Volume 14, Issue 1

This paper has been written in response to many requests for a practical guide to the use of the technique of sequential Monte Carlo in the fast numerical solving of large systems of linear equations . This method, which I have used with considerable success to solve such problems, improving the tricks of the trade as I learned more about it, has suffered from some neglect through the mathematical difficulty, for some of those who are more interested in using the tool than in thinking about it, of some of the theoretical aspects of rigorously proving its validity , which – at this juncture – is no longer in question. I hope that I have now closed this gap in the related literature.

We describe new variants of the Euler scheme and of the walk on spheres method for the Monte Carlo computation of Feynman–Kac representations. We optimize these variants using quantization for both source and boundary terms. Numerical tests are given on basic examples and on Monte Carlo versions of spectral methods for the Poisson equation. We especially introduce a new stochastic spectral formulation with very good properties in terms of conditioning.

For high dimensional transformations, the essential spectral radius of the Perron–Frobenius operator restricted to some natural space is generally smaller than the reciprocals of the radius of convergence of the dynamical zeta function. We will construct transformations whose essential spectral radius coincide with the reciprocal of the radius of convergence of the dynamical zeta functions, and construct new type of 2 dimensional low discrepancy sequences.

An approach is proposed for computing the percentile points of the maximum of k independent Student's t random variables. A simple program in R (a freely downloadable statistical software) is presented and its performance compared with other known approaches. Similar programs are presented for computing the percentile points of the componentwise maximum of k -variate t and k -variate normal vectors. The time efficiency of the programs is discussed.

We introduce a new pseudorandom number generator SSR (the Simplified Shift-Real random number generator) which generates the k -th random number nonrecursively (directly) based on chaos mappings on the interval [1, 2). We investigate properties of SSR random numbers and give the theoretical background of generation of random numbers. A practical integral (all-integer) version SSI of SSR, which is suitable for parallel computation, is also provided.