Since Arieh Iserles founded this book series in 1992, the goal was to publish one volume every year in which high quality survey papers are collected, written by specialists and only on invitation by the editorial board. The subjects in one volume cover different domains related to numerical analysis and scientific computing. Theory as well as practice of computational applied mathematics for applications in science and engineering are covered. In this way the collection forms a sliding window over the hot topics in the domain and after 20 years some evolution in the rise and decline of important subdomains become visible. The series has gained its well deserved status of an inevitable set on the shelves of any applied mathematical library that wants to keep their readers abreast with the state of the art. Perhaps electronic access is even better (another non-numerical change of attitude among researchers). For example the printed book has no colors, while the online version does.
The current volume fits the purpose of the series very well. There are seven papers ranging from about 80 to about 150 pages, worth a thick volume of 816 pages. Here is a brief summary.
Communication lower bounds and optimal algorithms for numerical linear algebra G. Ballard and E. Carson and J. Demmel and M. Hoemmen and N. Knight and O. Schwartz
With faster hardware on multi-core computers, the number of flops (floating point operations) is not a good criterion anymore to qualify the efficiency of a numerical algorithm. The bottleneck has become the communication between processors. This paper analyzes algorithms in linear algebra in view of this new paradigm. With successive BLAS versions, the problem was recognized already a while ago but communication optimal algorithms still allow for drastic improvement, even on the simplest operations like matrix multiplication. Algorithms ranging from direct methods for dense matrices to iterative methods for sparse matrices are analyzed in view of their communication cost.
Mathematical analysis of variational isogeometric methods L. Beirão da Veiga and A. Buffa and G. Sangalli and R. Vázquez
This surveys the theory of recently developed isogeometric approaches to solve partial differential equations (PDE). These methods allow the integration of finite elements and NURBS or spline-like methods. The paper focusses on T-splines and on elliptic and saddle point problems. T-splines were developed in the last decade for CAD applications. They can be made locally adaptive by adding T-points. At such a point, the control net shows a T-shaped connection. These points allow to reduce the number of control points, and can fit the pieces better in difficult regions, but book keeping of subdivision schemes will become more involved.
Topological pattern recognition for point cloud data Gunnar Carlsson
Persistent homology is a method used here to extract certain topological patterns from a point cloud. Here "persistent" refers to features at different resolution levels so that global, rather than local artifacts are recovered. Computation of homology on a simplicial complex and topological spaces is explained. Then signatures or `barcodes' are defined using finite metric spaces when these are considered as being sampled from a continuous space. Several examples are given where various versions of this technique can be used.
Numerical methods for kinetic equations G. Dimarco and L. Pareschi
Kinetic PDE describe the time evolution of a large set of particles. An overview of the current state of the art of numerical methods to solve these equations is given. Basic numerical techniques such as semi-Lagrangian methods (for transport equations), discrete velocity models (velocity discretization of Boltzmann equation and BGK models), and spectral methods are discussed. Fast (summation) methods, asymptotic preserving ideas and hybrid schemes are covered.
Stochastic finite element methods for partial differential equations with random input data Max D. Gunzburger and Clayton G. Webster and Guannan Zhang
For these methods, discretizations need an extra level of discretization for the stochastic variables. Several classical techniques exist like Galerkin methods in which the physical and probabilistic degrees of freedom are coupled or where they are uncoupled as in stochastic sampling and interpolatory-type collocation. That covers stochastic finite elements and Galerkin methods, Monte Carlo and stochastic approximation by polynomials and piecewise polynomials. These and other recent methods are reviewed with error estimates and complexity analysis.
Numerical tensor calculus Wolfgang Hackbusch
This is another hot topic in numerical linear algebra. Beyond two or three dimensional grids, linear algebra with matrices often becomes unfeasible because the matrices become too large and the problems can only be handled in the case of sparse matrices. So the challenge is to catch the principal subspaces in the tensor by extending as much as possible from the two dimensional matrix tools to a d-dimensional tensor. Filtering the essential components of a tensor can be obtained for example by a generalization of a singular value decomposition. This generalization is not unique and several effective representation of the tensor information are possible. Numerical computation of these representations and of tensor operations are discussed in this survey.
Numerical methods with controlled dissipation for small-scale dependent shocks Philippe G. LeFloch and Siddhartha Mishra
Modelling and approximation of discontinuous solutions of nonlinear hyperbolic equations with small-scale dependent shock waves is the subject of the last paper. Different classes of problems as well as different numerical methods are surveyed providing a guide to the literature of the last two decades.
Each of the papers is essential reading for the specialist in that particular domain. For every researcher or engineer in modelling and scientific computing, the whole collection is an excellent tool to be informed about what is going on in a neighboring field and/or to be instructed in a particular subject without the need to dig unguidedly through an avalanche of papers. Indeed, a topic of rising importance attracts many papers, and these are often chaotic, exploring dead-ends in different directions, and they are not always of the best quality. All the surveys in this series are written by the specialists which were selected not only because they are authorities, but also because of their expository skills and this shows clearly in the papers.