Dimension Reduction for Nonlinear Stochastic Systems

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Predictive computational models play a central role not only in engineering and the sciences, but also in society at large. This project addresses fundamental issues common to virtually all computational models. If a model is too simple, the underlying phenomenon will not be faithfully described; if a model is too complex, its predictive power will be minimal. There is, somewhere, a happy middle; this project is about how to find the right balance in terms of model complexity. How complex should a computational model be to be useful? Both the mathematical community and domain scientists have long been worried about the lower bound – the minimum complexity. Models have be faithful to the underlying biology, chemistry or physics but how much of the science should one include? Quantum and relativistic effects can safely be ignored on many problems but what should be included and what can be left out to describe complex chemical reactions or physiological models? This project is about finding the lowest complexity sufficient for the tasks at hand. Models of complex systems, however, present features that render the task of making prediction with quantified uncertainties challenging. Such features include high-dimensional uncertain input parameters, time and/or space dependent quantities of interest, inherent stochasticity, as well as computational expense of simulating complex models. Research in this project will bring about key advances in modeling under uncertainty by developing mathematical techniques and computational methods to address such challenges in models of complex systems. The overarching goal is to develop methods that allow domain scientists to simplify their models so as to facilitate forward uncertainty quantification and parameter estimation at reasonable costs. Students will be trained and mentored in the interdisciplinary aspects of this project. In addition, his project involves the development of new course material that reflects and addresses challenges in present-day scientific computing.The research in this project makes important contributions to computational modeling under uncertainty by developing mathematical theory and algorithms for (i) multiscale sensitivity analysis of stochastic compartment models, (ii) derivative- and variance-based sensitivity analysis methods for time-dependent stochastic systems, (iii) model complexity reduction in compartment models, and (iv) goal oriented multilevel dimension reduction for fast parameter estimation. The algorithms will be applied to models from biochemistry and physiology. A family of complex models of neurovascular coupling will be analyzed with the methods to be developed during the project, including dimension reduction, model complexity reduction and parameter estimation based on existing murine data. The model complexity reduction framework can be used in a broad range of models, where modelers can benefit from removing certain model components. Overall, by enabling dimension reduction in broad classes of models, the project bridges gaps in the processes of modeling, prediction under uncertainty, and parameter estimation: models with most essential components will be discovered and computational budget can be focused on quantifying uncertainty in only those model parameters that are most important to model output.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.