CDS&E: Collaborative Research: Surrogates and Reduced Order Modeling for High Dimensional Coupled Systems

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Increasingly, mathematical modeling of complex scientific systems plays a crucial role both in understanding and in making predictions about these systems. To understand the effects of different model assumptions and parameter values, one might need millions of computational simulations to fully probe a system's behavior. This computational burden is compounded by the fact that many complex systems are modeled not by just one computational model or algorithm, but rather by sets of sub-models and codes that need to interact with each other. This project aims to develop efficient and flexible approximations to such coupled computational models, to take the computational bottleneck out of the mathematical modeling-based scientific discovery process. The research will focus on a prototype coupled model of fluid flow and mechanical deformation that can be adapted to model both hydraulic fracturing and cartilage biomechanics. Students will be involved and trained in interdisciplinary aspects of the project. Computational simulation of systems of scientific and practical interest often requires coupling two or more mathematical models of physical phenomena. Such simulations typically depend on many input parameters, while validation of the models is constrained by limited available data. Numerical simulators of coupled mathematical models use approaches of full or loose coupling. Full coupling involves solving a single set of equations simultaneously, but due to computational complexity, feasible run times often necessitate simplified physics models. Conversely, loose coupling connects independent codes simulating distinct physical processes; it is often infeasible to run even loosely coupled simulations the large number of times required to perform uncertainty quantification. This project aims to develop methodology for Gaussian process emulation of high-dimensional coupled simulators and thus enable uncertainty quantification for challenging yet ubiquitous multi-physics models; the approaches will involve surrogate or reduced-order models of the governing model equations to allow quick approximation of the full physical simulator at different inputs. The efforts will entail developing (i) new ideas for dimension reduction, (ii) a novel method to emulate space-time fields, and (iii) an analysis of errors owing to both dimension reduction and emulation.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.