I was recently quizzed on the application of Monte Carlo
Uncertainty/Probabilistic techniques for Engineering applications of
turbomachinery, and that deserves some thought here.
While there are several practical applications of the MC technique for various
facets of the Sciences, perhaps the one that has fascinated me the most is its
use for the estimation of pi.
Pi needs no definition whatsoever. Several centuries ago,
Archimedes had spent a lifetime working on the geometric properties of circles
and had stumbled upon the elegant fact that all circles hold a constant ratio
between its perimeter and diameter, irrespective of the size of the circle. Archimedes
had used the concept of inscribing and superscribing circles of various sizes
in polygons of increasing number of sides to arrive at an estimate for pi. His
estimation was based on a limit at approximating the curved line as a series of
straight lines. Legend has it that Archimedes was able to scribe a 180-sided
polygon to arrive at the estimation that we use to date. (Legend also has it
that he was so obsessed with is circles that when the Romans came to take him
to the King, Archimedes enraged the soldier by telling him to not disturb his
circles. The enraged soldier ended up killing Archimedes with his sword).
Monte Carlo techniques work on the premise of allocating
randomized probabilities for occurrences of events, and then studying the behavior
of the system with these perturbed inputs. An underlying assumption in
implementing this technique is that the system under scrutiny can be
represented as a mathematical model. Given such a model, randomized
perturbations can be supplied as inputs, and the outputs can be obtained to
understand the sensitivity and uncertainties of the system.
To be cont'd.
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