Dimensional Collapse in Optimal Uncertainty Quantification
Author:Lan Huong Nguyen
- Houman Owhadi, Professor of Applied & Computational Mathematics and Control & Dynamical Systems, California Institute of Technology
- Tim Sullivan, Assistant Professor of Mathematics, University of Warwick, UK
In science and engineering, numerical modeling and experimental data help in studying various large-scale complex systems. Unfortunately, both of these involve uncertainty. In a lot of cases, the response function is non-deterministic and the information on inputs and response functions incomplete. For these systems Uncertainty Quantification (UQ) might seem computationally intractable. Optimal Uncertainty Quantification (OUQ) framework poses the UQ problems as optimization problems over infinite-dimensional spaces of probability measures on inputs and response functions. Finite-dimensional reduction theorems of Owhadi et al. show that computing sharp bounds on output uncertainties is possible. Working within the Mystic computational optimization framework, which facilitates solving OUQ problems, we exploit the numerical phenomenon of `dimensional collapse', where the optimization parameters (discrete probability measures) collapse to a lower-dimensional object. `Dimensional collapse' highly reduces dimension of the search space, and hence decreases the computational burden. We implement a code for detecting the collapse and applying corresponding changes to the optimizer by having optimizers generate (a) metadata that describes the collapse phenomena, and (b) `live code' that will alter the execution of the optimization to reflect the collapse.