Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications

Achdou, Yves; Barles, Guy; Ishii, Hitoshi; Litvinov, G. L.

doi:10.1007/978-3-642-36433-4

Cited by 58 publications

(99 citation statements)

References 18 publications

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“…where the second inequality is due to v ǫ ≤ 1. This yields (1). Since this argument is uniform for (t 0 , x 0 ) ∈ K, we deduce (56).…”

mentioning

confidence: 69%

“…Finally, (c) with d = 0.5 means that species u spreads faster than species v, i.e., c 1 = 2 > c 2 as discussed in Corollary 1. Due to the limitation of our methods, we can't get the asymptotic profiles of (1).…”

Section: Qian Liu Shuang Liu and King-yeung Lammentioning

confidence: 99%

“…Theorem 1.1 (Lewis et al [27,29]). Let (u, v) be the solution of (1) In this case, we say that the population u spreads at speed c LLW . Remark 1.2.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Asymptotic spreading of interacting species with multiple fronts Ⅰ: A geometric optics approach

Liu¹,

Liu²,

Lam³

2020

Discrete &Amp; Continuous Dynamical Systems - A

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This is part two of our study on the spreading properties of the Lotka-Volterra competition-diffusion systems with a stable coexistence state. We focus on the case when the initial data are exponential decaying. By establishing a comparison principle for Hamilton-Jacobi equations, we are able to apply the Hamilton-Jacobi approach for Fisher-KPP equation due to Freidlin, Evans and Souganidis. As a result, the exact formulas of spreading speeds and their dependence on initial data are derived. Our results indicate that sometimes the spreading speed of the slower species is nonlocally determined. Connections of our results with the traveling profile due to Tang and Fife, as well as the more recent spreading result of Girardin and Lam, will be discussed.

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“…where the second inequality is due to v ǫ ≤ 1. This yields (1). Since this argument is uniform for (t 0 , x 0 ) ∈ K, we deduce (56).…”

mentioning

confidence: 69%

Section: Qian Liu Shuang Liu and King-yeung Lammentioning

confidence: 99%

See 1 more Smart Citation

Asymptotic spreading of interacting species with multiple fronts Ⅰ: A geometric optics approach

Liu¹,

Liu²,

Lam³

2020

Discrete &Amp; Continuous Dynamical Systems - A

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“…In this paper we drop the assumption of periodicity, which is usually adopted in the MFG literature to avoid technical issues, and move to the setting of reflecting boundary, for which, to the best of our knowledge, no existence results are available; Achdou and Capuzzo-Dolcetta carried out numerical analysis on stationary models in [1,2] with such conditions at the boundary.…”

Section: Introductionmentioning

confidence: 99%

Multi-population Mean Field Games systems with Neumann boundary conditions

Cirant

2015

Journal de Mathématiques Pures et Appliquées

109

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“…However, typical treatments of such equations on an unbounded domain (e.g. in Barles [1] or Evans [18]) assume that the solutions are bounded and uniformly continuous, whereas our solution v exhibits explosive behaviour near the boundary of U . In Yong and Zhou [39] solutions are only assumed to be continuous, but the Hamiltonian is assumed to grow at most linearly in the spatial variable x.…”

Section: Uniqueness For the Hjb Equationmentioning

confidence: 99%

Parameter Uncertainty in the Kalman--Bucy Filter

Allan¹,

Cohen²

2019

SIAM J. Control Optim.

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In standard treatments of stochastic filtering one first has to estimate the parameters of the model. Simply running the filter without considering the reliability of this estimate does not take into account this additional source of statistical uncertainty. We propose an approach to address this problem when working with the continuous time Kalman-Bucy filter, by making evaluations via a nonlinear expectation. We show how our approach may be reformulated as an optimal control problem, and proceed to analyse the corresponding value function. In particular we present a novel uniqueness result for the associated Hamilton-Jacobi-Bellman equation.

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Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications

Cited by 58 publications

References 18 publications

Asymptotic spreading of interacting species with multiple fronts Ⅰ: A geometric optics approach

Asymptotic spreading of interacting species with multiple fronts Ⅰ: A geometric optics approach

Multi-population Mean Field Games systems with Neumann boundary conditions

Parameter Uncertainty in the Kalman--Bucy Filter

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