Arieh Iserles

Arieh Iserles: Making Computers Respect Geometry

A planet orbiting a star conserves its energy exactly, forever. Simulate that orbit on a computer using a standard numerical method and the energy will drift — the planet spiralling slowly into the sun or out into the dark, not because physics says so but because the arithmetic leaks. Arieh Iserles belongs to the group of mathematicians who decided this was unacceptable and set about building algorithms that obey the laws their equations encode.

From Jerusalem to the Cam

Iserles was born on 2 September 1947 and studied at the Hebrew University of Jerusalem and Ben-Gurion University of the Negev. His doctoral work was on numerical methods for stiff ordinary differential equations — a subject that sounds narrow and is anything but.

Stiffness is the pathology that appears whenever a system contains processes running on wildly different timescales: a chemical reaction where one step takes microseconds and another takes hours. Naive numerical methods are forced to crawl at the pace of the fastest process even when nothing interesting is happening there, and the computation becomes unaffordable. Solving stiffness properly was one of the central practical problems of twentieth-century scientific computing, and it is a good training ground for someone who will spend his career on the mismatch between continuous mathematics and discrete machines.

Cambridge, With One Detour

He came to Cambridge as a Junior Research Fellow at King's College in 1978, became a Senior Research Fellow there in 1982, and spent 1986–87 as an associate professor at the University of Arizona. Then he returned and stayed.

He was a University Lecturer at Cambridge from 1987 to 1995, Reader in Numerical Analysis from 1995 to 1999, and since 1999 has been Professor of Numerical Analysis of Differential Equations in the Department of Applied Mathematics and Theoretical Physics — DAMTP, the department Dirac and Hawking worked in. From 2010 to 2015 he directed the Cambridge Centre for Analysis.

Geometric Integration

The central idea of Iserles's research life is that a differential equation carries structure, and a good numerical method should preserve it.

Some systems conserve energy. Some conserve volume in phase space. Some evolve on a curved manifold — a rotating rigid body, for instance, whose state lives on the group of rotations and can never leave it, no matter how long you wait. Conventional numerical methods know nothing of this. They approximate the next step and accept a small error, and over a long simulation those small errors accumulate into qualitative nonsense: the rotation matrix stops being a rotation, the energy drifts, the orbit is wrong in a way no amount of extra precision will fix.

Geometric numerical integration attacks this by designing methods that respect the invariants exactly — symplectic integrators for Hamiltonian systems, volume-preserving methods, and schemes built so that a system living on a Lie group stays on it. Iserles's own contributions cluster around the machinery for this: approximations of the matrix exponential, particularly methods that map a Lie algebra correctly onto its Lie group, and the analysis of isospectral flows and Lie–Poisson structures.

Oscillation, the Other Hard Case

His second major territory is highly oscillatory phenomena — problems where the solution wiggles very fast.

This is a genuine nightmare for standard numerical analysis, and for an unintuitive reason. Ordinary quadrature works by sampling; if the function oscillates faster than you sample, you learn nothing, and the classical response is to sample faster, which costs more the worse the problem gets. Iserles worked on effective discretisation methods for highly oscillatory integrals — a family of techniques where, remarkably, high oscillation can be turned from an obstacle into an advantage, with accuracy improving as frequency rises rather than collapsing. He extended the approach to differential equations and to the spectra of operators, and developed exponential integrators for equations with several interacting scales of oscillation.

His full research range takes in ordinary and partial differential equations, approximation theory, orthogonal polynomials, functional equations and computational dynamics.

The Gatekeeper

Iserles has been unusually central to the institutions of his field. He is managing editor of *Acta Numerica*, the annual review volume that is essentially the field's statement of where it stands, and editor-in-chief of the *IMA Journal of Numerical Analysis*, alongside board positions elsewhere. From 1997 to 2000 he chaired the Society for Foundations of Computational Mathematics.

And he wrote *A First Course in the Numerical Analysis of Differential Equations*, published by Cambridge University Press and now in its second edition — for a great many graduate students worldwide, the book that taught them the subject.

Why Arieh Is Called a Genius

The intellectual quality on display is a specific kind of inversion: repeatedly taking the property that makes a problem hard and turning it into the property that makes the solution work. The clearest case is highly oscillatory integration, where the standard framing treats rapid oscillation as the enemy to be out-sampled, and the methods Iserles worked on get *better* as oscillation increases. Geometric integration involves the same move at a higher level — stop treating a differential equation as a formula to be approximated and start treating it as a geometric object whose structure is the thing to be preserved. Reframings of that kind are rarer and harder than technical virtuosity, and they change what a whole community thinks it is doing.

The recognition, honestly weighed, is substantial but service-heavy. The Onsager Medal came in 1999. The David Crighton Medal followed in 2012, awarded explicitly "for services to mathematics and the mathematics community." The SIAM Prize for Distinguished Service to the Profession came in 2014. Two of those three are, by their own citations, about service rather than discovery.

So the counter-case is real. No theorem carries Iserles's name in the way a Runge–Kutta or a Newton method does. Geometric integration was built by a community — Hairer, Lubich, Wanner, Marsden, McLachlan and others alongside him — and he is one important contributor, not its solitary author. His largest measurable influence may be editorial and pedagogical: running *Acta Numerica*, running the *IMA Journal*, writing the standard first course. No accessible source records anyone calling him a genius. He is an outstanding applied mathematician and an exceptional institution-builder, and the second half of that sentence is doing more work than admirers usually admit.

Legacy

Iserles remains at DAMTP, in the department where he has worked since 1987.

The programme he helped drive has quietly won. Symplectic and structure-preserving integrators are now standard in molecular dynamics, celestial mechanics, plasma physics and accelerator design — anywhere a simulation has to run for a very long time and still be believed at the end. The insight underneath is philosophical as much as technical: a numerical method is not a neutral instrument for approximating an equation. It is a small dynamical system in its own right, and if you build it carelessly, it will tell you a story about a universe that does not exist.

Achievements

• David Crighton Medal — 2012
• Fields: mathematical analysis