Computational Challenges in the Numerical Approximation of Nonlinear Functionals and Functional Differential Equations
Abstract
The fundamental importance of functional differential equations (FDEs) has been recognized in many areas of mathematical physics, such as fluid dynamics (Hopf equation of turbulence), quantum field theory (SchwingerÐDyson equations) and statistical physics (equations for generating functionals and effective FokkerÐPlanck equations). However, no effective numerical method has yet been developed to compute their solution. The main objective of this proposal is to fill this gap, and provide a new mathematical framework to approximate numerically nonlinear functionals and the solution to FDEs. This is a long-standing open problem in mathematical physics, and a timely effort motivated in particular by recent advances in numerical methods for high-dimensional systems and parallel computing algorithms. The three-year research plan consists of theoretical and numerical developments, as well as a general software framework that will implement the proposed algorithms. Demonstration examples will be drawn from problems in fluids mechanics, statistical physics, and stochastic dynamical systems. The proposed work will have a significant and broad impact in many areas of mathematical physics and engineering, as it will set the foundations of new theoretical and computational methods to solve FDEs on a computer. The new simulation capability will provide answers to many open questions in computational science, such as determining the solution to the Hopf equation of turbulence, solving optimal control problems by direct discretization of functional derivatives, or determining macroscopic properties, phase transitions, and critical phenomena in a great variety of statistical systems related to materials science.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Feb 25, 2019
- Source ID
- W911NF1810309
Entities
People
- Daniele Venturi
Organizations
- Army Contracting Command
- United States Army
- University of California, Santa Cruz