Functional differential equations are capable of describing the complete statistical properties of a turbulent flow or predicting the electronic structure of superconducting materials. For nearly seventy years, the math behind them has been so complex they were impossible to solve, and their approximation was simply unfeasible using conventional techniques.
In 2018, UC Santa Cruz Applied Mathematics Professor Daniele Venturi published a Physics Reports paper entitled “The numerical approximation of nonlinear functionals and functional differential equations,’’ in which he addresses this problem.
“There was a long-standing open problem in computational mathematics,’’ Professor Venturi said. “On the one hand we have these beautiful equations which can describe exactly the dynamics of infinite-dimensional random systems such as turbulent flows, or many-electron systems in condensed-matter physics. On the other hand, we currently do not know how to solve these equations effectively.’’
Solving these equations could have an enormous impact, particularly with respect to materials science, condensed-matter physics, and toward improving the accuracy of large-scale simulations such as weather forecasts.
“We did not really expect this research would generate such an avalanche of activities in so many different areas of computational math,’’ Professor Venturi added while describing two recent ArXiv papers he published with his Ph.D. students Alec Dektor and Abram Rodgers.
These activities include a proposed algorithm which will compute the numerical solution to the full BGK model of the Boltzmann Transport Equation, a notoriously hard to solve nonlinear partial differential equation in six variables plus time. On November 25, Venturi and his co-authors Arnout Boelens and Daniel Tartakovsky (Stanford University) will present a “Solution of the BGK model of the Boltzmann Transport Equation using Alternating Least Squares” to the 72nd Annual meeting of the American Physical Society. A paper is also forthcoming.
The Army Research Office, an element of U.S. Army Combat Capabilities Development Command's Army Research Laboratory, has been funding this research since 2018. Dr. Joseph Myers, division chief, mathematical sciences division at Army Research Office, said that “Nonlinear functionals and functional equations are of fundamental importance in many areas of science, and this research will work to provide new mathematical tools to approximate them effectively, which will be increasingly important in applications.’’
Professor Venturi recently received tenure at the University of California Santa Cruz. For more information about his lab please see: https://venturi.soe.ucsc.edu.