where f is a real-valued function of the s-dimensional vector \underline{x} and C^s denotes the s-dimensional hypercube [0,1]^s. For effective lattice methods the value of s is assumed to be somewhere between 10 and 20. Assumptions are also made on f; it must be periodic of period 1 in each variable and sufficiently smooth. Although periodicity sounds very restrictive it is shown how, by suitable non-linear transformations, a non-periodic integrand may be pushed into this form.
Such integrals arise in a variety of contexts; the authors mention atomic physics, quantum chemistry and statistical mechanics. There are doubtless others where such integrals need to be evaluated. Although one dimensional quadrature formulae have been with us for centuries, significant progress on evaluating multiple integrals had to await the advent of the modern computer. The use of Monte Carlo methods, for example, to evaluate multiple integrals dates back to the late 1940s. The study of lattice methods, which are the subject of this book, began in about 1960 and, after a bit of a lapse, was rejuvenated in the last decade due, in no small part, to the efforts of the senior author, Professor I.~H. Sloan. The basic approach is readily stated. Let us recall that in one-dimension the N-point rectangle rule is given simply by
Research monographs, written by authors who have contributed substantially to the subject, often tend to be technically formidable. One has the feeling that such authors write primarily for the cognoscente . The present authors, bearing in mind that their audience will comprise chemists, physicists and statisticians, have not fallen into this trap. They have written for such an audience and have taken great pains to ensure that their mathematical arguments are readily understood. For example, in Chapter 3, in case the reader has forgotten, they devote 8 pages to introducing some elementary group theory, with illustrative examples taken from the lattice rules already introduced for the simplest two-dimensional case. Both the authors' enthusiasm for the topic and their eagerness to help the reader share this enthusiasm, come through repeatedly.
The book is divided into eleven chapters together with three appendices. The first three chapters give a gentle introduction to the topic with examples for the case when s = 2. The next six chapters introduce the reader to rank 1 rules, then to lattice rules of higher rank and, in particular, maximal rank lattice rules. There is a substantial chapter on non-periodic integrands and the last two chapters attempt to justify all the analysis that has gone before by first of all discussing the practical implementation of lattice rules and then comparing them with other methods for evaluating such integrals. The comparison of various integration rules is fraught with difficulties. A rule which looks impressive for one sort of integrand might be quite hopeless with another. The authors have tested 4 different multiple integration rules against a package of 6 different integrands each having distinct characteristics. The 4 multiple integration packages are called:
The authors' conclusion is that, based on various criteria, the two most successful methods are COPY and ADAPT and the authors conclude the book with the modest claim, ``... the numerical results presented here suggest that 2^s copy lattice rules as implemented in COPY make a useful addition to the armoury of methods available for multiple integration''.
This book is a timely addition to the literature on quadrature methods for
multiple integration. No one who is even remotely concerned with this
problem should fail to purchase this book. It is one of the best
mathematical ``reads'' that this reviewer has come across for a long time and
the authors are to be congratulated on such a splendid production which will
bring readers up to date with this subject.
The University of Tasmania
INVERSE PROBLEMS IN THE MATHEMATICAL SCIENCES
Mathematically, the formal concept of an inverse problem (or, alternatively, an improperly posed or an ill-posed problem) is quite new, though the processes it aims to model are as old as civilization itself, as encapsulated in the above quotation. For example, it was a slow and challenging process to establish that our Earth is not the centre of our Solar System or the Universe. In addition, it was the measurement of the travel times of seismic waves which led Lord Kelvin to first conclude that our Earth was basically solid, and, subsequently, Inge Lehman to discover that it had a liquid outer core and inner solid core. The earlier view (before Kelvin), inferred from the volcanic evidence, was of an Earth consisting of a solid crust on a molten interior.
