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Monte Carlo Analysis
Importance Sampling Monte Carlo
• It should produce high-quality random numbers uniformly distributed in the interval
(0, 1).
• The computations should be fast. Separate threads of random numbers are allowed to be
generated in parallel.
• It should be portable (Unix and Windows platforms).
To start this generator, its initial state needs to be specified. Knuth introduced an algorithm that
uses a single integer, which can take on approximately 10 9 different values, in such a way that a
given seed is guaranteed to produce at least 10 21 different starts before colliding with those
produced by a different seed.
Importance Sampling Monte Carlo
The Importance Sampling Monte Carlo (ISMC) method is dedicated to the estimation of failure
probabilities (or the computation of parametric yield in circuit design terms) and, conversely, to
the estimation of the thresholds corresponding to user-defined yield.
Note
Estimation of low failure probabilities is often called a “high sigma” problem in the EDA
literature. The terms “low probability estimation” and “rare events simulation” are used in
this section to mean the same thing.
Rare events can be defined as the regions of the input parameter space that have a very small
volume with respect to the measure induced by the joint probability of the parameters. The
associated probabilities of interest are close to zero, and may range from 10 −4 to 10 −10 . In this
context, Monte Carlo analysis is used as a verification tool to check that the probability of
exceeding a performance specification is acceptably small.
The Monte Carlo algorithm is an iterative simulation method that may be described briefly as a
random exploration of the space of input parameters. The fundamental problem with this
technique is that the sample size has to be large when the probabilities to estimate are small. For
example, estimating a probability of around 10 −K (where K > 2) to a target accuracy of 10% will
involve more than 10 −(K + 2) simulations of the circuit performances.
The ISMC method is implemented using an adaptive procedure based on importance sampling,
and has the following properties:
• It is as insensitive as possible to the dimension of the problem.
• It requires no prior knowledge to function. Specifically, it functions if nothing is known
about the output distribution and its tails.
• It does not need to be manually tuned to work with a given cell or circuit.
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Eldo® User's Manual, 15.3