Algorithm for Hamilton–Jacobi Equations in Density Space Via a Generalized Hopf Formula

Chow, Yat Tin; Li, Wuchen; Osher, Stanley; Yin, Wotao

doi:10.1007/s10915-019-00972-9

Cited by 29 publications

(35 citation statements)

References 38 publications

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“…The results of our calculations are shown in Figure 3. As expected, the approximated functional derivative is continuously Fréchet differentiable 15 , and therefore the result (69) holds.…”

Section: Approximation Of Nonlinear Functionalssupporting

confidence: 71%

“…Next, we study convergence of the first-order functional derivative (15). This is relatively straightforward given the convergence result we just obtained in Theorem 6.1.…”

Section: Approximation Of Fréchet and Functional Derivatives In Compamentioning

confidence: 94%

“…It was shown in [69, p. 76] that the analytical solution of the FDE (118) in the function space (106) is 15 The functional (114) admits continuous Fréchet derivatives to any desired. In particular, we have 113), and for test functions θ with spectra (110) (power law decay) and (111) (exponential decay).…”

Section: Analytical Solutionmentioning

confidence: 99%

“…For instance, functionals were used by Wiener for the description of Brownian motion [76], by Hohenberg and Kohn [30] to reduce the dimensionality of the Schrödinger equation in many-body quantum systems (density functional theory [46? ]), by Hopf to describe the statistical properties of turbulence [31,1,43,41], and by Bogoliubov to model systems of interacting bosons in superfluid liquid helium [8,61]. Other applications of nonlinear functionals can be found in [68,15,47,46,36,64,48,75,35,40].…”

Section: Introductionmentioning

confidence: 99%

“…In some cases, it is possible to reformulate such optimization problems in terms of a nonlinear Hamilton-Jacobi FDE in probability density space. The standard form of such equation is [15] ∂F ([ρ], t) ∂t…”

Section: Introductionmentioning

confidence: 99%

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The numerical approximation of nonlinear functionals and functional differential equations

Venturi

2018

Physics Reports

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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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Section: Approximation Of Nonlinear Functionalssupporting

confidence: 71%

“…Next, we study convergence of the first-order functional derivative (15). This is relatively straightforward given the convergence result we just obtained in Theorem 6.1.…”

Section: Approximation Of Fréchet and Functional Derivatives In Compamentioning

confidence: 94%

Section: Analytical Solutionmentioning

confidence: 99%