Stability estimates for the X-ray transform of tensor fields and boundary rigidity

By Plamen Stefanov, Gunther Uhlmann. 2004

163 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/902866297c1c3c05fdc62b273bfcff78b4a4ca05

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In Computational Theory and Mathematics

In Mathematical Physics

In Mechanics of Materials

Citations in the corpus

Sensitivity analysis of an inverse problem for the wave equation with caustics (2012, result): “Moreover, our result is fundamentally different from those in the literature where Lipschitz, Hölder or logarithmic estimates are derived; see, for example, [6], [1], [33], [25], [26], [27], [28], [19] and [14].”
Rigidity of broken geodesic flow and inverse problems (2007, background): “The classical boundary rigidity problem is the following (see [13], [14], [17], [19], [30], [35], [36], [37], [40], [41]): Assume that we know the distances dist(x, y) of boundary points x, y ∈ ∂M.”
The X-Ray Transform for a Generic Family of Curves and Weights (2007, background): “different choice of H and a coordinate system on it. On the other hand, injectivity of N�,w,α is equivalent to injectivity of I�,w,α , and the latter is equivalent to injectivity of I�,w restricted to supp α ;s ee [ 20 ], and this property is independent of the choice of H and the coordinates on it ”
Inverse problems: seeing the unseen (2014, methodology): “In [179], stability estimates for s-injectivemetrics [see (99) below] were shown and sharp estimates about the recovery of a 1-form f = f j dx j and a function f from the associated Ig f which is defined by Ig f (γ ) = ∫ fi (γ (t))γ̇ i (t) dt.”
Local and global boundary rigidity and the geodesic X-ray transform in the normal gauge (2017, methodology): “This does not work for the non-linear problem, however: once the coordinates are fixed, we can make the tensor g1 − g2 solenoidal relative to a reference metric by enforcing δsz(g1 − g2) = 0, changing g1 − g2 by a potential term, but this eliminates the identity I(g1 − g2) = 0 where I is the general”
The geodesic X-ray transform with fold caustics (2010, methodology): “A direct calculation — see [Stefanov and Uhlmann 2004] and Theorem 5.”