AMS Colloquium 11-15-13

One of the largest issues facing the ever-growing field of large-scale computational research is how to deal with the nearly infinite number of parameters needed to exactly model a physical reality. With no efficient or cost-effective way to handle the enormous numbers of tests needed to even begin to satisfy requirements needed to model reality, it is inevitable that models will need to be parameterized. However, this causes a problem known as the “Curse of Dimensionality,” in which for every unavoidable parameter used in a model, there is an unavoidable uncertainty associated with the assumption(s) used to define said parameter. As part of his current research, Dr. Tan Bui-Thanh of the University of Texas presented on how he is working on how to deal with computational problems involving these discretization errors.

As Dr. Bui-Thanh explained, most every physically real scenario falls victim to this “Curse of Dimensionality” when modeling of the situation is attempted. This is because no matter how large-scale the computation is or how many data points are used, a ‘perfect’ model can never be achieved due to the infinite number of points in the situation. Because the data is finite and limited, there will always be some amount of error present in a mathematical model. Bui-Thanh was trying to answer the question, “How can we work with these uncertainties and errors?”

He approached a solution to this by beginning with a simple chart of error versus the amount of tests run (or data points collected) and solving for the known value at the point of highest uncertainty. This causes the uncertainty at and closely around the tested point to drop dramatically, creating a new point of highest uncertainty. That point is also tested for an exact value, and the process is repeated until the maximum uncertainty drops to an acceptable level. Bui-Thanh referred to this tactic as a “sample-then-reduce” method and demonstrated with sample data how much more efficiently this technique can lower total error to acceptable levels against a more brute-force approach of simply taking a massive number of samples.

In the end, Dr. Bui-Thanh concluded that the optimum approach to dealing with large-scale computations is to set up as much of the model as possible on an infinite level. He claims that discretization and parameterization should be the final steps to completing a model in order to keep assumptions and, consequently, initial uncertainty to a minimum. From there, the “sample-then-reduce” tactic should then be used to lower the model’s total uncertainty to its minimum.

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