Abstract
The Newsletter is a key element in further enhancing the value of the journal to the inverse problems community. So why not be a part of this exciting forum by sending to our Bristol office material suitable for inclusion under any of the categories mentioned above. Your contributions will be very welcome.
Forthcoming event
Conference announcement
2nd International Conference on Identification in Engineering Systems, University of Wales Swansea, 29 - 31 March 1999
Parameter estimation and system identification are used extensively to obtain dynamic models of engineering systems. A large number of methods have been developed and a huge amount of experience gained in their application, particularly in control engineering and structural dynamics. Although aspects of the methods are different there is a substantial overlap in the methodology and practice used in the identification. Following the success of the first conference held in March 1996 in Swansea, this conference will provide a forum for researchers and practitioners in the art and science of identification from a range of disciplines and provide a further impetus to the cross-fertilization of ideas in this area. Further details, including the procedure for the submission of abstracts, are available at URL:
http://www.swan.ac.uk/mecheng/ies99/
or contact Dr M I Friswell (e-mail: m.i.friswell@swansea.ac.uk, fax: +44 (0)1792 295676).
Book reviews
75 Years of Radon Transform
S Gindikin and P Michor (ed)
1994 Boston, MA: International 303 pp ISBN 1-57146-008-X $42.00
In 1917 Johann Radon published his now celebrated formula for reconstructing a function (in many applications, this means the density of an object) from line integrals through it in different directions [1]. This paper now stands out as one of the most fundamental contributions of all time to the subject of inverse problems. It inspired much fundamental work in mathematics, such as a notable series of papers by Fritz John in the 1930s. However, the inventors of computerized tomography (CT) for medical imaging could not trace any of this when they needed exactly this inversion in the 1960s. The full significance of Radon's pioneering contribution did not become universally recognized until the 1970s.
In 1917, early on in his career, Johann Radon was an assistant and a Privatdozent at the University of Vienna. He would later spend 25 years at various German universities before finishing his career as Dean, and later Rector, at his alma mater, the University of Vienna. In 1992, the 75th anniversary of the transform was celebrated at the University of Vienna with a conference. This book forms the proceedings of that conference.
The book has three major parts:
Biographical contributions. This part contains, among other items, a biography written by Radon's daughter Brigitte Burkovics, reminiscences by one of his students and by Fritz John, and a description by Allen Cormack about how he approached the line-integral inversion problem when developing CT.
Scientific contributions. This part constitutes the bulk of the book; it contains selected presentations from the technical sessions of the conference - not easy reading for the mathematically faint-hearted.
Reprinted papers. We find here some notes by the Editor of the volume, followed by a now classical (1938) contribution by Fritz John, and the original 1917 paper by Radon.
One thing the book does not contain (in line with the mathematical theme of the conference) is any summary of how the transform - in its many forms of implementation - is now used throughout a vast array of applications.
For a general scientific audience, the first part contains fascinating historical reading, and the third part provides for pleasant casual browsing (as well as for in-depth study). The second part is strictly for specialists in the mathematical aspects of the Radon transform.
Reference
[1] Radon J 1917 Über die Bestimmung von Functionen durch ihre integralwerte längs gewisser Mannigfaltigkeiten Ber. Sächsische Akad. der Wissenschaften, Leipzig, Math.-Nat. Kl. 69 262 - 77
B Fornberg University of Colorado, Boulder
Inverse Problems of Wave Propagation and Diffraction Proceedings, Aix-les-Bains, France 1996, Lecture Notes in Physics, Volume 486
G Chavent and P C Sabatier (ed)
1997 Berlin: Springer 379 pp ISBN 3-540-62865-7 DM106.00, öS774.00, sFr96.50, £ 41.00, $79.00
This book is a collection of lectures that were presented at the `Conference on Inverse Problems of Wave Propagation and Diffraction'. This meeting was organized by two renowned French experts in inverse problems, Guy Chavent and Pierre C Sabatier, in Aix-les-Bains in September 1996.
