What kind of humongous tasks do supercomputers do? What type of mathematical models can be so complex that it requires a computer close to $1B?
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For weather and climate forecasting, imagine the atmosphere broken up into a 4000 x 4000 x 100 (vertical) grid, and the model state (temperature, humidity, cloud content, dust content, sometimes chemical species content, wind direction) has to be represented in each one on say a 5-minute time step. Some models are ensembles, which may have 20 or 50 or more different versions of the same simulation running, with small variations. You have to do the math to propagate everything forward in time, which is relatively straightforward. And you also have to do the math to “assimilate” thousands of ground, radar, balloon, aircraft, and satellite observations to try to make the model agree with those observations as best you can…this is a huge linear algebra equation which involves solving an equation with millions of terms. The numbers can vary depending on the geographical extent, the resolution, the time horizon, etc. but that should give you an idea.
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