In this paper we consider the construction, analysis, implementation and application of exponential integrators. The focus will be on two types of stiff problems. The first one is characterized by a Jacobian that possesses eigenvalues with large negative real parts. Parabolic partial differential equations and their spatial discretization are typical examples. The second class consists of highly oscillatory problems with purely imaginary eigenvalues of large modulus. Apart from motivating the construction of exponential integrators for various classes of problems, our main intention in this article is to present the mathematics behind these methods. We will derive error bounds that are independent of stiffness or highest frequencies in the system.Since the implementation of exponential integrators requires the evaluation of the product of a matrix function with a vector, we will briefly discuss some possible approaches as well. The paper concludes with some applications, in which exponential integrators are used.
The subject of inverse problems in differential equations is of enormous practical importance, and has also generated substantial mathematical and computational innovation. Typically some form of regularization is required to ameliorate ill-posed behaviour. In this article we review the Bayesian approach to regularization, developing a function space viewpoint on the subject. This approach allows for a full characterization of all possible solutions, and their relative probabilities, whilst simultaneously forcing significant modelling issues to be addressed in a clear and precise fashion. Although expensive to implement, this approach is starting to lie within the range of the available computational resources in many application areas. It also allows for the quantification of uncertainty and risk, something which is increasingly demanded by these applications. Furthermore, the approach is conceptually important for the understanding of simpler, computationally expedient approaches to inverse problems.
We discuss the finite element approximation of eigenvalue problems associated with compact operators. While the main emphasis is on symmetric problems, some comments are present for non-self-adjoint operators as well. The topics covered include standard Galerkin approximations, non-conforming approximations, and approximation of eigenvalue problems in mixed form. Some applications of the theory are presented and, in particular, the approximation of the Maxwell eigenvalue problem is discussed in detail. The final part tries to introduce the reader to the fascinating setting of differential forms and homological techniques with the description of the Hodge–Laplace eigenvalue problem and its mixed equivalent formulations. Several examples and numerical computations complete the paper, ranging from very basic exercises to more significant applications of the developed theory.
A classical mathematical proof is constructed using pencil and paper. However, there are many ways in which computers may be used in a mathematical proof. But ‘proof by computer’, or even the use of computers in the course of a proof, is not so readily accepted (the December 2008 issue of the Notices of the American Mathematical Society is devoted to formal proofs by computer).In the following we introduce verification methods and discuss how they can assist in achieving a mathematically rigorous result. In particular we emphasize how floating-point arithmetic is used.
The need to evaluate a function f(A) ∈ ℂn×n of a matrix A ∈ ℂn×n arises in a wide and growing number of applications, ranging from the numerical solution of differential equations to measures of the complexity of networks. We give a survey of numerical methods for evaluating matrix functions, along with a brief treatment of the underlying theory and a description of two recent applications. The survey is organized by classes of methods, which are broadly those based on similarity transformations, those employing approximation by polynomial or rational functions, and matrix iterations. Computation of the Fréchet derivative, which is important for condition number estimation, is also treated, along with the problem of computing f(A)b without computing f(A). A summary of available software completes the survey.
Numerical methods are usually constructed for solving mathematical problems such as differential equations or optimization problems. In this contribution we discuss the fact that numerical methods, applied inversely, were also important in establishing these models. We show in detail the discovery of the laws of planetary motion by Kepler and Newton, which stood at the beginning of modern science. The 400th anniversary of the publication of Kepler's laws (1609) is a good occasion for this investigation.
We introduce basic ideas of binary separation by a linear hyperplane, which is a technique exploited in the support vector machine (SVM) concept. This is a decision-making tool for pattern recognition and related problems. We describe a fundamental standard problem (SP) and show how this is used in most existing research to develop a dual-based algorithm for its solution. This algorithm is shown to be deficient in certain aspects, and we develop a new primal-based SQP-like algorithm, which has some interesting features. Most practical SVM problems are not adequately handled by a linear hyperplane. We describe the nonlinear SVM technique, which enables a nonlinear separating surface to be computed, and we propose a new primal algorithm based on the use of low-rank Cholesky factors.It may be, however, that exact separation is not desirable due to the presence of uncertain or mislabelled data. Dealing with this situation is the main challenge in developing suitable algorithms. Existing dual-based algorithms use the idea of L1 penalties, which has merit. We suggest how penalties can be incorporated into a primal-based algorithm. Another aspect of practical SVM problems is often the huge size of the data set, which poses severe challenges both for software package development and for control of ill-conditioning. We illustrate some of these issues with numerical experiments on a range of problems.