The Dirichlet-to-Neumann map for a semilinear wave equation on Lorentzian manifolds

Hintz, Peter; Uhlmann, Günther; Zhai, Jing

doi:10.48550/arxiv.2103.08110

Cited by 3 publications

(4 citation statements)

References 44 publications

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“…where O j ⊂ (0, T ) × ∂Ω is a small open neighborhood of γ x j ,ξ j (t o j ) with t o j defined in (14). Then following the same ideas of scattering control as in [38,Proposition 3.2], we can choose a boundary source f j and set v (k) j be the solution to the boundary value problem (17) with f j and b (k) , h (k) , such that…”

Section: The Recovery Of the One-form And The Nonlinearitymentioning

confidence: 99%

Nonlinear acoustic imaging with damping

Zhang¹

2023

Preprint

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In this paper, we consider an inverse problem for a nonlinear wave equation with a damping term and a general nonlinear term. This problem arises in nonlinear acoustic imaging and has applications in medical imaging and other fields. The propagation of ultrasound waves can be modeled by a quasilinear wave equation with a damping term. We show the boundary measurements encoded in the Dirichlet-to-Neumann map (DN map) determine the damping term and the nonlinearity at the same time. In a more general setting, we consider a quasilinear wave equation with a one-form (a first-order term) and a general nonlinear term. We prove the one-form and the nonlinearity can be determined from the DN map, up to a gauge transformation, under some assumptions.This theorem shows the unique recovery of the one-form b(x) and the nonlinear coefficients β m+1 for m ≥ 1, from the knowledge of the DN map, up to a gauge transformation, under our assumptions. On the one hand, without knowing the potential h(x), one can recover b(x) up to an error term 2̺ −1 d̺, where ̺ ∈ C ∞ (M ) with ̺| ∂M = 1. On the other hand, with the assumptions on h(x), we can show this error term is given by some ̺ ∈ C ∞ (Ω) with ̺| ∂Ω = 1, which corresponds to a gauge transformation, for more details see Section 2. We point out it would be interesting to consider the recovery of the potential h(x) from the DN map but this is out of scope for this work.The inverse problems of recovering the metric and the nonlinear term for a semilinear wave equation were considered in [54], in a globally hyperbolic Lorentzian manifold without boundary. The main idea is to use the multi-fold linearization and the nonlinear interaction of waves. By choosing specially designed sources, one can expect to detect the new singularities produced by the interaction of distorted plane waves, from the measurements. The information about the metric and the nonlinearity is encoded in these new singularities. One can extract such information from the principal symbol of the new singularities, using the calculus of conormal distributions and paired Lagrangian distributions. Starting with [54,53], there are many works studying inverse problems for nonlinear hyperbolic equations, see

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Section: The Recovery Of the One-form And The Nonlinearitymentioning

confidence: 99%

Nonlinear acoustic imaging with damping

Zhang¹

2023

Preprint

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“…Especially the works [14,15,19] introduced the concept of measurement function (the term measurement function was coined in [20]) that allowed to apply the higher order linearization method also for inverse problems of boundary value problems for nonlinear wave equations. The works [11,12,21] study inverse problems for boundary value problems for nonlinear wave equations by using the aforementioned method. We refer the reader to the works [1,4,5,6,7,17,23,24,29,30,31] for more examples of inverse problems for nonlinear wave type and hyperbolic equations, and to [12] for additional references.…”

mentioning

confidence: 99%

“…The works [11,12,21] study inverse problems for boundary value problems for nonlinear wave equations by using the aforementioned method. We refer the reader to the works [1,4,5,6,7,17,23,24,29,30,31] for more examples of inverse problems for nonlinear wave type and hyperbolic equations, and to [12] for additional references. The very recent work [28] does a numerical study of an inverse problem for a nonlinear elliptic equation by using the higher order linearization method.…”

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confidence: 99%

An inverse problem for a semi-linear wave equation: a numerical study

Lassas¹,

Liimatainen²,

Potenciano-Machado³

et al. 2022

Preprint

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We consider an inverse problem of recovering a potential associated to a semi-linear wave equation with a quadratic nonlinearity in 1 + 1 dimensions. We develop a numerical scheme to determine the potential from a noisy Dirichlet-to-Neumann map on the lateral boundary. The scheme is based on the recent higher order linearization method [20]. We also present an approach to numerically estimating two-dimensional derivatives of noisy data via Tikhonov regularization. The methods are tested using synthetic noisy measurements of the Dirichlet-to-Neumann map. Various examples of reconstructions of the potential functions are given.

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“…In the study of this problem, the non-linear interaction of linearized waves produced by suitable sources in U × R + can be used to produce "artificial" microlocal point sources at the points y ∈ M , including the unknown region M \ U where the original source f vanishes, see [40,54,57,59,64,80,83]. The wave fronts that are produced by these point sources and are observed in U determine the distances d(x, y) for the points x ∈ M and y ∈ U .…”

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confidence: 99%

Reconstruction and interpolation of manifolds II: Inverse problems for Riemannian manifolds with partial distance data

Fefferman¹,

Ivanov²,

Lassas³

et al. 2021

Preprint

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We consider how a closed Riemannian manifold and its metric tensor can be approximately reconstructed from local distance measurements. In the part 1 of the paper, we considered the construction of a smooth manifold in the case when one is given the noisy distances d(x, y) = d(x, y) + εx,y for all points x, y ∈ X, where X is a δ-dense subset of M and |εx,y| < δ. In this paper we consider a similar problem with partial data, that is, the approximate construction of the manifold (M, g) when we are given d(x, y) for x ∈ X and y ∈ U ∩X, where U is an open subset of M . As an application, we consider the inverse problem of determining the manifold (M, g) with non-negative Ricci curvature from noisy observations of the heat kernel G(y, z, t) with separated observation points y ∈ U and source points z ∈ M \ U on the time interval 0 < t < 1.

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The Dirichlet-to-Neumann map for a semilinear wave equation on Lorentzian manifolds

Cited by 3 publications

References 44 publications

Nonlinear acoustic imaging with damping

Nonlinear acoustic imaging with damping

An inverse problem for a semi-linear wave equation: a numerical study

Reconstruction and interpolation of manifolds II: Inverse problems for Riemannian manifolds with partial distance data

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