It is well-known that scientific discovery is base on the ``Hypothesis-and-Test'' (or, equivalently, the trial-and-error) ansatz. But the knowledge which such discovery has accumulated about the world around us, is often taken for granted. It is either forgotten or not appreciated how, as a result of long, arduous and often dangerous endeavours by numerous individuals, such knowledge has been accumulated. By first observing some regular pattern of events (such as the oscillatory nature of the seasons as well as the repeating pattern of day and night), and then using this information to elucidate, through an evolving trial-and-error process, knowledge about the underlying phenomenon was uncovered. Examples include the discovery of the helical structure of DNA, and the fact that our Earth rotates on an axis which is not vertical to the plane of the elliptical orbit on which it moves about the sun.
The same basic process of discovery continues today. Though the essence has not changed, the framework in which it is performed has. In the light of the knowledge which has already been accumulated and the ways in which that knowledge has been applied and developed, one now has a situation where sophisticated instrumentation and computers can be used to collect and analyse, along with the assistance of statistics and mathematical modelling, the carefully verified observations. Microscopic and astronomic data and patterns, as well as the more traditional macroscopic, can be used as input to the trial-and-error process which is now performed with greater and greater finesse.
Within this overall process, including the trial-and-error manipulations, an ever increasingly important role is being played by mathematics, especially in terms of its own increasing sophistication. There are various ways in which this can be explained, but, in one way or another, they relate back to questions connected with the analysis of indirect measurements where some effect of the phenomenon of interest is all that is observed . In such situations, mathematics can be used to model the relationship between the (indirect) measurements performed on the phenomenon of interest and the properties which define and characterise the phenomenon under investigation.
Often, it is only necessary to solve the forward problem of putting an assumed structure of the phenomenon into the chosen model to determine the corresponding effect. In other situations, however, it is necessary to start with specific measurements performed on the phenomenon and use them, in conjunction with the chosen model, to recover information about the structure and status of the particular phenomenon under investigation. For obvious reasons, such formulations are often referred to as inverse problems .
In general, inverse problems arise naturally in any situation where conclusions must be derived from indirectly obtained information (measurements). This occurs in all aspects of problem solving including, for example, forensic science, detective work, and decision making, as well as situations where direct observation is impossible (structure of the Earth and Sun), an unrealistic alternative (unstructured systematic searching in geophysical exploration), or something which is to be avoided whenever possible (surgery for purely diagnostic purposes; destructive testing in engineering). For such reasons, inverse problems are often characterized as indirect measurement problems , though in fact they define a much larger class of problems.
The consequences associated with using indirect measurements to recover the required information is reflected in the structure of the mathematics which model inverse problems. This structure formalizes how the properties of primary interest have been modified and smoothed to produce the indirect measurements. Intuitively, it characterizes the extent to which the required information about the phenomenon is retained in the indirect measurements. Formally, the equations which model inverse problems fail at least one of Hadamard's three conditions for a problem to be properly posed which are:
Consequently, inverse problems (and hence, indirect measurement problems) are often classified mathematically as improperly posed problems .
Thus, from scientific, industrial and mathematical points of view, inverse problems are too important to be shrouded in mystery. This book of Charles (Chuck) Groetsch goes a long way to assist with introducing the mathematics of inverse problems to a wider audience. Among other things, it adopts a point of view similar to that outlined above and goes on to discuss the history and the mathematical framework in some detail. In addition, it clearly shows the important role that mathematics has and will continue to play in the solution of practical and theoretical inverse problems.
After a short Introduction which consists, in part, of a brief historical survey, Chapter 2 is devoted to a survey and analysis of applications which reduce to the solution of integral equations of the first kind. It accounts for about one third of the book. It starts with a brief introduction to the concept of a first kind integral equation and ends with a brief summary of some key properties. As applications which reduce to the solution of a first kind integral equation, the author gives 19 examples which include the hanging cable, geological prospecting, pressure gauges, vibrating strings, irrigation, temperature probes, immunology, spectrocopy, tomography and radiotherapy. It is interesting that many reduce, in one way or another, to the solution of an Abel integral equation. Chapter 3 examines model identification problems for differential equations. It includes examples from compartmental modelling, hydraulics, structures and diffusion. Together, these two Chapters account for half of the book and contain, in terms of the wide spectrum of examples treated, a wealth of information about the practical nature of inverse problems. They form the core and the strength of the book.
