The Carleman-based contraction principle to reconstruct the potential of nonlinear hyperbolic equations

Nguyen, Dinh-Liem; Nguyen, Vu; Truong, Trung

doi:10.48550/arxiv.2204.06060

Cited by 3 publications

(6 citation statements)

References 34 publications

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“…In particular, the convergence of our numerical method is rigorously proved in H 2 . This result is a significant improvement in comparison to the main theorem in [30,33] which shows a similar convergence in H 1 only. These results is verified rigorously by our numerical tests.…”

Section: Introductionmentioning

confidence: 63%

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Global reconstruction of initial conditions of nonlinear parabolic equations via the Carleman-contraction method

Le¹

2022

Preprint

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We propose a global convergent numerical method to reconstruct the initial condition of a nonlinear parabolic equation from the measurement of both Dirichlet and Neumann data on the boundary of a bounded domain. The first step in our method is to derive, from the nonlinear governing parabolic equation, a nonlinear systems of elliptic partial differential equations (PDEs) whose solution yields directly the solution of the inverse source problem. We then establish a contraction mapping-like iterative scheme to solve this system. The convergence of this iterative scheme is rigorously proved by employing a Carleman estimate and the argument in the proof of the traditional contraction mapping principle. This convergence is fast in both theoretical and numerical senses. Moreover, our method, unlike the methods based on optimization, does not require a good initial guess of the true solution. Numerical examples are presented to verify these results.

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Section: Introductionmentioning

confidence: 63%

“…One of our contributions in this paper is that we rigorously prove the convergence of the iterative scheme to the true solution of the nonlinear PDEs in H 2 (Ω) N . It is an important improvement in comparison with the main theorem in [30,33] which prove the convergence in H 1 only.…”

Section: The Main Theoremsmentioning

confidence: 97%

Global reconstruction of initial conditions of nonlinear parabolic equations via the Carleman-contraction method

Le¹

2022

Preprint

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“…This cut off technique was used to solve several different types of inverse problems; see e.g. [12,13,14,23,25,32,36,31]. In contrast, proving the convergence of (3.13) as N → ∞ is extremely challenging.…”

Section: A Numerical Methods To Solve Problem 11mentioning

confidence: 99%

“…This paper belongs to a series of works to solve inverse problems for nonlinear partial differential equations [23,31,34]. In particular, we aim to globally solve an inverse source problem for nonlinear parabolic equations.…”

Section: Introductionmentioning

confidence: 99%

The Carleman-Newton method to globally reconstruct a source term for nonlinear parabolic equation

Abhishek¹,

Le²,

Nguyen³

et al. 2022

Preprint

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We propose to combine the Carleman estimate and the Newton method to solve an inverse source problem for nonlinear parabolic equations from lateral boundary data. The stability of this inverse source problem is conditionally logarithmic. Hence, numerical results due to the conventional least squares optimization might not be reliable. In order to enhance the stability, we approximate this problem by truncating the high frequency terms of the Fourier series that represents the solution to the governing equation. By this, we derive a system of nonlinear elliptic PDEs whose solution consists of Fourier coefficients of the solution to the parabolic governing equation. We solve this system by the Carleman-Newton method. The Carleman-Newton method is a newly developed algorithm to solve nonlinear PDEs. The strength of the Carleman-Newton method includes (1) no good initial guess is required and (2) the computational cost is not expensive. These features are rigorously proved. Having the solutions to this system in hand, we can directly compute the solution to the proposed inverse problem. Some numerical examples are displayed.

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“…The reader can find many other versions of Carleman estimates in [7,30,29,41,46]. These estimates are used to solve inverse problems; see e.g., [23,34,39].…”

Section: A Piece-wise Carleman Estimatementioning

confidence: 99%

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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The Carleman-based contraction principle to reconstruct the potential of nonlinear hyperbolic equations

Cited by 3 publications

References 34 publications

Global reconstruction of initial conditions of nonlinear parabolic equations via the Carleman-contraction method

Global reconstruction of initial conditions of nonlinear parabolic equations via the Carleman-contraction method

The Carleman-Newton method to globally reconstruct a source term for nonlinear parabolic equation

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

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