What impressed me about this book was the clear orientation of the vast majority of the papers towards applications. The range of these applications is quite broad. In terms of fields of study, it varies from `electromagnetic' to `acoustic' inverse problems. Applications naturally dictate a broad variety of mathematical models as well as the appropriate mathematical tools. Regarding the models, the majority of authors consider the frequency domain regime, thus dealing with the Helmholz-like equation. Some authors, however, also work with the inverse problems for the hyperbolic (i.e. time dependent) equations. The goal of almost all the papers is clear and can be appreciated: to develop and test effective numerical methods for the inverse problems under consideration.
The problems studied here can mainly be split into two categories, which have rather blurry boundaries: inverse obstacle problems and inverse coefficient problems. In the first class, one is recovering the shape and location of an obstacle given scattered data. Often, this is an impenetrable obstacle. In the second class, the goal is to find an internal structure of a penetrable object, not only its shape. As a result, the second class of problems amounts to the determination of an unknown coefficient(s) of a PDE. An interesting observation is that the majority of authors discussing the inverse obstacle problems are profoundly influenced by, I would say, a classic book by D Colton and R Kress: Inverse Acoustic and Electromagnetic Scattering Theory.
A point which is made, implicitly or explicitly, in almost all the papers is the importance of reasonable a priori constraints on the solution of an inverse problem. While a wide variety of numerical approaches is discussed (given the broad scope of inverse problems under consideration), this is perhaps the only point which is common to all the approaches. This falls well into the fundamental Tikhonov's principle of an a priori choice of an appropriate compact set of solutions.
Here are some examples of these a priori constraints. The paper by D Colton describes a fresh idea to recover only support of a penetrable anomaly rather than the value of the unknown coefficient within it. In this way the original nonlinear problem is reduced to a linear one, which can be solved in an elegant way using the fact that the fundamental solution of the Helmholz-like PDE has a singularity at the source location. The paper by F Natterer represents another example of a novel and effective numerical approach which was developed using the concept of a priori constraints on solutions. An intruiging core idea of this paper is to solve a classical ill-posed Cauchy problem for the elliptic equation in a well-posed fashion, assuming a priori that the grid size should not be too small. This, in turn, leads to a rapid image reconstruction algorithm, in 3D, though only in the high frequency regime.
While the works of Colton, Natterer and some other authors consider the cases of
full view illumination and measurements of the target medium, which is quite
acceptable in medical imaging, other authors consider the limited view case, which
adds up to the ill-posedness of the problems under consideration. This is certainly
true for a number of works devoted to inverse seismic problems, which are
notoriously challenging. The difficulty of these problems is clearly demonstrated in
the paper by L Fatone, P Maponi, C Pignotti and F Zirilli, in which the 1D inverse
problem for the 2D (in space) hyperbolic equation is studied. Probably the major
challenge, even in the 1D case, consists of figuring out discontinuities of the
speed 
) in the layered medium, given
measurements of the backscattered data on the surface. Although the paper by A
Litman, D Lesselier and F Santosa deals with an electromagnetic, rather than with a
seismic, inverse problem, it still works with the incomplete data collection. The
goal of these authors is to reconstruct the shape and location of a defect given
electrical parameters of this object and incomplete view measurements of the
scattered electrical field. This problem is solved by introducing a level set
function which describes the shapes of a family of defects (including the correct
one) and which satisfies a certain Hamilton - Jacobi-type PDE. This PDE, in turn, is
connected with a least-squares cost functional.
The tutorial paper by M Bertero, P Boccacci and M Piana stays somewhat outside of
the common scope of the other papers. Nevertheless, this is a very interesting work.
It clarifies a rather ambiguous issue of the achievable resolution R versus
the wavelength 
. The common view is that
. The authors show, however, that this is
true only in the far-field data. Contrary to this, in the case of the near-field
data, resolution can be very much less than the wavelength, because of the
evanescent waves, whose intensity decays exponentially with the distance from the
source.
It is impossible for me to comment on all the papers included in this book. My overall comment, however, is that each of the works presented in this book is interesting in its own right, and I enjoyed reading all of them, thanks to the contributors and the editors. In my opinion, this book represents a state of the art (as of 1996) collection of numerical and some theoretical approaches to inverse problems of wave propagation.
