.

This article reviews the application of various notions from the theory of dynamical systems to the analysis of numerical approximation of initial value problems over long-time intervals. Standard error estimates comparing individual trajectories are of no direct use in this context since the error constant typically grows like the exponential of the time interval under consideration.

Instead of comparing trajectories, the effect of discretization on various sets which are invariant under the evolution of the underlying differential equation is studied. Such invariant sets are crucial in determining long-time dynamics. The particular invariant sets which are studied are equilibrium points, together with their unstable manifolds and local phase portraits, periodic solutions, quasi-periodic solutions and strange attractors.

Particular attention is paid to the development of a unified theory and to the development of an existence theory for invariant sets of the underlying differential equation which may be used directly to construct an analogous existence theory (and hence a simple approximation theory) for the numerical method.