In Chapter 4, the author introduces some basic mathematical concepts connected with function spaces and operator theory, so that the mathematical treatment of inverse problems, which follows in Chapter 5, can be put on a formal footing. Chapter 5 is devoted to an examination of many of the methods proposed for the solution of inverse problems. It contains a detailed and useful discussion of regularization. The book concludes, in Chapter 6, with an Annotated Bibliography which adds greatly to its value as a source book for a general audience.
It has been written for students and faculty with a basic knowledge of integral and differential equations, though some of the examples (such as the exponential growth model) could be used as material to motivate undergraduate students about the nature of inverse problems. The book could be used as an introduction to applied functional analysis and operator theory for advanced mathematics undergraduate students, or engineering and science postgraduate students. However, as explained in the Preface, it was written more for the teachers (and advanced students), rather than the undergradute students who would like to explore on their own.
It was a pleasure to read such a well written book, though the treatment
is a little uneven in that the definition of a first kind Fredholm
equation comes before the examples. There are some minor typos, like
Stirling's approximation for n! on page 103. However, overall, it is
an excellent book for what it aims to achieve, and is strongly
recommended to all who would like to know more about inverse problems.
Division of Mathematics and Statistics
CSIRO
Canberra
KAC ALGEBRAS AND DUALITY OF LOCALLY COMPACT GROUPS
The theory of duality for the category of locally compact abelian groups is one of the great triumphs of modern harmonic analysis: the pairing G \leftrightarrow \widehat{G} = Hom ( G , {T} ) gives is an exact correspondence between the category of locally compact abelian groups and its dual category, which is also the category of locally compact abelian groups. In the 1940s a great deal of effort was put into finding analogous results for the category of non abelian locally compact groups. However, the results of Tannaka, Krein and Stinespring were unable to match the beauty and scope of the abelian (Pontryagin) theory.
In the 1960s it was suggested by Kac, and later by Takesaki that the right category to study might be a category of Hopf algebras which contained the category of locally compact groups. For technical reasons, the most suitable candidate appears to be a special category of Hopf algebras called Kac algebras. Precisely, a Kac algebra K is a co-involutive Hopf-von Neumann algebra with a Haar weight. Roughly speaking K is a quadruple (M ,\Gamma , \kappa , \varphi ) where M is a von Neumann algebra, \Gamma: M \rightarrow M \otimes M is a co-product, \kappa : M \rightarrow M is a co-involution, and \varphi is a faithful, semi-finite normal weight on the positive elements of M which is left co-invariant. In the locally compact group case we obtain an abelian Kac algebra where M = L^\infty (G), which has coproduct \Gamma (f) (s,t) = f(st), co-involution \kappa (f) (s) = f ( s^{-1} ) and Haar weight \varphi (f) = \int f(s) . ds, where f \in L^\infty (G) and ds is Haar measure on G.
The book begins by developing the abstract theory of Kac algebras K, and their representation theory. The notion of a dual Kac algebra \widehat{ K} is introduced, and it is shown that the second dual \widehat{\widehat{ K} is isomorphic to the original Kac algebra K. The class of Kac algebras is then given the structure of a category; duality and equivalence functors are given between certain subcategories of Kac algebras and the category of locally compact groups. In the final chapter consequences of this theory are explored in the context of unimodular, compact, discrete and finite-dimensional Kac algebras; in particular the duality results of Krein are shown to be a special case of this new approach.
The theory of Kac algebras has now reached the stage where an
overview is required. This book provides a rich bibliography
charting the development from the outset, which makes it
accessible to readers with a only a modest understanding of von
Neumann algebras. A suggested improvement might be the
inculsion of an index of notation, so that one may pick a way
through the many symbols used more easily. Sadly the theory is
not yet sufficiently developed to properly handle Quantum Groups,
hopefully this can be addressed in the future. I recommend this
book to any researcher who is interested in learning more about
the theory of Kac algebras and their links with duality theory for
locally compact groups.
The University of Newcastle