The Numerical Approximation of Nonlinear Functionals and Functional Differential Equations

Abstract

The fundamental importance of functional differential equations has been recognized in many areas of mathematical physics, such as fluid dynamics (Hopf characteristic functional equation), quantum field theory (Schwinger-Dyson equations) and statistical physics (equations for generating functionals and effective Fokker-Planck equations). However, no effective numerical method has yet been developed to compute their solution. The purpose of this report is to fill this gap, and provide a new perspective on the problem of numerical approximation of nonlinear functionals and functional differential equations.

142 pages, 41 figures

References in corpus (3)

• Co-occurrence of resonant activation and noise-enhanced stability in a model of cancer growth in the presence of immune response
• The Lundgren-Monin-Novikov Hierarchy: Kinetic Equations for Turbulence
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• The Turbulent Dynamo
• Recent Developments in Theories of Inhomogeneous and Anisotropic Turbulence
• Data-driven closures for stochastic dynamical systems
• Data-Driven Discovery of Coarse-Grained Equations
• A new scalable algorithm for computational optimal control under uncertainty
• On the estimation of the Mori-Zwanzig memory integral
• Tensor methods for the Boltzmann-BGK equation
• Parallel numerical tensor methods for high-dimensional PDEs
• Dynamically orthogonal tensor methods for high-dimensional nonlinear PDEs
• Dynamic tensor approximation of high-dimensional nonlinear PDEs
• Generalized Langevin equations for systems with local interactions
• Stability analysis of hierarchical tensor methods for time-dependent PDEs
• Tensor rank reduction via coordinate flows
• Hypoellipticity and the Mori-Zwanzig formulation of stochastic differential equations
• Remarks on the principles of statistical fluid mechanics