This research project is concerned with the behavior in time of complex systems subject to random influences. Complex systems with a large number of parameters and agents are ubiquitous -- think, for example, of the traffic on a network of highways or of bio-chemical systems, which involve many types of molecules and many chemical reactions. For such systems, the issue of sensitivity analysis presents special challenges: sensitivity analysis consists in understanding the effects of parameters on the future behavior of the system -- for example, how the closure of one lane on a busy highway influences the traffic, possibly many miles away and several hours later. Understanding these effects is at the heart of the field of predictive modeling and poses formidable challenges, which often exceed the capabilities of modern computers. The Principal Investigators will develop tools, both conceptual and computational, to tackle such issues by using, in a complementary fashion, techniques from different fields of science. This includes methods from information theory (which quantifies how information is lost in physical processes), from statistical mechanics (which allows one to understand the global behavior of very large systems), and from numerical analysis (which is the science that allows one to simulate and reproduce the behavior of complex systems on a computer).
Understanding non-equilibrium systems driven by external forces, boundary effects, and multi-physics presents numerous mathematical and computational challenges, especially if the system is large and involves many parameters. A paradigmatic example is a bio-chemical reaction network, which often involves multiple time scales, feedback loops, hundreds of species, and hundreds of parameters. At the center of this project is the development of mathematical and numerical tools for the analysis and simulation of such non-equilibrium high-dimensional stochastic systems. The research blends together concepts from information theory, statistical mechanics, numerical analysis, and probability theory in order to develop novel algorithms, as well as novel tools to assess existing algorithms for complex systems. The issue of sensitivity analysis plays a central role in the work. One of the novelties here is the systematic use and development of information-theoretic tools for the analysis and uncertainty quantification of stochastic systems, with a strong emphasis on the long-time behavior of the systems. Concepts such as the relative entropy and the Fisher information matrix, especially applied to the time histories of the system, are particularly well suited. Another important feature of the project concerns the fact that non-equilibrium systems do not respect invariance under time reversal. Measuring quantitatively this symmetry breaking is one the main themes in modern non-equilibrium statistical mechanics, and the Principal Investigators will use concepts developed in this project to create efficient algorithms to do this for complex systems. Finally, the research team will investigate the role of memory or delay effects, which are unavoidable in complex systems with multiple spatial or temporal scales.
