Numerical viscosity solutions to Hamilton-Jacobi equations via a Carleman estimate and the convexification method

Klibanov, Michael V.; Nguyen, Loc H.; Tran, Hung V.

doi:10.1016/j.jcp.2021.110828

Cited by 11 publications

(14 citation statements)

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“…The use of Carleman estimates quickly became a powerful tool in many areas of PDEs, especially in both theoretical and numerical methods for inverse problems, see e.g., [5,4,16,7,9,19,26,29]. Carleman estimates were used in cloaking [28] and in the area of computing solution to Hamilton-Jacobi equations [20,27]. We recall a useful Carleman estimate which is important for us in the proof of the main theorem in this paper.…”

Section: A Carleman Estimatementioning

confidence: 99%

“…We specially draw the reader's attention to different forms of Carleman estimates for all three main kinds of differential operators (elliptic, parabolic and hyperbolic) and their applications in inverse problems and computational mathematics [4,5,16,29]. It is worth mentioning that some Carleman estimates hold true for all functions v satisfying v| ∂Ω = 0 and ∂ ν v| Γ = 0 where Γ is a part of ∂Ω, see e.g., [20,32], which can be used to solve quasilinear elliptic PDEs partly given boundary data.…”

Section: A Carleman Estimatementioning

confidence: 99%

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The Carleman-contraction method to solve quasi-linear elliptic equations

Nguyen¹

2022

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We propose a globally convergent numerical method to compute solutions to a general class of quasi-linear PDEs. Combining the quasi-reversibility method and a suitable Carleman weight function, we define a map of which fixed point is the solution to the PDE under consideration. To find this fixed point, we define a recursive sequence with an arbitrary initial term using the same manner as in the proof of the contraction principle. Applying a Carleman estimate, we show that the sequence above converges to the desired solution. On the other hand, we also show that our method delivers reliable solutions even when the given data are noisy. Numerical examples are presented.

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Section: A Carleman Estimatementioning

confidence: 99%

Section: A Carleman Estimatementioning

confidence: 99%

The Carleman-contraction method to solve quasi-linear elliptic equations

Nguyen¹

2022

Preprint

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“…The second drawback is that, in general, the distance between the true solution v 0 to (2.4) and the computed solution v 0 min is not known. In this paper, we generalize the convexification method in [23,32] to compute the "best fit" solution to (2.9) and (2.12). By convexification, we mean that we let a Carleman weight function involve in the functional J, defined in (2.13).…”

Section: A Change Of Variablementioning

confidence: 99%

“…Recently, we, in [23,32], developed the convexification method to solve (1) the inverse scattering problems and (2) a general class of Hamilton-Jacobi equations in a bounded domain. The efficiency of the convexification method was rigorously proved.…”

mentioning

confidence: 99%

“…However, the version of the convexification method above requires both Neumann and Dirichlet boundary conditions of the solution, which are not always available. Therefore, in order to apply the convexification method in [23,32] without requesting the knowledge of the boundary conditions, we have to develop a new version. The key of the success involves a new piece-wise Carleman estimate and a new mismatch functional with a suitable Carleman weight function.…”

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confidence: 99%

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The Carleman convexification method for Hamilton-Jacobi equations on the whole space

Le¹,

Le²,

Nguyen³

2022

Preprint

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We propose a new globally convergent numerical method to solve Hamilton-Jacobi equations in R d , d ≥ 1. This method is named as the Carleman convexification method. By Carleman convexification, we mean that we use a Carleman weight function to convexify the conventional least squares mismatch functional. We will prove a new version of the convexification theorem guaranteeing that the mismatch functional involving the Carleman weight function is strictly convex and, therefore, has a unique minimizer. Moreover, a consequence of our convexification theorem guarantees that the minimizer of the Carleman weighted mismatch functional is an approximation of the viscosity solution we want to compute. Some numerical results in 1D and 2D will be presented.

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