AMC-SS: Computational Algorithms and Reduced Models for Stochastic PDEs

University Of Southern California, Los Angeles CA

Investigators

Abstract

Recent advances in the ability to manipulate matter across many scales, including the nanoscale, have contributed significantly towards developing computers and sensors whose capabilities could not be imagined only a decade ago. This has, in turn, brought closer the promise of computational experiments as true surrogates for physical experiments. The benefits of this development are very clear: products can be designed to their smallest detail before expensive production begins; complex systems such as airplanes and chemical plants can be faithfully designed and certified while bypassing the very expensive "physical testing" phase; what-if scenarios involving hypothetical disasters such as terrorist attacks or meteor encounters can be preempted or at least be readied for. An essential component towards fulfilling this promise of "computational reality" is the realization that reality is variable: every time a wave is observed on the shore, every time an earthquake is measured, a soil specimen dug out from the earth, different and unique features are observed of the wave, the quake, or the soil, respectively. Then, a challenge to "computational reality" is the ability to reproduce this real-world scatter. The proposed research addresses this very issue by modeling the unknown root cause of this variability using the mathematical theory of probabilities. Thus, uncertainties that contribute to the observed variability in nature become an intrinsic part of the predictive model, endowing it with the ability to more realistically reproduce reality. As significant contributions to this overarching problem, two issues are specifically addressed in the present research. First, it is vital that the particular form of uncertainty with which the model is endowed does indeed correspond to that observed in reality. Thus, in the first component of the present research, theory and algorithms for constructing models of stochastic processes that are consistent with experimental observations will be developed. These models will be developed such that they can be efficiently embedded into computational algorithms currently in use by state-of-the-art predictive tools. Stochastic representations pioneered by the PI will be used to that end. Secondly, it must be noted that any effort at capturing the variability in nature is fraught with complexity, not the least of which is the burden of enumerating, in some sense, all possible states of nature. The second component of the present research addresses this complexity through innovative computational algorithms that can efficiently and faithfully reproduce natural variability. This will be done by capitalizing on a certain structure both in the underlying physics as well as in the mathematical form assumed to govern the physical behavior of interest. Both new models and new algorithms will be developed to tackle this problem. This research provides a significant contribution towards enabling rational risk management and resource allocation for complex systems whose accurate behavior requires the large-scale computational solution of complex mathematical equations.