US 6,289,296 B1 · Statistical simulation method and corresponding simulation system responsive to a storing medium in which statistical simulation program is recorded

Abstract

A method for simulating a stochastic phenomenon, particularly one employing a Monte Carlo method, includes a step for generating random numbers having a nonuniform density function ρ(x) based on an ideal model of a chaotic dynamic system in the form Xn+1=f(Xn). The ideal model of chaotic map F(x) is a class of an algebraic map derived from an addition theorem of an elliptic function. The nonuniform density function ρ(x) is expressed by an algebraic function of a dynamical variable x. Examples of the ideal model of the chaotic map F(x) include a generalized Ulam-von Neumann map and a generalized cubic map. The nonuniform density function ρ(x) of the chaotic maps has the form Xn+1=F(Xn). The class of chaotic dynamical systems is ideal in the sense that the invariant measure ρ(x)dx satisfies the ergodic property required by the Monte Carlo method. Thus, a statistical simulation can be performed in the same manner as the conventional Monte Carlo method. Compared to the usual Monte Carlo method, the computing speed is increased while the computing error is decreased. Thus, a time for a convergence of a computation result can be reduced, whereby problems, which have not been heretofore able to be treated, can be solved.

Claims2

Dependent claims

In a statistical simulation method for statistically performing a computation by generating random numbers and constructing a stochastic phenomenon for one of the computation of an expected value of a physical quantity and a many-body problem, the improvement comprising the step of generating chaotic random numbers having a non-uniform density function ρ(x) using a chaotic dynamical system of the form

Xn+1=F(Xn)
wherein F(X), which represents the chaotic dynamical system, is expressed by an algebraic function derived from an addition theorem of an elliptic function having a generalized Ulan-von Neumann map in the form:
F ⁡ ( x ) = 4 ⁢ x ⁡ ( 1 - x ) ⁢ ( 1 - l ⁢ ⁢ x ) ⁢ ( 1 - m ⁢ ⁢ x ) 1 + A ⁢ ⁢ x 2 + B ⁢ ⁢ x 3 + C ⁢ ⁢ x 4
where:
A=−2(l+m+lm)
B=8lm
C=l2+m2−2lm−2l2m−2lm2+l2m2−∞<m≦l<1
and constructing said stochastic phenomenon for one of the computation of an expected value of a physical quantity.

5 more elements

Dependencies are inferred from the claim text.

Statistical simulation method and corresponding simulation system responsive to a storing medium in which statistical simulation program is recorded

Life of this patent

Statistical simulation method and corresponding simulation system responsive to a storing medium in which statistical simulation program is recorded

Life of this patent

Abstract

Claims2