## Sobol’s Monte Carlo

**T**he name of Ilya Sobol is familiar to researchers in quasi-Monte Carlo methods for his Sobol’s sequences. I was thus surprised to find in my office a small book entitled *The Monte Carlo Method* by this author, which is a translation of his 1968 book in Russian. I have no idea how it reached my office and I went to check with the library of Paris-Dauphine around the corner [of my corridor] whether it had been lost: apparently, the library got rid of it among a collection of old books… Now, having read through this 67 pages book (or *booklet* as Sobol puts it) makes me somewhat agree with the librarians, in that there is nothing of major relevance in this short introduction. It is quite interesting to go through the book and see the basics of simulation principles and Monte Carlo techniques unfolding, from the inverse cdf principle [established by a rather convoluted proof] to importance sampling, but the amount of information is about equivalent to the Wikipedia entry on the topic. From an historical perspective, it is also captivating to see the efforts to connect physical random generators (such as those based on vacuum tube noise) to shift-register pseudo-random generators created by Sobol in 1958. On a Soviet Strela computer.

While Googling the title of that book could not provide any connection, I found out that a 1994 version had been published under the title of A Primer for the Monte Carlo Method, which is mostly the same as my version, except for a few additional sections on pseudo-random generation, from the congruential method (with a FORTRAN code) to the accept-reject method being then called von Neumann’s instead of Neyman’s, to the notion of constructive dimension of a simulation technique, which amounts to demarginalisation, to quasi-Monte Carlo [for three pages]. A funny side note is that the author notes in the preface that the first translation [now in my office] was published without his permission!

December 12, 2016 at 6:38 pm

Sobol’ also wrote some part of the 1964 book ‘Methods of Statistical Testing’, edited by Yu. A. Schreider. That book came out in the same year as Hammersley and Handscomb. Sobol’s chapter(s) include material on sequential inversion of the multivariate CDF (inverse Rosenblatt transformation) and generating Brownian motion by sampling the skeleton of the process. [Like the Brownian bridge construction.]

His main QMC book from the 1960s has not (to my knowledge) been translated and it is hard to get hold of. Some lecture notes of his written in German circulated in the 1980s or so, but I don’t know how to find them.

December 12, 2016 at 6:50 pm

Thank you, Art. I have never read anything else by Sobol and do not mean to be dismissive of his work: just this booklet popping in my office by chance and an opportunity to get a glance at the Soviet perception of Monte Carlo methods in the 1960’s…!