# The 2016 ANZIAM Medal

**Detailed Citation for Frank Robert de Hoog’s Award of the ANZIAM Medal**

**(The full citation to be published in the AustMS Gazette)**

Frank Robert de Hoog (or simply Frank to colleagues and friends) epitomises a worthy recipient of the ANZIAM medal because his contributions to applied, computational and industrial mathematical research are nationally and internationally famous, while his contributions to ANZIAM have had a significant impact especially in the development of the student support scheme.

**Contributions to Applied, Computational and Industrial Mathematics Research**

Frank de Hoog commenced his studies at the University of Western Australia in 1966 and graduated with first class honours in Mathematics in 1970. It was clear to his lecturers that he was gifted mathematically. During this period, his interest in applied and computational mathematics was greatly influenced by Professor J. J. Mahoney FAA, Dr Neville Fowkes and Dr Jack Williams who introduced him to applied and computational mathematics. In fact, he started a PhD at the University of Western Australia on the numerical solution of ordinary differential equations under the supervision of Dr Williams. However, with Dr Williams’ return to the UK, Frank accepted a PhD scholarship at the Australian National University to work with Professor M. R. Osborne FAA on the numerical solution of ordinary differential equations. Because of the work that Richard Weiss was pursuing under the supervision of Bob Anderssen, Frank switched his focus to the numerical solution of integral equations. In hindsight, this change proved to be very important because of the success with which he, in collaboration with Richard Weiss, solved a number of important open problems.

As a direct result of the high regard that the research he performed as a PhD student established for him internationally, he was offered an Assistant Professorship in Mathematics at UCLA. With Richard Weiss at CALTEC, their research turned to the very challenging problem of the numerical analysis of singular ordinary differential equation. The resulting publications, because they resolved key issues that others working in the area had failed to resolve, immediately established international reputations for both as numerical analysts, and opened doors to an appointment for Richard Weiss in Vienna and to a return to the ANU for Frank de Hoog. Before accepting his position at CSIRO’s Division of Mathematics and Statistics, he returned to earlier interests in integral equations and commenced his collaboration with his PhD student John Paine and Bob Anderssen on the numerical determination of the eigenvalues of ordinary differential equations. The result that a simple algebraic correction formula could account for the differences between the algebraic estimates and the actual differential eigenvalues was a startling new result, the proof of which involved technically challenging and complex considerations. It has greatly influenced the subsequent design of algorithms for the computation of differential eigenvalues.

It was the significance, depth and breadth of this research which led to Frank being awarded the Australian Mathematical Society’s Medal in 1988.

It was Frank's move to CSIRO that restimulated his strong interest in applied and industrial mathematics. From that point on, his research has had a very strong applied and industrial mathematical, as well as computational, impact. The success of that research has been recognized by his fairly rapid promotion to a Chief Research Scientist position on the basis of scientific merit, and his recent appointment as CSIRO Fellow. The impressive fact about this research is its phenomenal breadth, ranging over a very broad spectrum of subjects in both applied and computational mathematics, as well as its non-trivial depth and penetrating mathematical insight. There are many highlights in the research that Frank has undertaken during his more than 35 years with the CSIRO. Some, but not all, significant examples include:

(i) Laplace Transform Inversion. For many constant coefficient differential equations which arise in applications, it is relatively simple to determine the Laplace transform of the solution. The numerical inversion of such transforms is a very popular topic in numerical analysis and, thereby, a very challenging area in which to make a contribution that now has a science citation index score of more than 593. In part, this relates to the fact that Frank's contribution is the basis for standard algorithms in numerical analysis libraries including IMSL.

(ii) Smoothing Spline Optimization. The major limitation of the early methods for data smoothing with splines was the complexity of the computation of the regularization using cross-validation. In this research, it was shown how to reduce the complexity from an order *n*-squared activity to an order *n* deliberation. It is now the basis for all standard algorithms for smoothing spline fitting to observational data.

(iii) Fast Methods for Toeplitz Matrices. Various authors have proposed ways to exploit the special structure of a Toeplitz matrix in order to derive fast algorithms. It was a very competitive subject that required deep mathematical understanding and insight to identify an alternative strategy which improved on the earlier alternatives. Though other methods were proposed, it was the first stable numerical method for positive definite Toeplitz matrices.

(iv) Mineral Separation. In designing an industrial device, it is not only a matter of engineering but of matching the engineering to the operational parameters of the device. Dr de Hoog developed a mathematical model for the Kelsey Centrifugal Jig of Geologics which allowed such matching to be performed. As a result of this collaboration, these jigs are now used around the world to perform mineral separation that was not possible in the past. These jigs are responsible for the recovery of minerals with a value in excess of half a billion dollars a year. (Further details are available at http://www.rochemt.com.au/11285.htm )

(v) Sheet Metal Rolling. The efficient production of sheet metal is a highly competitive situation. The faster the rolling can be performed, the higher the profits. However, the faster the rolling the more sophisticated must be the algorithms controlling the operation of the rolling. The work that Dr de Hoog and colleagues have accomplished has given BHP a competitive edge internationally.

(vi) Mode Coupling in the Vibration of Beams and Shells. The mode coupling that occurs in beams and shells is quite complex, and the earlier publications that aimed to explain this phenomenon were complex and cumbersome. The importance of this work is that it identified the key issues involved in a way that yielded a clear simple explanation.

(vii) Conditioning and Dichotomy of Boundary Value Problems. This research established for the first time that there are uncoupled boundary conditions that have similar conditioning to the most general situations. It thereby laid a theoretical framework for the formulation of robust algorithms for two-point boundary value problems.

**Contributions to Industrial Mathematics Study Groups**

In addition to the above contributions, it is important to acknowledge that Frank played a key role, along with others, in the successful establishment of the Mathematics-in-Industry Study Group Meetings, which are now an important part of the R&D image of ANZIAM. The first Study Group Meeting was organized by the then CSIRO Division of Mathematics and Statistics, with Noel Barton and Frank playing lead roles along with strong support from Kerry Landman and Terry Speed. The Study Group’s successful implementation would not have been achieved with the speed and impact that it has if it were not for the dedicated and insightful commitment of Frank, Noel, Kerry and Terry, as well as others.

In many ways, one of Frank’s great personal characteristics, which is the basis for his highly successful collaborative research endeavours, is his understanding, fair and unselfish approach to colleagues. An example is his work on the winding of coils, which arose initially through the Mathematics-in-Industry Study Group framework. In collaboration with CSIRO and industrial colleagues, he developed and applied theory which established explicitly the winding stress required to achieve specified residual stresses in the wound coils. As an immediate consequence of such endeavours, the damage due to excess stress was minimized while the stability of the wound coil was maximized. These results have become the basis for the processing of coils in both the aluminium (Comalco) and steel industries (BHP/Blue Scope), and contributed directly to the cost-efficiency with which aluminium and steel sheets are now rolled and coiled.

**Contributions to ANZIAM**

Frank has been a regular attendee since 1976 invariably giving a talk. He was a key organizer for the meeting in Merimbula in 1984.

As an extension of his strong support for students in CSIRO, in terms of student support schemes and internship programs, Frank garnered financial support for ANZIAM from CSIRO on the understanding that the funds would be exclusively used to support student participation at ANZIAM. This has turned into the CSIRO-ANZIAM Student Support Scheme and has become a feature of AustMS conferences as well as those of ANZIAM.

The Selection Committee unanimously agreed that, for all these contributions, Frank de Hoog be awarded the ANZIAM Medal for 2016.

Dr Robert Anderssen

Professor Robert McKibbin

Professor Kerry Landman