Mathematical modeling of random phenomena in Physics and Engineering:
analysis and efficient numerical simulations



Background

    Analyzing random phenomena, for instance the effect of random perturbations on deterministic dynamical systems, is of paramount importance in a variety of problems, being rather ubiquitous. Within such difficult problems, an explicit analytical solution can be hardly found, thus pushing to the search for asymptotic or numerical methods. Effective numerical methods for integrating stochastic dynamical systems (that is systems of stochastic differential equations, SDEs) are based, as a rule, on the generation of suitable sequences of random numbers, which are often called ‘‘pseudorandom’’, to stress that ideal randomness cannot be reproduced on real computers, but it is highly inefficient.


                  Scatter random points on the plane generated with pseudorandom and quasi-random number   .
                         Note that the quasi-random numbers are spread more uniformly over the plane


                               


                                    On the right, a sequence of quasi-random numbers of the Van der Corput-type.
                                      Pick an integer, convert into binary, inverse, and convert back into decimal.

An efficient method for solving SDEs

     An alternative choice to the pseudorandom numbers does exist, and is represented by the so-called quasi-random" numbers. They have been exploited already in some applications in physics and in financial mathematics. These sequences of numbers are built using strictly deterministic uniformly distributed numbers, which are strongly correlated. In spite of such a correlation, these numbers have been successfully used in some problems, instead of their pseudorandom analog. When this can be done, the advantage is great. We have shown that stochastic dynamical systems can be solved numerically adopting sequences of quasi-random numbers. This can done very efficiently provided that some care is paid to their implementation. In particular, a reordering strategy should be used to destroy correlations which affect such sequences. It was also shown that such an action, in general, may not suffice, in view of periodicity or quasi-periodicity which characterize such sequences. This difficulty can be overcome by a careful choice of the sample size being used in the numerical simulations.

 

 

                     

Numerical error made in computing the second moment of a 1D brownian motion versus time, using pseudorandom sequences of numbers, quasi-random, and quasi-random with reordering.

Selected publications

• J.A. Acebrón, R. Spigler, Fast simulations of stochastic dynamical systems, J. Comput. Phys., 208 (2005) 116-115
• J.A. Acebrón, R. Spigler, Second harmonics effects in random Duffing oscillators, SIAM J. Appl. Math., 66 (2005) 266-285