# Weidlinger Associates

1984 – 2010

In 1984 I moved to Northern California, where I took a permanent job with Weidlinger Associates. Around 1985 I initiated the Geophysical Inversion Project that would be the umbrella under which most of my future research and development at Weidlinger Associates would be done. With financing from the SBIR programs at NSF and DOE we started a comprehensive attack on the problem of modeling 3D complex geological formations with Oil Exploration as our target, but also with extensive applications to Earthquake Seismology, Geothermal Energy and other Earth Sciences disciplines. The program required non-government contributions and tangible technological output. We were pioneers in forming an Industry Consortium of Petroleum and Computer companies to provide guidance, focus, and early private financing for this project. Then, later on, when some mature software had been produced, we teamed with Don Larson and Bob Limbaugh from Geoquest International, which changed to Geoquest Technology and more recently to GXT, to bring exploration geophysics software products to the market place. After five years of successful marketing, more than 150 licenses of 3D modeling products (3D AIMS) were installed worldwide, and a multimillion dollar business was created.

My last publication on deferred correction per se and a personal view of the historical developments in the field appeared in:

Then a series of computational geophysics papers started to appear. Three of them show us again at the forefront of technological innovation, by signaling early adoption of microcomputer hardware:

*“Microcomputer-Based Two-Point Ray-Tracing Algorithms for 2-D Media”, with G. Wojcik, Geophysics, 50:1195 (1985)*

The last one contains our first attempt to conduct serious geological modeling, and it stemmed from consulting work with Weidlinger and a business relationship with Kim Tech, a Denver, CO, geophysical software firm.

In our continuous collaboration with José Rial we used an earlier implementation of our 3D modeling and ray tracing code on a real seismological application resulting in:

An expository presentation and an essay were written around this time:

Then came a long list of ray tracing, modeling and inversion papers:

*“Two Point Ray Tracing in Complex 3-D Media”. Exp. Abst. SEG Annual Meeting. II: 1056-1060 (1988)*

*“Direct and Inverse Modeling Three-Dimensional Complex Geology”. J. of Comp. Math. 7: 182-192, Beijing, China (1989)*

This last paper was the result of a very interesting trip to mainland China, invited by the Academia Sinica, and marvelously hosted by Professor Lin Qun. We visited Beijin, Xian, Chengdu, Guangsho, and upon exit from China, Hong Kong.

The next works represent my collaboration with Steve Wright in various aspects of common interest:

In 1989, and as a sub-product of the technology we had been developing, we started a project to simulate the ultrasound inspection of nuclear reactor parts. The project was initially financed by the Electric Power Research Institute, and it was lead by Jeremy Isenberg. Mathew Koshy first and then Laura Carcione and Hua Song were the main developers. A number of products were created and are being marketed internationally.

Another sign of early adoption of winning technological strategies is exemplified by:

*“Distributed Computing Applications in Forward and Inverse Geophysical Modeling”, Expanded Abstracts SEG 61 Annual Meeting, Houston, Texas. Vol. I, pp. 349-352 (1991)*

Together with J. Black from IBM, this was the first public application of Beowulf clusters to exploration geophysics. Twenty years later this is about the only valid parallel technology in the market.

Some of this work was commented upon in January 1991, on a long article that the New York Times devoted to D. Gelertner, the inventor of the language LINDA that we were using for distributed computing (pre PVM, MPI, etc.). It might have been this notoriety that moved the Unabomber to send a package bomb that almost killed Professor Gelertner some time later.

For a little while, this 13 million elements full elastic wave propagation simulation held the record for a large calculation. It took six hours on a Fujitsu VT100 machine at a sustained 0.5 Gigaflop rate.

With the sponsorship of the NSF SBIR program we undertook a very ambitious project to put the fusion of multiple data sets on a solid footing. We used as a basic tool concepts of multi-objective optimization and produced two reports:

*“Multi-Objective Inversion (MOI) User’s Guide”, with D. Woods, and M. Koshy. Weidlinger Associates Geophysical Inversion Project, IV:167-202 (1991)*

*“Numerical Algorithms for Cooperative Imaging of Complex Geological Regions”, with D. Woods. Final Report to NSF of Phase II project (1993)*

Further work is required to make these ideas practical, but the next technical developments have been strongly influenced by part of the work we did in fusion. Later on we will also make some interesting contributions to the calculation of Pareto fronts.

This work extends the use of SVD based Levenberg-Marquardt type nonlinear least squares methods to large dimensional problems. It combines ideas of multigrid, domain decomposition, local model parametrization, and chaotic iterations, to produce a distributed computing approach to solve this type of problems. It also opens the way to the application of global optimization techniques to large dimensional problems.

Through the years we have demonstrated this technology in many different systems, as a portable implementation of a three-dimensional tomographic inversion that can be applied to large-scale realistic industrial problems.

An expository paper was written by invitation:

Of course, although we have concentrated on elastic ray tracing, everything holds true for acoustic ray tracing as we have shown in:

Further contributions to Oil Exploration are contained in:

*“Raytrace Modeling of a Blocky Version of the SEG Salt Model”. With T. Kunz. SEG 66 Annual Meeting, Houston. Extended Abstracts (1996)*

*“Parallel, Asynchronous Global Optimization Techniques for Large Inversion Problems”. Invited paper, IV International Conference on Approximation and Optimization. Caracas, Venezuela (1997)*

We started to apply optimization techniques to other problems in Engineering, which resulted in some publications:

*“Characterization of Piezoelectric Materials by Non-Linear Least Squares Inversion of Finite Element Calculations”. With D. Powell, G. Wojcik and S. Ayter. US Navy Workshop on Acoustic Transduction Materials and Devices, Penn State (2000)*

This work has now been consolidated within a NSF-SBIR sponsored project entitled “A Toolbox for Optimal Design“, which aims to link Weidlinger large scale finite element and seismic ray tracing simulation programs in a toolbox format, in order to facilitate the rapid deployment of simulation based optimization techniques to other applications.

