Data processing: structural design – modeling – simulation – and em – Modeling by mathematical expression
Reexamination Certificate
1998-03-31
2001-09-11
Teska, Kevin J. (Department: 2123)
Data processing: structural design, modeling, simulation, and em
Modeling by mathematical expression
C708S250000, C380S046000
Reexamination Certificate
active
06289296
ABSTRACT:
BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates to a statistical simulation method and corresponding simulation system for statistically computing an expected value of a physical quantity or the like, and a storing medium in which a statistical program is recorded. The present application claims priority under 35 U.S.C. §119 of Japanese Patent Application No. 9-082723, filed Apr. 1, 1997, the entire disclosure of which is incorporated herein by reference.
2. Description of the Related Art
The Monte Carlo method is well known as a statistical simulation method for statistically computing an expected value of a physical quantity or the like. This Monte Carlo method has been heretofore widely used in, for example, the device modeling field. TED (“Gunn Effect Device”), MOSFET, MESFET, bipolar transistors, hetero-structure devices, Schottky diodes, photo-detectors and the like are known as devices developed by the use of this method. It will be appreciated that the Monte Carlo method can also be used for simulation of PN junction devices, heterojunction devices, or the like. It will also be noted that the Monte Carlo method has applications to other many-body problems having numerous collected uncertainties, e.g., operations research and research in the sociology field.
It will be appreciated that the above-described Monte Carlo method is a method for computing or simulating a stochastic phenomenon by generating random numbers and constructing the stochastic phenomenon. As disclosed in U.S. Pat. No. 5,301,118, which reference is incorporated herein by reference, Monte Carlo analysis is a predictive technique, which can be used, for example, to determine variation of assemblies based on probabilistic modeling. Monte Carlo analysis is performed by establishing a range for each individual component tolerance, for example a range of USL-LSL expressed in terms of a probability density function. A random sampling fitting a mathematically defined distribution is taken from within this range, and the response of the circuit or system is evaluated. The output values can then be further analyzed using traditional statistical methods.
Monte Carlo analysis, i.e., analysis performed according to the Monte Carlo method, uses a random number generator to perform the distribution sampling. Therefore, it will be appreciated that Monte Carlo simulation can be employed to simulate large sample sizes on digital computers. In particular, Monte Carlo analysis is especially useful where complex assemblies can not be readily or realistically analyzed by traditional linear methods, such as root- sum- of -squares analysis or worst case analysis. It will also be appreciated that Monte Carlo analysis can be useful where the completed assemblies needing analysis are costly or time consuming to manufacture.
Assuming that this Monte Carlo method is applied to a many-particle simulation, this method is based on the assumption that a motion of a certain particle (an electron or a molecule) can be tracked over a sufficiently long time whereby information on the behavior of the whole electronic system or the entire gas system can be obtained. For example, in a free traveling process of a certain particle (carrier), an inherent stochastic distribution function can be used to establish conditions for determining the mean free path, whereby the random numbers are continuously generated. An elapse of time of the carrier is measured, whereby the motion of the particle is simulated.
The Monte Carlo method is based on the ergodic hypothesis that “a physical average is equal to a longtime average”. That is, this method succeeds in using ergodicity of a deterministic equation (1) for generating the random numbers.
Formula (1)
Xn+
1
=F
(
Xn
) (1)
A possession of the ergodicity by the equation (1) herein means that an invariant measure (dx) is present corresponding to a physical quantity Xn, where Xn &egr; M, and where the equation (1) is defined in terms of a physical space M. The dynamical system of equation (1) has ergodicity if the relationship defined in formula (2) holds. It will be appreciated that the formula (2) means that an integral of a function Q(x) with respect to the invariant measure &mgr;(dx) on a phase space M is equal to a time integral of Q(Xi)(wherein Q(Xi)is a Lebesgue integrable function of Xi). The formula (2) represents the above-mentioned assumption by an expression. According to the formula (2), in the Monte Carlo method, a space integral on the phase space M on the right side of the formula (2) is easily computed because the invariant measure &mgr;(dx) represents the underlying stochastic phenomenon.
Formula (2)
lim
N
→
∞
1
N
∑
i
=
1
N
Q
(
xi
)
=
∫
M
Q
(
x
)
μ
(
ⅆ
x
)
Stated another way, the term &mgr;(dx) is the invariant measures on M and Q(x) is a regular function, which makes the right hand side of formula (2) finite. Q(x) is a Lebesgue integrable function of x. Thus, Monte Carlo simulations can be thought of as evaluating the right hand side of formula (2) by approximately calculating the left hand side of formula (2) under the assumption of ergodicity.
As described above, the Monte Carlo method is a method for statistically simulating the expected value of a physical quantity, or the like, on a computer. This method has been widely used. However, an environment, in which a large-sized computer can be employed, is generally required for application of the Monte Carlo method, although this method can sometimes be successfully performed on more readily available, i.e., less powerful, computers when the simulated system is relatively simplistic. It will be appreciated that the conventional Monte Carlo method must generate the random numbers having a uniform invariant measure &mgr;(dx) for each step in the simulation process. Moreover, since the generated uniform random numbers are “pseudo-random numbers having a finite period”, it will be noted that the computing speed cannot be advantageously improved to any great extent nor can the computing error be reduced by any appreciable extent. Thus, the time required for the simulation process to produce convergence of the computation result cannot be reduced. In many cases, even if a large-sized computer can be dedicated to the simulating system, the Monte Carlo method cannot be advantageously employed.
SUMMARY OF THE INVENTION
The present invention was motivated by a desire to solve the above-identified problems. Thus, one object of the present invention is to provide a statistical simulation method capable of performing a statistic simulation in the same manner as the Monte Carlo method.
Another object of the present invention is to provide a statistical simulation method capable of performing a statistical simulation while greatly improving the computing speed of the simulation with respect to the speed achieved by the Monte Carlo method.
Still another object of the present invention to is provide a statistical simulation method capable of performing a statistical simulation while reducing computing error to a level below that associated with the Monte Carlo method.
Yet another object of the present invention is to provide a statistical simulation method capable of performing a statistical simulation while reducing the time to convergence of a computation result with respect to the convergence time associated with the Monte Carlo method.
Another object of the present invention is to provide a statistical simulation method capable of performing a statistical simulation which heretofore has not been possible using the conventional Monte Carlo method.
Although the objects of the present invention are discussed above with respect to a statistical simulation method, it will be appreciated that the present invention is not so limited. Thus, computer systems programmed with instructions embodying the statistical simulation method and storage or storing media which store computer readable instructions for converting a g
Griffin & Szipl, P.C.
Phan Thai
Teska Kevin J.
The Institute of Physical and Chemical Research (Riken)
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