M V Klibanov University of North Carolina, Charlotte
Inverse Nodal Problems: Finding the Potential from Nodal Lines Memoirs of the American Mathematical Society, Number 572
O H Hald and J R McLaughlin
1996 Providence, RI: American Mathematical Society 148 pp ISBN 0-8218-0486-3 £ 30.00
This book describes new and sometimes unexpected properties of an eigenvalue problem for an elliptic equation, in potential form, on a rectangle with Dirichlet boundary conditions. First the Sturm - Liouville problem for a Laplacian is considered, and eigenvalues (and thus eigenfunctions) can be easily found. The authors studied the behaviour of eigenvalues and proved that for most rectangles almost all eigenvalues are separated from other eigenvalues by a specified gap; they also found other useful features.
Next the authors describe properties of the same problem with a sufficiently smooth potential function, using perturbation theory. The main result is the proof that the potential is uniquely determined, up to an additive constant, by a subset of the nodal lines of the eigenfunctions. The authors present a formula that yields an approximation to the potential at a dense set of points.
The book is interesting to read; it could be recommended to all specialists in mathematical physics and inverse problems.
A Yagola Moscow State University
Formulas in Inverse and Ill-Posed Problems
Yu E Anikonov
1997 Utrecht: VSP 240 pp ISBN 90-6764-216-9 £ 82.00
The rather unusual title of this book is actually quite apt; it is a collection of representation formulae for a wide variety of inverse problems, but primarily those of evolution type.
Lest the reader assume that the holy grail has at last been found, I should point
out that not all of the formulae give explicit representations; indeed for most of
the inverse problems that are inherently nonlinear the representation is an implicit
one. Also, many of the problems are not one of the standards one finds in the
current literature or motivated by practical considerations (at least as far as the
reviewer is aware). The book's first example is illustative: to recover the
coefficient 
in the parabolic equation
, 
given an initial condition 
and boundary values on 
. The overposed data considered are the values of the Fourier transform 
, 
for each 
. While this allows the
coefficient to be (formally) recovered, it is not the sort of data that one expects
to provide in diffusion problems.
Despite this, the book is a mine of information and useful ideas. It would certainly repay anyone interested in a particular inverse problem to examine it. It is not a book one reads but one that is consulted with the understanding that, while it will very likely not contain ready-made answers, there is reasonable hope that it might provide some insight. Proofs of the claims and justification of the formulae are rarely complete, but there is an extensive list of references, although these are certainly not representative of the areas claimed to be covered in the text. To amplify the last remark, more than two thirds of the citations are to the Russian literature; half of these are by the book's author. Of the remaining third, half are to books or articles published before 1975 and only 20% to works of the last ten years. Thus the recent explosion in the inverse problems literature on a worldwide basis over the last decade or so is largely unaddressed.
Who will or should buy this book? As the above remarks indicate, it is not really suitable as a textbook for graduate courses, at least not for programmes in western Europe or the USA. I suspect that the market here is the research libraries, as I don't imagine many individuals will find it to be a must-have item for their bookshelves, and the very high cost per page will not help in this regard.
W Rundell Texas A&M University, College Station
Inverse Problems in Wave Propagation The IMA Volumes in Mathematics and its Applications, Volume 90
G Chavent, G Papanicolaou, P Sacks and W Symes (ed)
1997 Berlin: Springer 498 pp ISBN 0-387-94976-3 DM118.00, öS861.40, sFr104.00, £ 48.50, $96.95
This book is based on the proceedings of a two-week workshop which was an integral part of the 1994 - 1995 IMA programme `Waves and Scattering'. This workshop took place from 6 - 17 March 1995. There are 24 papers. This book represents most applications in inverse wave propagation problems, together with fundamental mathematical investigations of the relation between waves and scatterers. The following contributions are included.
