Spectral methods for nonlinear functionals and functional differential equations
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We develop a rigorous convergence analysis for finite-dimensional approximations of nonlinear functionals, functional derivatives, and functional differential equations (FDEs) defined on compact subsets of real separable Hilbert spaces. The purpose this analysis is twofold: first, we prove that under rather mild conditions, nonlinear functionals, functional derivatives and FDEs can be approximated uniformly by high-dimensional multivariate functions and high-dimensional partial differential equations (PDEs), respectively. Second, we prove that such functional approximations can converge exponentially fast, depending on the regularity of the functional (in particular its Fréchet differentiability), and its domain. We also provide sufficient conditions for consistency, stability and convergence of functional approximation schemes to compute the solution of FDEs, thus extending the Lax-Richtmyer theorem from PDEs to FDEs. Numerical applications are presented and discussed for prototype nonlinear functional approximation problems, and for linear FDEs.
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