The Quasi-reversibility Method to Numerically Solve an Inverse Source Problem for Hyperbolic Equations

Le, Thuy T.; Nguyen, Loc H.; Nguyen, Thi-Phong; Powell, William

doi:10.1007/s10915-021-01501-3

Cited by 26 publications

(13 citation statements)

References 50 publications

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“…Corollary 2 Estimate (28) affirmatively answers the question how the fixed point of Φ approximate the true solution u * . In fact, when λ and β are fixed, using (25) and (26) and Remark 3, we obtain the following estimate with respect to the noise level…”

Section: A Fixed Point Theoremmentioning

confidence: 92%

“…They were first used to prove the unique continuation principle, see e.g., [6,34]. 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].…”

Section: A Carleman Estimatementioning

confidence: 99%

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

Nguyen¹

2022

Preprint

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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 Fixed Point Theoremmentioning

confidence: 92%

Section: A Carleman Estimatementioning

confidence: 99%

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

Nguyen¹

2022

Preprint

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“…Although the rigorous study of the asymptotic behavior of (2.6) as N large is missing, we do not experience any difficulty in our numerical study. We refer the readers to [15,30,32,23,34,39,35] for the successful use of similar approximations when the basis {Ψ n } is given in [19].…”

Section: An Approximate Cauchy Problem For Problem 11mentioning

confidence: 99%

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

Nguyen¹,

Nguyen²,

Truong³

2022

Preprint

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We develop an efficient and convergent numerical method for solving the inverse problem of determining the potential of nonlinear hyperbolic equations from lateral Cauchy data. In our numerical method we construct a sequence of linear Cauchy problems whose corresponding solutions converge to a function that can be used to efficiently compute an approximate solution to the inverse problem of interest. The convergence analysis is established by combining the contraction principle and Carleman estimates. We numerically solve the linear Cauchy problems using a quasi-reversibility method. Numerical examples are presented to illustrate the efficiency of the method.

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“…This basis was first introduced to solve the electrical impedance tomography problem with partial data in [9]. Afterward, it is widely used in our research group to solve a variety kinds of inverse problems, including ill-posed inverse source problems for elliptic equations [35], parabolic equations [25] [29] and hyperbolic equations [27], nonlinear coefficient inverse problems for elliptic equations [38], and parabolic equations [20,40,32,39,26], transport equations [14] and full transfer equations [37].…”

Section: A Special Orthonormal Basismentioning

confidence: 99%

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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The Quasi-reversibility Method to Numerically Solve an Inverse Source Problem for Hyperbolic Equations

Cited by 26 publications

References 50 publications

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

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

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

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

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