Recovery of coefficients for a weighted p-Laplacian perturbed by a linear second order term

Cârstea, Cătălin I.; Kar, Manas

doi:10.1088/1361-6420/abcea1

Cited by 17 publications

(10 citation statements)

References 29 publications

(38 reference statements)

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“…Our main strategy is to use the linearization technique of Isakov and others [28,29,30,31,32] in dealing with nonlinear equations to decompose the inverse problem to the semilinear radiative transport equation ( 1) into an inverse coefficient problem for the linear transport equation where we reconstruct σ a and σ s by the result of Bal-Jollivet-Jungon [6], and an inverse source problem for the linear transport equation where we reconstruct the two-photon absorption coefficient σ b . This is the same type of strategy that have been successfully employed to solve many inverse problems for nonlinear PDEs recently; see for instance [4,11,13,14,15,18,20,25,26,33,35,36,38,39,40,41,43,44,45,46,47,50,58,61,62,63] and reference therein.…”

Section: Inverse Problems In the Radiative Transport Regimementioning

confidence: 99%

“…We denote the corresponding data by H T and H T . Then we have that u (1) = u (1) and u (2) and u (2) are solutions to (13) with σ b and σ b , respectively.…”

Section: Based On Propositionmentioning

confidence: 99%

“…Proof. From the data ( 14) and the fact that u (2) , and u (2) are solutions to (13) with the same σ a , we have that…”

Section: Based On Propositionmentioning

confidence: 99%

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Inverse transport and diffusion problems in photoacoustic imaging with nonlinear absorption

Lai¹,

Ren²,

Zhou³

2021

Preprint

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Motivated by applications in imaging nonlinear optical absorption by photoacoustic tomography (PAT), we study in this work inverse coefficient problems for a semilinear radiative transport equation and its diffusion approximation with internal data that are functionals of the coefficients and the solutions to the equations. Based on the techniques of first-and secondorder linearization, we derive uniqueness and stability results for the inverse problems. For uncertainty quantification purpose, we also establish the stability of the reconstruction of the absorption coefficients with respect to the change in the scattering coefficient.

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Section: Inverse Problems In the Radiative Transport Regimementioning

confidence: 99%

“…We denote the corresponding data by H T and H T . Then we have that u (1) = u (1) and u (2) and u (2) are solutions to (13) with σ b and σ b , respectively.…”

Section: Based On Propositionmentioning

confidence: 99%

See 1 more Smart Citation

Inverse transport and diffusion problems in photoacoustic imaging with nonlinear absorption

Lai¹,

Ren²,

Zhou³

2021

Preprint

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“…For instance, the coefficients of a quasilinear elliptic equation in divergence form are reconstructed from the the Dirichlet to Neumann (DN) map defined on the boundary (see [10] for details). In [11], similar analysis has been carried out for the perturbed weighted p-Laplacian equation to recover the coefficients from prescribed DN-map.…”

Section: Introductionmentioning

confidence: 99%

Reconstruction for the time-dependent coefficients of a quasilinear dynamical Schr{ö}dinger equation

Nakamura¹,

sarkar²,

Vashisth³

2022

Preprint

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We study an inverse problem related to the dynamical Schrödinger equation in a bounded domain of R n , n ≥ 2. Since the concerned non-linear Schrödinger equation possesses a trivial solution, we linearize the equation around the trivial solution. Demonstrating the well-posedness of the direct problem under appropriate conditions on initial and boundary data, it is observed that the solution admits ε-expansion. By taking into account the fact that the terms O(|∇u(t, x)| 3 ) are negligible in this context, we shall reconstruct the time-dependent coefficients such as electric potential and vectorvalued function associated with quadratic nonlinearity from the knowledge of input-output map using the geometric optics solution and Fourier inversion. H r,s (Σ) = L 2 (0, T ; H r (∂Ω)) ∩ H s (0, T ; L 2 (∂Ω))2020 Mathematics Subject Classification. 35R30.

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“…The enclosure method introduced by Ikehata uses Complex Geometrical Optics (CGO) solution in place of point sources (see [56,57]). The authors in [64] have treated a nonlinear problem (linear plus a nonlinear term) which is a particular case of the model proposed in this article.…”

Section: Introductionmentioning

confidence: 99%

Monotonicity Principle in tomography of nonlinear conducting materials ^*

et al. 2021

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We treat an inverse electrical conductivity problem which deals with the reconstruction of nonlinear electrical conductivity starting from boundary measurements in steady currents operations. In this framework, a key role is played by the Monotonicity Principle, which establishes a monotonic relation connecting the unknown material property to the (measured) Dirichlet-to-Neumann operator (DtN). Monotonicity Principles are the foundation for a class of non-iterative and real-time imaging methods and algorithms. In this article, we prove that the monotonicity principle for the Dirichlet Energy in nonlinear problems holds under mild assumptions. Then, we show that apart from linear and p-Laplacian cases, it is impossible to transfer this monotonicity result from the Dirichlet Energy to the DtN operator. To overcome this issue, we introduce a new boundary operator, identified as an average DtN operator.

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Recovery of coefficients for a weighted p-Laplacian perturbed by a linear second order term

Abstract: This paper considers the inverse boundary value problem for the equation ∇ ⋅ (σ∇u + a|∇u| p−2∇u) = 0. We give a procedure for the recovery of the coefficients σ and a from the corresponding Dirichlet-to-Neumann map, under suitable regularity and ellipticity assumptions.

Cited by 17 publications

References 29 publications

Inverse transport and diffusion problems in photoacoustic imaging with nonlinear absorption

Inverse transport and diffusion problems in photoacoustic imaging with nonlinear absorption

Reconstruction for the time-dependent coefficients of a quasilinear dynamical Schr{ö}dinger equation

Monotonicity Principle in tomography of nonlinear conducting materials ^*

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Recovery of coefficients for a weighted p-Laplacian perturbed by a linear second order term

Abstract: This paper considers the inverse boundary value problem for the equation ∇ ⋅ (σ∇u + a|∇u| p−2∇u) = 0. We give a procedure for the recovery of the coefficients σ and a from the corresponding Dirichlet-to-Neumann map, under suitable regularity and ellipticity assumptions.

Cited by 17 publications

References 29 publications

Inverse transport and diffusion problems in photoacoustic imaging with nonlinear absorption

Inverse transport and diffusion problems in photoacoustic imaging with nonlinear absorption

Reconstruction for the time-dependent coefficients of a quasilinear dynamical Schr{ö}dinger equation

Monotonicity Principle in tomography of nonlinear conducting materials *

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Product

Resources

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Monotonicity Principle in tomography of nonlinear conducting materials ^*