Stability of the Gel'fand inverse boundary problem via the unique continuation

Burago, Dmitri; Ivanov, S. V.; Lassas, Matti; Lu, Jingzhou

doi:10.48550/arxiv.2012.04435

Cited by 4 publications

(16 citation statements)

References 22 publications

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“…It was shown in [57] that inverse boundary spectral problems are equivalent to inverse problems for the wave and heat equations in the time domain. The stability of the solutions of these inverse problems have been analyzed in [1,20,25,43,81]. Numerical methods to solve the Gel'fand's inverse problems have been studied in [13,40,41].…”

Section: Earlier Results and Related Inverse Problemsmentioning

confidence: 99%

The Gel'fand's inverse problem for the graph Laplacian

Blåsten¹,

Isozaki²,

Lassas³

et al. 2021

Preprint

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We study the Gel'fand's inverse boundary spectral problem of determining a finite weighted graph. Suppose that the set of vertices of the graph is a union of two disjoint sets: X = B ∪ G, where B is called the set of the boundary vertices and G is called the set of the interior vertices. We consider the case where the vertices in the set G and the edges connecting them are unknown. Assume that we are given the set B and the pairs (λj, φj|B), where λj are the eigenvalues of the graph Laplacian and φj|B are the values of the corresponding eigenfunctions at the vertices in B. We show that the graph structure, namely the unknown vertices in G and the edges connecting them, along with the weights, can be uniquely determined from the given data, if every boundary vertex is connected to only one interior vertex and the graph satisfies the following property: any subset S ⊆ G of cardinality |S| 2 contains two extreme points. A point x ∈ S is called an extreme point of S if there exists a point z ∈ B such that x is the unique nearest point in S from z with respect to the graph distance. This property is valid for several standard types of lattices and their perturbations.

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Section: Earlier Results and Related Inverse Problemsmentioning

confidence: 99%

The Gel'fand's inverse problem for the graph Laplacian

Blåsten¹,

Isozaki²,

Lassas³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Under such type of conditions, invariant stability results for various inverse problems have been proven in [5,29,75,76]. In particular, for the inverse interior spectral problem on manifolds with non-trivial topology, stability results have been obtained in [5,9], see also [14] for analogous results for manifolds with boundary. In the latter, it was shown that the convergence of the boundary spectral data implies the convergence of the manifolds with respect to the Gromov-Hausdorff distance.…”

Section: 34mentioning

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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“…Tataru's unique continuation theorem is crucial for the Boundary Control method in solving inverse problems for linear equations, see e.g. [3,7,13,14,26,28,29]. The unique continuation for linear systems of hyperbolic equations was studied in [21], and the result can be applied to the time-dependent classical elasticity system and the Maxwell system.…”

Section: Introductionmentioning

confidence: 99%

“…Inspired by Tataru's ideas in [44], the quantitative stability of the unique continuation for the wave operator was obtained by [11,12] and [30] independently. An explicit stability of the unique continuation on Riemannian manifolds with boundary was recently obtained by [14], with U being a subset of the boundary. In this paper, we study the explicit stability of the unique continuation for a linear system of hyperbolic equations on Riemannian manifolds with boundary, with U being an interior open subset of the manifold.…”

Section: Introductionmentioning

confidence: 99%

Quantitative unique continuation for the elasticity system with application to the kinematic inverse rupture problem

Hoop¹,

Lassas²,

Lu³

et al. 2022

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We obtain explicit estimates on the stability of the unique continuation for a linear system of hyperbolic equations. In particular our result applies to the elasticity and Maxwell systems. As an application, we study the kinematic inverse rupture problem of determining the jump in displacement and the friction force at the rupture surface, and we obtain new features on the stable unique continuation up to the rupture surface.

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Stability of the Gel'fand inverse boundary problem via the unique continuation

Cited by 4 publications

References 22 publications

The Gel'fand's inverse problem for the graph Laplacian

The Gel'fand's inverse problem for the graph Laplacian

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

Quantitative unique continuation for the elasticity system with application to the kinematic inverse rupture problem

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