An Example of a BIG Numerical Problem

A very simple computational model of pollution in the air over the U.S. might try to compute the concentration of three pollutants at each point in a 3000 x 1500 x 25 grid.

From physics and chemistry a mathematical model of the interaction of these three pollutants can be derived. This mathematical model is a system of 3 time-dependent, nonlinear, partial differential equations.

A numerical method for this problem would compute values of each of the three unknowns so that the mathematical equations are approximately satisfied at each of the grid points.

So the total number of unknowns is

3 x 3000 x 1500 x 25 ~= 3 x 10 8

The classic algorithm requires O (n3) flops, where n is the number of unknowns. Hence, we need more than 1025 flops to solve the problem. On a teraflop machine, this would take at least

1025/1012 = 1013 seconds (nearly) = 317,000 years

Unfortunately, since the problem is time-dependent, we actually need to solve one of the above problems for each of hundreds or thousands of time steps!

flops: floating point operations

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Last Updated 04/16/2000