R A Albanese discusses wave propagation issues in medicine and environmental health. The uniqueness of one-dimensional reconstruction for orthogonal and oblique incidence is touched upon. J G Berryman shows that the inverse problem with the data serving as constraints is most easily analysed when it is possible to segment the solution space into regions that are feasible (satisfying all the constraints) and infeasible (violating some of the constraints). The variational structure of three inverse problems has been investigated. R W Brookes and K P Bube investigate the numerical convergence of a finite-difference method in space and time for one-dimensional models in reflection seismology, and discuss the order of convergence in relation to the source wavelet. M D Collins discusses topics in ocean acoustic inverse problems, including remote sensing and localization problems. Efficient techniques are discussed for the solution of the forward problem, including poro-elastic media, and techniques for solving global scale acoustic problems have been adapted. D Colton discusses in general the three-dimensional electromagnetic inverse scattering problem for inhomogeneous objects and in particular an example in medical imaging. A two-dimensional problem is solved numerically with the dual space method. E Croc and Y Dermenjian study the generalized modes in an acoustic strip, which is a simplified model of a seismic experiment, where both source and receiver are situated in a well. G Eskin and J Ralston consider the inverse problems for Schrödinger operators with magnetic and electric potentials. A Faridani shows old and new results in computed tomography, which entails the reconstruction of a function f from line integrals off. Problems of uniqueness, reconstruction formulae for the x-ray transform, filtered backprojection, error estimates and incomplete data problems are treated in detail. D J Foster, R G Keys and D P Schmitt propose a theoretical basis for interpreting amplitude versus offset (AVO) inversion. It defines the background seismic response and characterizes anomalous events by distance from this background. F A Grunbaum and S K Patch investigate how many parameters one can solve for in diffuse tomography. Some bounds of the range of the parameters are given. J G Harris constructs some physically reasonable models of acoustic imaging. In particular, to investigate small surface-breaking cracks, the leaky Rayleigh wave is used in the imaging mechanism. V Isakov analyses the reconstruction of the diffusion and of the principal coefficient of a hyperbolic equation, in particular the diffusion coefficient, by the use of beam solutions and the recovery of discontinuity of the wave speed. V G Khajdukov, V I Kostin and V A Tcheverda consider the r-solution and its applications in a linearized waveform inversion for a layered background. Y Kurylev and A Starkov deal with an inverse boundary problem for a second-order elliptic operator and its nonstationary counterparts. The uniqueness of the reconstruction of a density parameter is proved and a direct procedure of its reconstruction using directional moments is described. Ching-Ju Ashraf Lee and J R McLaughlin solve an inverse nodal problem for a rectangular membrane using the ratio method and the method of parameter identification. Changmei Liu derives a uniqueness theorem for any ball and for two balls, when the scattering amplitudes for some independent incident directions are given. J R McLaughlin, P E Sacks and M Somasundaram discuss the inverse problem of the determination of acoustic parameters using far-field data, by using the properties of the interior eigenvalues, and show how the sound speed depending on the vertical coordinate can be determined. G Hanamura and G Uhlmann describe a layer stripping algorithm in elastic impedance tomography, in which an elastic tensor has to be reconstructed from measurements of the displacement and the traction at the boundary of the domain. G Nolet uses the WKBJ or path integral approximation to the solution of the elastodynamic equations in a slightly heterogeneous earth to partition the inverse problem for a large set of observed seismograms along different wavepaths. R L Nowack discusses an inverse method for the analysis of refraction and wide-angle seismic data and illustrates it through the inversion of a shallow crustal structure. F R Pijpers presents a brief overview of applications of inversions within astronomy, including a recent modification of the Backus and Gilbert method. E L Ritman, J H Dunsmuir, A Faridani, D V Finch, K T Smith and P J Thomas present an example in which local reconstruction extends the capability of a micro-CT scanner beyond the physical limits imposed by global tomographic reconstruction techniques. J Sylvester discusses the layer stripping problem. Some theorems are cited as evidence that it provides a productive theoretical method which yields new insights into an old problem. M E Taylor studies estimates for approximate solutions to acoustic inverse scattering problems, in particular the recovery of the near-field wave from the scattering amplitude and the consequences of linearization of the inverse problems.
This book is certainly an important contribution to the theory and application of inverse wave propagation problems and is a good reference work that should be present in the library.
P M van den Berg Delft University of Technology