We have also experimented with distributed computing across the Internet. The following paper describes our experience in solving one of the previous problems by combining a finite element solver in California with an optimization code in Illinois.

The next interesting pieces of work provide tools to fit scattered data on irregular regions with holes, discontinuities and noise, using tensor product B-splines:

With Professor Gene Golub we undertook a revision of the evolution and applications of our 1973 paper on separable least squares. We were pleasantly surprised to see the wide impact this work has had in a number of important application fields, such as: chemistry, mechanical systems, neural networks, parameter estimation and approximation, telecommunications, electrical and electronic engineering, dynamical systems, environmental sciences, robotics and vision, and medical and biological imaging. As a result we wrote an invited review paper:

This paper was the most down-loaded paper of the Inverse Problems Journal for two years on a row, although it could be argued that it was not in the main stream of research associated with that journal. One of the most important and widespread applications of this technology has been to Nuclear Magnetic Resonance Spectroscopy. There is a project at the Universidad Autónoma de Barcelona, Spain, which maintains a tool based on VARPRO that is used by more than 780 research groups in 53 countries to perform routinely time-domain analysis of in vivo NMR data.

The Research Project that I initiated in 2003 has helped extend and consolidate a number of previous developments in Optimization and Large Scale Modeling. This project was sponsored by NSF and the Air Force and resulted in a number of publications and commercial software.

This last paper presents our own implementation of a fast Neural Network training algorithm based on VARPRO that is used, for instance, to produce inexpensive surrogates of large scale simulations. These techniques have been successfully employed in the following work on aircraft design optimization.

In the next two papers we retake our original work on Multiobjective Optimization with the aim of making it practical for large scale expensive functionals and constraints. Much inspiration and impetus came from the work on Aircraft Design Optimization that I carried with my colleagues of the Stanford Aeronautics Department.

The next paper continues our work on collocation for PDE’s, but now using radial functions. This work incorporates successfully our solution of ill-conditioned least squares problems and even uses VARPRO, the separable nonlinear least squares problem solver, to provide an adaptive algorithm.

In connection with the use of VARPRO in MRI Spectroscopy George Fleming, a physicist from Yale wrote a paper calling the attention of his colleagues to its possible application to problems in Quantum Chromodynamics. Later on we started a collaboration that has produced a couple of papers:

Some work on fast solutions of the wave equation can be found in:

I have organized several International Conferences through the years and edited a number of their Proceedings :

Numerical Methods, Lecture Notes in Mathematics 1005 (with A. Reinoza). Proceedings of a Meeting held in Caracas, Venezuela, June 14-18 (1982). Springer-Verlag, Berlin (1982)

Special Issue in Computational Geophysics. Applied Numerical Mathematics 4 (1988)

Special Issue: Applied and Computational Topics in Partial Differential Equations (with J. Castillo). Papers from the Second Panamerican Workshop in Applied and Numerical Mathematics, held in Gramado, Brazil, 8-12 September 1997. Journal of Computational and Applied Mathematics 103 and Applied Numerical Mathematics 30 (1999)

Special Issue on Applied and Computational Mathematics (with J. Castillo). Papers from the Third Panamerican Workshop in Applied and Computational Mathematics, held in Trujillo. Peru, 24-28 April 2000. Applied Numerical Mathematics 40 (2002)

Special Issue on Applied and Computational Mathematics (with J. Castillo). Papers from the Fourth Panamerican Workshop in Applied and Computational Mathematics, held in Córdoba, Argentina, 1-5 July 2002. Applied Numerical Mathematics 47 (2003)

Special Issue on Applied and Computational Mathematics (with J. Castillo). Papers from the Fifth Panamerican Workshop in Applied and Computational Mathematics, held in Tegucigalpa, Honduras, 21-25 June 2004. Mathematics and Computers in Simulation 73, Issues 1-4 (2006)

Special Issue on Applied and Computational Mathematics (with G. Scherer and J. Castillo). Papers from the Sixth Panamerican Workshop in Applied and Computational Mathematics, held in Huatulco, Mexico 23-28 July, 2006. Mathematics and Computers in Simulation 79, Issue 6 (2009)

Special Issue on Applied and Computational Mathematics (with G. Scherer and J. Castillo). Papers from the Seventh Panamerican Workshop in Applied and Computational Mathematics, held in Choroni, Venezuela June 6-11, 2010. Mathematical and Computer Modelling Vol. 57, Numbers 9-10 (2013)

In the realm of community service I was an Editor for SIAM Journals and SEG Geophysics for many years. I am currently Senior Editor of The Journal of Computational Acoustics, Editor of Applied Numerical Mathematics, Matematica Aplicada e Computacional, and International Journal of Computing Sciences and Mathematics.

I have also been a Consulting Professor of Computer Sciences at Stanford University during many years and since 1997 I am also an Adjoint Professor of Mathematics at San Diego State University. I was a member of the Computer Sciences Doctorate Commission of the Instituto Tecnólogico de Buenos Aires, Argentina for 4 years, starting in 2005.

June 24th, 2009 I retired officially from Weidlinger Associates, although they kindly let me use my office, and I am Principal Emeritus. In 2016 Weidlinger merged with Thornton-Tomasetti and it is now a 1200 strong predominantly civil engineering firm with offices around the world.