A new release, version 4.0, is available of the NLEVP MATLAB toolbox, which provides a collection of nonlinear eigenvalue problems. The toolbox has become a standard tool for testing algorithms for solving nonlinear eigenvalue problems.
When it was originally released in 2008, the toolbox contained 26 problems. The new release contains 74 problems. It is now distributed via GitHub and is available at https://github.com/ftisseur/nlevp.
Last February the SIAM Computational Science and Engineering (CSE19) conference took place in Spokane, WA, USA. We organized a two-part minisymposium on recent Advances in Analyzing Floating-point Errors in Computational Science (see links to part 1 and part 2). Below is the list of the eight talks along with the slides which we have made available.
On the occasion of the Gilles Kahn prize award ceremony, I was asked to write an article about my PhD thesis for the popular science blog Binaire from the French newspaper Le Monde (the French version of the article is available here). You can read the English translation below.
Can you tell what the following problems have in common: predict tomorrow’s weather, build crash-resisting cars, scan the bottom of the oceans searching for oil? These are all difficult problems that are too costly to be tackled physically. Importantly, they can also be described by a fundamental tool of mathematics: linear equations. Therefore, the solution of these physical problems can be numerically simulated by solving instead systems of linear equations.
You probably remember from math class in high school how tedious solving these systems could get, even when they had a small number of equations. In practice, it is actually quite common to face systems with thousands or even millions or equations. While computers can fortunately solve these systems for us, the computational cost of the solution can become very high for such large numbers of equations.
To respond to this need, great quantities of resources and money have been dedicated to the construction of supercomputers of great computational power, equipped with a large number of computing units called “cores”. For example, while your personal computer is likely to have less than a dozen cores, the most powerful supercomputer in the world has several millions of these cores. Nevertheless, the size of the problems that we must tackle today is so great that even these supercomputers are not sufficient.
To take up this challenge, I have worked during my PhD thesis on new algorithms to solve systems of linear equations whose computational cost is greatly reduced. More precisely, a crucial property of these algorithms is that their cost grows slowly with the number of equations: this is referred to as their complexity. Methods of very low complexity (so-called “hierarchical”) have been proposed since the 2000s. However, these hierarchical methods are quite complex and sophisticated, which makes them unable to attain high performance on supercomputers: that is, their high reduction of the theoretical complexity is translated into only very modest gains in terms of actual computing time.
For this reason, my PhD thesis focused on another method (so-called “Block Low Rank”), that is better suited than hierarchical methods for high performance computing. My first achievement was to compute the complexity of this method, which was previously unknown. I proved that, even if its complexity is slightly higher than than of hierarchical methods, it is still low enough to tackle systems of very large dimensions. In the second part of my thesis, I worked on efficiently implementing this method on supercomputers, so as to translate this theoretical reduction into actual time gains.
By significantly reducing the cost of solving systems of linear equations, this work allowed us to solve several physical problems that were previously too large to be tackled. For instance, it took less than an hour to solve a system of 130 million equations arising in a geophysical application, using a supercomputer equipped with 2400 cores.
SIAM and the Association for Computing Machinery (ACM) jointly award the SIAM/ACM Prize in Computational Science and Engineering every two years at the SIAM Conference on Computational Science and Engineering for outstanding contributions to the development and use of mathematical and computational tools and methods for the solution of science and engineering problems. With this award, SIAM and ACM recognize Dongarra for his key role in the development of software and software standards, software repositories, performance and benchmarking software, and in community efforts to prepare for the challenges of exascale computing, especially in adapting linear algebra infrastructure to emerging architectures.
When asked about his research for which the prize was awarded, Dongarra said “I have been involved in the design and development of high performance mathematical software for the past 35 years, especially regarding linear algebra libraries for sequential, parallel, vector, and accelerated computers. Of course, the work that led to this award could not have been achieved without the help, support, collaboration, and interactions of many people over the years. I have had the good fortune of working on a number of high profile projects: in the area of mathematical software, EISPACK, LINPACK, LAPACK, ScaLAPACK, ATLAS and today with PLASMA, MAGMA, and SLATE; community de facto standards such as the BLAS, MPI, and PVM; performance analysis and benchmarking tools such as the PAPI, LINPACK benchmark, the Top500, and HPCG benchmarks; and the software repository netlib, arguably the first open source repository for publicly available mathematical software.”
This article was extracted from SIAM News. Further information is available here.
The Numerical Linear Algebra Group at the University of Manchester is seeking two Research Associates to work with Professor Nick Higham on developing and analyzing numerical linear algebra algorithms for current and future high-performance computers.
Topics for investigation include linear equations, linear least squares problems, eigenvalue problems, the singular value decomposition, correlation matrix problems, and matrix function evaluation. This work will exploit multiprecision arithmetic (particularly the fast half precisions available on some recent and forthcoming processors) and techniques such as acceleration and randomization. It will involve using rounding error analysis, statistical analysis, and numerical experiments to obtain new understanding of algorithm accuracy and efficiency.
One of the posts is associated with Professor Higham’s Royal Society Research Professorship. The other post is associated with the EPSRC project Inference, Computation and Numerics for Insights into Cities (ICONIC), which involves Imperial College (Professor Mark Girolami), the University of Manchester (Professor Nick Higham), the University of Oxford (Professor Mike Giles), and the University of Strathclyde (Professor Des Higham).
The closing date is February 11, 2019. For the advert and more details see here.