Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for this type of system. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.

This review concerns the computation of curvature-dependent interface motion governed by geometric partial differential equations. The canonical problem of mean curvature flow is that of finding a surface which evolves so that, at every point on the surface, the normal velocity is given by the mean curvature. In recent years the interest in geometric PDEs involving curvature has burgeoned. Examples of applications are, amongst others, the motion of grain boundaries in alloys, phase transitions and image processing. The methods of analysis, discretization and numerical analysis depend on how the surface is represented. The simplest approach is when the surface is a graph over a base domain. This is an example of a sharp interface approach which, in the general parametric approach, involves seeking a parametrization of the surface over a base surface, such as a sphere. On the other hand an interface can be represented implicitly as a level surface of a function, and this idea gives rise to the so-called level set method. Another implicit approach is the phase field method, which approximates the interface by a zero level set of a phase field satisfying a PDE depending on a new parameter. Each approach has its own advantages and disadvantages. In the article we describe the mathematical formulations of these approaches and their discretizations. Algorithms are set out for each approach, convergence results are given and are supported by computational results and numerous graphical figures. Besides mean curvature flow, the topics of anisotropy and the higher order geometric PDEs for Willmore flow and surface diffusion are covered.

Random matrix theory is now a big subject with applications in many disciplines of science, engineering and finance. This article is a survey specifically oriented towards the needs and interests of a numerical analyst. This survey includes some original material not found anywhere else. We include the important mathematics which is a very modern development, as well as the computational software that is transforming the theory into useful practice.

Recent developments in numerical methods for solving large differentiable nonlinear optimization problems are reviewed. State-of-the-art algorithms for solving unconstrained, bound-constrained, linearly constrained and non-linearly constrained problems are discussed. As well as important conceptual advances and theoretical aspects, emphasis is also placed on more practical issues, such as software availability.

We present the field of computational chemistry from the standpoint of numerical analysis. We introduce the most commonly used models and comment on their applicability. We briefly outline the results of mathematical analysis and then mostly concentrate on the main issues raised by numerical simulations. A special emphasis is laid on recent results in numerical analysis, recent developments of new methods and challenging open issues.

In convection-diffusion problems, transport processes dominate while diffusion effects are confined to a relatively small part of the domain. This state of affairs means that one cannot rely on the formal ellipticity of the differential operator to ensure the convergence of standard numerical algorithms. Thus new ideas and approaches are required.

The survey begins by examining the asymptotic nature of solutions to stationary convection-diffusion problems. This provides a suitable framework for the understanding of these solutions and the difficulties that numerical techniques will face. Various numerical methods expressly designed for convection-diffusion problems are then presented and extensively discussed. These include finite difference and finite element methods and the use of special meshes.

We review level set methods and the related techniques that are common in many PDE-based image models. Many of these techniques involve minimizing the total variation of the solution and admit regularizations on the curvature of its level sets. We examine the scope of these techniques in image science, in particular in image segmentation, interpolation, and decomposition, and introduce some relevant level set techniques that are useful for this class of applications. Many of the standard problems are formulated as variational models. We observe increasing synergistic progression of new tools and ideas between the inverse problem community and the ‘imagers’. We show that image science demands multi-disciplinary knowledge and flexible, but still robust methods. That is why the level set method and total variation methods have become thriving techniques in this field.

Our goal is to survey recently developed techniques in various fields of research that are relevant to diverse objectives in image science. We begin by reviewing some typical PDE-based applications in image processing. In typical PDE methods, images are assumed to be continuous functions sampled on a grid. We will show that these methods all share a common feature, which is the emphasis on processing the level lines of the underlying image. The importance of level lines has been known for some time. See, e.g., Alvarez, Guichard, Morel and Lions (1993). This feature places our slightly general definition of the level set method for image science in context. In Section 2 we describe the building blocks of a typical level set method in the continuum setting. Each important task that we need to do is formulated as the solution to certain PDEs. Then, in Section 3, we briefly describe the finite difference methods developed to construct approximate solutions to these PDEs. Some approaches to interpolation into small subdomains of an image are reviewed in Section 4. In Section 5 we describe the Chan–Vese segmentation algorithm and two new fast implementation methods. Finally, in Section 6, we describe some new techniques developed in the level set community.