Numerical Linear Algebra Group

Dr Stefan Güttel, Professor Nick Higham, and Professor Françoise Tisseur have been awarded a new 30-month project with Arup, a multidisciplinary engineering firm operating in all areas of built environment.

This Knowledge Transfer Partnership (KTP), funded by Arup and Innovate UK, aims to embed new matrix eigenvalue solvers into Arup’s next generation software for structural engineering simulation.

The Numerical Linear Algebra  (NLA) group team will be working with Dr Stephen Hendry and Dr Ramaseshan Kannan of Arup, along with a KTP Associate, for which the position is advertised here.

The project builds on a long history of collaboration between Arup and the NLA Group, which has  previously led to the development of “model stability analysis” in Arup’s flagship structural engineering simulation package, Oasys GSA (see the paper What is Your Structural Model Not Telling You?).

The November 2019 edition of SIAM News contains an article by Research Associate Srikara Pranesh about the growing importance of low precision floating-point arithmetic. Sri describes the opportunities provided by recent hardware and explains how new algorithms are being derived to exploit low precision arithmetic.  To read the article click the image below.

Also see Sri’s recent blog post Simulating Low Precision Floating-Point Arithmetics.

Photo provided by @ICIAMnews

Professor Nick Higham delivered an invited talk at the International Congress on Industrial and Applied Mathematics (ICIAM) on July 19, 2019, in Valencia, Spain. His talk “Exploiting Low Precision Arithmetic in the Solution of Linear Systems” reported work by Higham and his colleagues over the last three years to use the fast half precision arithmetic available on accelerators such as GPUs to speed up the solution of linear systems.  A video of the talk is available here and the slides are available here.

The talk was summarized in the Friday 19th July edition of the ICIAM2019 newsletter:

“Nick Higham (University of Manchester) advocated in his invited lecture using arithmetics of different precision at different stages of computations in order to design algorithms that are faster, require less communications and consume less energy. The main motivation is that last-generation GPUs may be up to eight times faster when they perform arithmetic operations in half precision than when they do in single precision.

The general philosophy is doing the bulk of the computations in half precision, and then perform some kind of clean-up refinement of the solution in higher precisions. As an archetypal example to illustrate this line of thought Higham chose how to accelerate the solution of linear systems via Gaussian elimination. He showed that if the LU factorization is computed in half precision, followed by iterative refinement using a mix of half, double and quadruple precision, the solution can be sped up significantly without accuracy loss. This works in principle for systems with moderate condition number, but even ill-conditioned systems can be dealt with by performing the iterative refinement via GMRES on a suitably pre-conditioned system.”

Photo credit Françoise Tisseur

Professor Françoise Tisseur delivered the Olga Taussky-Todd Lecture at the International Congress on Industrial and Applied Mathematics (ICIAM) on July 15, 2019, in Valencia, Spain. The honour of giving this lecture is conferred every four years on a woman who has made outstanding contributions in applied mathematics and/or scientific computation.

Her talk “Challenges in the numerical solution of nonlinear eigenvalue problems” (slides available here in PDF form) concerned eigenvalue problems defined in terms of matrix-valued functions of a scalar parameter, which occur in many areas of science and engineering. She described recent advances in numerical methods for solving such problems, which include methods for approximating by a rational or polynomial matrix function and methods for linearizing rational and polynomial eigenproblems. Pitfalls that the methods must overcome were illustrated through simple but realistic practical examples.

Her talk followed the opening ceremony, which included traditional Valencian dancing and an address to the approximately 4000 attendees by King Felipe VI of Spain (video clip), who spoke of the good health of Spanish mathematics and the importance of mathematics for driving innovation.

Congratulations to Massimiliano Fasi for being awarded his PhD, which was supervised by Nick Higham. We asked him a few questions about his thesis, title Computing Matrix Functions in Arbitrary Precision Arithmetic.

Please can you introduce yourself and tell us a bit about your experience before attending University of Manchester?
I was born in Assisi, a smallish town in central Italy chiefly known for having been home to Saint Francis. I thought I would become a philologist, and during my teen years I studied mostly linguistics and ancient languages, while taking plenty of courses at the local music conservatory. After much pondering, I decided to start my university career with a scientific discipline and I chose computer science. I found that I really enjoyed the subject, and after graduating from the University of Perugia I pursued a Master’s degree at the University of Bologna and, at the same time, one at the École Normale Supérieure of Lyon.

What is your thesis on?
My thesis investigates the computation of matrix functions in arbitrary precision arithmetic, and is in fact a collection of results that are somewhat connected to this topic. We started by revisiting several well-known techniques for two very general problems: the evaluation of rational functions at a matrix argument and the solution of rational matrix equations. We developed new numerical methods and provided some theoretical insights, and used these as building blocks to design multiprecision algorithms to evaluate matrix functions that are of interest in applications. We focused mainly on the matrix exponential and the matrix logarithm, but the machinery we developed is more general and can be used for a much broader range of problems.

Why did you choose to study your PhD in Manchester?
Numerical analysis was my favourite module as an undergraduate student. I decided to do an internship with the lecturer teaching that course, and the subject happened to be the matrix Lambert W function. I’ve been interested in matrix functions ever since, and when the time came to choose where to do a PhD, I thought that it would be great to join the strongest linear algebra group I was aware of. I was lucky enough to be admitted at The University of Manchester, where I could do research with Professor Nick Higham, who is a leading expert in this field.

How did you find Manchester?
Manchester is a lively city and is well connected to the rest of the world. The postgraduate community I was part of is very dynamic, and somewhat larger than I was used to. A PhD can be rather difficult at times, but the many people I had the chance to meet here made my Mancunian experience unforgettable.

What’s your next step?
My long-term plan is to remain in academia, and I will try to follow the path that leads to a permanent position. For the moment, I am a research associate in the group here in Manchester, and am trying my best to deepen my understanding of the current trends in numerical linear algebra.

Massimiliano Fasi

Professor Nick Higham has been awarded the London Mathematics Society’s Naylor Prize and Lectureship for his leadership in numerical linear algebra, numerical stability analysis, and communication of mathematics.

The Naylor Prize and Lectureship is awarded every two years in memory of Dr V. D. Naylor. The grounds for the award may include work in, and influence on, and contributions to applied mathematics and/or the applications of mathematics, and lecturing gifts.

The full prize citation is available here.

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.

Further details are given in An Updated Set of Nonlinear Eigenvalue Problems. The collection will grow and contributions are welcome.

The following table shows the 22 new problems in version 4.0 of the toolbox .