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Maintained by Bernadette Watts and Deanna Ceballos
Maintained by Bernadette Watts and Deanna Ceballos
To Roll the Dice or Not; Fusing Statistical, Probabilistic, and Deterministic Analysis Tools
Don Estep
Colorado State University
Approaches to quantification of the effects of uncertainties and errors can be divided crudely into two schools of thought; one based on using probability and statistics and the other based on deterministic analysis tools. While the communities working in these directions share some common goals, there are still unfortunate gulfs between them. I believe that this hampers progress on some fundamental problems in predictive modeling. Ultimately, quantification of uncertainties and errors on predictions obtained from a multiphysics, multiscale model will require a fusion of statistics, probability, and deterministic analysis tools.
I will illustrate these thoughts with four examples. First, I will consider the density estimation problem. I will briefly recall its importance for UQ and sensitivity analysis and briefly explore some computational issues. I will then explain how deterministic analysis tools can be applied effectively to the computational problem of estimating densities in the context of differential equations. Then I will turn the tables and explain how probabilistic sampling methods can be applied effectively to adaptive error control for solving differential equations. Third, I will address the conventional wisdom that numerical errors from solving differential equations are different than other sources of error, e.g. numerical errors are considered to be controllable and deterministic. I will expose this line of thought to discretization of multiphysics, multiscale problems and explain why the assumption that numerical errors are controllable and predictable is not supportable and also make a case for the need for a probabilistic description of numerical error. Finally, I will briefly consider predictive data assimilation. I will explain some dangers that arise if a purely statistical point of view is adopted and argue for the need to include deterministic properties of the processes described by the model in the assimilation.