Ulam’s Monte Carlo method

When someone says that ‘everything happens for a reason’ they betray their deterministic wiring. Yet connecting cause and effect is a tricky business, and never more so than when probability is involved. Our gut feelings are often wildly at odds with where ‘reason’ should lead us.

The problem is that randomness is everywhere. Some of it is inherent (that infuriating quantum weirdness), and some of it deterministic (the equations quickly degenerate into gazillions of possible outcomes). The arrival of digital computers made it much easier to model that randomness, thanks in part to the feverish thinking of a mathematician involved in the design of The Bomb.

Stanisław Ulam was born in Lemberg (later Lwów, now Lviv) in what was then called Galicia, the intersection of Austria-Hungary, Poland and Ukraine. He studied at the Lwów Polytechnic Institute, famous for its mathematicians, where he gained a PhD. Visits to a who’s who of Europe’s mathematicians included a meeting with Janos (John) von Neumann, who invited Ulam to spend the summer at the Institute for Advanced Study in Princeton, US. Ulam spent the next few years shuttling between Europe and Harvard working on ergodic theory, a branch of mathematics that explores the long-term averages of dynamical systems. Emerging from the statistical physics of Boltzmann, it suggests, for example, that one can extract the average behaviour of a system by tracking the dynamics of single points within it. The theory would come to link fundamental ideas in thermodynamics like entropy and mixing with topology and geometry.

In August 1939, Ulam’s father presciently put Stanisław and his younger brother on a ship to the US. Left behind, his family, along with much of Lwów’s Jewish community, would be slaughtered in the Holocaust.

Core problems

In 1943, the physicist Hans Bethe invited Ulam to join the Manhattan Project. He was soon at work on the problem of how to focus the implosions required to set off the fission of the plutonium core of an atomic weapon. The hydrodynamic equations to describe the core of the bomb were fiendishly complex to solve; Ulam joked that he had sunk so low as a mathematician that he actually had to work with numbers with decimal points.

With von Neumann, Ulam successfully identified ways to cut through the difficulties, but the work made them even more aware of the need for faster computation to solve such problems iteratively.

When the war ended, Ulam got a job at the University of Southern California. In 1946, he developed severe encephalitis; he fell into a coma. Risky surgery and massive doses of antibiotics stopped the infection but he spent many weeks in hospital, initially unable even to speak. He played solitaire to pass the time and wondered how to predict the chance of winning. Every card he drew led to more branches in the tree of choices. The space of possibility was just too vast.

A winning strategy

But what if one took a sample of, say, 100 hands of cards, chosen at random, ran each game and used the outcomes of this subset to estimate the likelihood of winning? Talking to von Neumann, they both realised that neutron chain reactions were a similar kind of problem. While sampling approaches had been tried before, they were beyond tedious when done by hand. Perhaps computers could transform the process? When a friend from Los Alamos came to visit, Ulam explained his thinking, reminiscing about an uncle who, before the war, had enthusiastically visited the casinos of Monte Carlo. The name stuck.

The key to the method’s success was how the sample was chosen: randomly, but with a weighting determined by the dynamics of the system. The starting points of a calculation for a liquid or a gas, for example, might be weighted by the Boltzmann distribution and then the atoms moved at random, step by step, iterating to minimise the energy and predict physical properties with remarkable accuracy.

Ulam would use the method to develop the hydrogen bomb, working with Edward Teller, but the generality of the approach would cause it to spread across the physical and then social sciences, and on into finance where Monte Carlo simulations are used to gauge risk and inform pricing.

The deterministic Albert Einstein famously said that ‘God does not play dice’. For mere mortals, the sign of being an adult is how we deal with uncertainty. Ulam and his colleagues helped us grow up.