Category Archives: news

For a Few Equations More

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.

A system of linear equations (under matrix form).

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.

The Inteprid supercomputer, equipped with 164000 cores © Argonne National Laboratory

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.

Celebrating the Centenary of James H. Wilkinson’s Birth

by Sven Hammarling and Nick Higham


September 27, 2019 is the 100th anniversary of the birth of James Hardy Wilkinson—the renowned numerical analyst who died in 1986. We are marking this special anniversary year in several ways:

The tag wilkinson lists all the posts in this series.

Jack Dongarra Awarded SIAM/ACM Prize in Computational Science and Engineering

Congratulations to Jack Dongarra who recently received the SIAM/ACM Prize in Computational Science and Engineering.

Jack Dongarra will receive the SIAM/ACM Prize in Computational Science and Engineering at the SIAM Conference on Computational Science and Engineering (CSE19) held February 25 – March 1, 2019 in Spokane, Washington. He will receive the award and deliver his prize lecture, “The Singular Value Decomposition: Anatomy of an Algorithm, Optimizing for Performance,” on February 28, 2019.

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.

Professor Jack Dongarra

Professor Jack Dongarra

Numerical Algorithms for High-Performance Computational Science, Royal Society, London, April 8-9, 2019

This 2-day scientific discussion meeting is being held at the Royal Society, London, UK, and is organized by Jack Dongarra, Laura Grigori and Nick Higham.

The programme consists of invited talks and contributed posters. The invited speakers are

  1. Guillaume Aupy, INRIA Bordeaux
  2. Erin Carson, Charles University, Prague
  3. George Constantinides, Imperial College
  4. Steve Furber, FRS, University of Manchester
  5. Mike Heroux, Sandia National Laboratories
  6. Tony Hey, Science and Technology Facilities Council
  7. David Keyes, King Abdullah University of Science and Technology, Saudi Arabia
  8. Doug Kothe, Director of the DOE Exascale Computing Project (ECP)
  9. Satoshi Matsuoka, RIKEN Center for Computational Science, Tokyo
  10. Tim Palmer, FRS, Oxford University
  11. Jack Poulson, Hodge Star Scientific Computing, Toronto
  12. Anna Scaife, University of Manchester
  13. John Shalf, Lawrence Berkeley National Laboratory
  14. Rick Stevens, Argonne National Laboratory
  15. Michela Taufer, University of Delaware
  16. Kathy Yelick, Lawrence Berkeley National Laboratory,

The deadline for submission of poster abstracts is Friday March 1, 2019 (extended from Monday 4 February 2019).

Attendance is free, but places are limited and advance registration is essential.

For more information, including the programme and registration, see

Research Associate in Numerical Linear Algebra Group – 2 posts

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.


Beyer Chair in Applied Mathematics

The School of Mathematics is seeking to appoint an outstanding mathematical scientist to the Beyer Chair in Applied Mathematics.

The Beyer Chair is a senior professorial position in Applied Mathematics, established in 1881 by an endowment from the industrialist Charles Frederick Beyer.

Applicants will have a distinguished track record of research in one or more areas of Applied Mathematics, defined in its broadest sense, and will play a leading role in the life of the School, by providing inspiring leadership in research, and also through appropriate teaching and service activities.

The closing date is March 22, 2019. For advert and more details see here.

NLA Group Talks at SIAM Conference on Computational Science and Engineering 2019

Members of the Numerical Linear Algebra Group will be giving nine presentations at the upcoming SIAM Computational Science and Engineering (CSE) conference. They are also organizing the two-part minisymposium Advances in Analyzing Floating-point Errors in Computational Science.

The conference will be held at the Spokane Convention Center, Washington, USA, from 25th February to 1st March, 2019.

Here are the dates and times where members of our group will be giving their talks:

Tuesday 26 February
9:45 – 10:05 Sven Hammarling
Standardization of the Batched Blas

Wednesday 27 February
9:45 – 10:05 Theo Mary
A New Approach to Probabilistic Roundoff Error Analysis
14:15 – 14:35 Pierre Blanchard
Algorithm Based Error Analysis for Mixed Precision Matrix Factorizations

Thursday 28 February
8:45 – 9:15 Jack Dongarra
SIAM/ACM Prize in Computational Science and Engineering: The Singular Value Decomposition: Anatomy of an Algorithm, Optimizing for Performance
9:45 – 10:05 Françoise Tisseur
NLEVP: A Collection of Nonlinear Eigenvalue Problems 
10:35-10:55 Steven Elsworth
The Rational Krylov Toolbox
14:15 – 14:35 Nick Higham
Exploiting Half Precision Arithmetic in Solving Ax=b
15:05 – 15:25 Jack Dongarra
Experiments with Mixed Precision Algorithms in Linear Algebra
16:10 – 16:30 Mawussi Zounon
Distributed Tasking in the PLASMA Numerical Library

Friday 1 March
9:45 – 11:25 Srikara Pranesh
Domain Decomposition Method for High Dimensional Stochastic Systems

More information on CSE19 is available here.

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