Plamen Stefanov

Thermoacoustic tomography with variable sound speed

We study the mathematical model of thermoacoustic tomography in media with a variable speed for a fixed time interval [0, T] so that all signals issued from the domain leave it after time T. In the case of measurements on the whole boundary, we give an explicit solution in terms of a Neumann series expansion. We give almost necessary and sufficient conditions for uniqueness and stability when the measurements are taken on a part of the boundary.

Boundary rigidity and stability for generic simple metrics

Let (M, g) be a Riemannian manifold with boundary. Denote by pg the distance function in the metric g. We consider the inverse problem of whether pg(x,y), known for all x, y on <9M, determines the metric uniquely. This problem arose in geophysics in an attempt to determine the inner structure of the Earth by measuring the travel times of seismic waves. It goes back to Herglotz [H] and Wiechert and Zoeppritz [WZ]. Although the emphasis has been in the case that the medium is isotropic, the anisotropic case has been of interest in geophysics since it has been found that the inner core of the Earth exhibits anisotropic behavior [Cr]. In differential geometry this inverse problem has been studied because of rigidity questions and is known as the boundary rigidity problem. It is clear that one cannot determine the metric uniquely. Any isometry which is the identity at the boundary will give rise to the same measurements. Furthermore, the boundary distance function only takes into account the shortest paths, and it is easy to find counterexamples to unique determination, so one needs to pose some restrictions on the metric. Michel [Mi] conjectured that a simple metric g is uniquely determined, up to an action of a diffeomorphism fixing the boundary, by the boundary distance function pg(x,y) known for all x and y on dM. We recall Definition 1.1. We say that the Riemannian metric g is simple in M, if dM is strictly convex w.r.t. g, and for any x G M, the exponential map exp^ : exp~1(M) ? M is a diffeomorphism.

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

We study the boundary rigidity problem for domains in Rn: is a Riemannian metric uniquely determined, up to an action of diffeomorphism fixing the boundary, by the distance function g.x; y/ known for all boundary points x andy? It was conjectured by Michel that this was true for simple metrics. In this paper, we study the linearized problem first which consists of determining a symmetric 2-tensor, up to a potential term, from its geodesic X-ray integral transformIg . We prove that the normal operator Ng D I g Ig is a pseudodifferential operator provided that g is simple, find its principal symbol, identify its kernel, and construct a microlocal parametrix. We prove hypoelliptic type of stability estimate related to the linear problem. Next we apply this estimate to show that unique solvability of the linear problem for a given simple metric g, up to potential terms, implies local uniqueness for the non-linear boundary rigidity problem near that g.

An Efficient Neumann Series-Based Algorithm for Thermoacoustic and Photoacoustic Tomography with Variable Sound Speed

We present an efficient algorithm for reconstructing an unknown source in thermoacoustic and photoacoustic tomography based on the recent advances in understanding the theoretical nature of the problem. We work with variable sound speeds that also might be discontinuous across some surface. The latter problem arises in brain imaging. The algorithmic development is based on an explicit formula in the form of a Neumann series. We present numerical examples with nontrapping, trapping, and piecewise smooth speeds, as well as examples with data on a part of the boundary. These numerical examples demonstrate the robust performance of the Neumann series-based algorithm.

Thermoacoustic tomography arising in brain imaging

We study the mathematical model of thermoacoustic and photoacoustic tomography when the sound speed has a jump across a smooth surface. This models the change of the sound speed in the skull when trying to image the human brain. We derive an explicit inversion formula in the form of a convergent Neumann series under the assumptions that all singularities from the support of the source reach the boundary.

Quasimodes and resonances: Sharp lower bounds

We prove that, asymptotically, any cluster of quasimodes close to each other approximates at least the same number of resonances, counting multiplicities. As a consequence, we get that the counting function of the number of resonances close to the real axis is bounded from below essentially by the counting function of the quasimodes.

Electrochemical deposition of thin zirconia films on stainless steel 316 L

Zirconia films (ZrO2) have been deposited electrochemically on stainless steel SS 316 L in a non-aqueous electrolyte based on absolute ethyl alcohol and ZrCl4. Scanning electron microscopy (SEM) studies have shown that the film consists of crystallites with a spheroidal shape, forming agglomerates with a very large surface area. Their sizes vary within the range of 0.1–0.5 μm. The layer composition is very close to the stoichiometric ZrO2, as has been determined by X-ray photoelectron spectroscopy (XPS).

Integral geometry of tensor fields on a class of non-simple Riemannian manifolds

We study the geodesic X-ray transform IΓ of tensor fields on a compact Riemannian manifold M with non-necessarily convex boundary and with possible conjugate points. We assume that IΓ is known for geodesics belonging to an open set Γ with endpoints on the boundary. We prove generic s-injectivity and a stability estimate under some topological assumptions and under the condition that for any (x, ξ) ∈ T*M, there is a geodesic in Γ through x normal to ξ without conjugate points.

Neumann resonances in linear elasticity for an arbitrary body

We study resonances (scattering poles) associated to the elasticity operator in the exterior of an arbitrary obstacle inR3 with Neumann boundary conditions. We prove that there exists a sequence of resonances tending rapidly to the real axis.

Reconstruction of the coefficients of the stationary transport equation from boundary measurements

We study the inverse problem of recovering the absorption coefficient and the collision kernel in the stationary linear Boltzmann equation in a bounded domain from the albedo operator on the boundary. We show that under some conditions on the coefficients that guarantee well-posedness of the direct problem, the inverse problem has a unique solution. Moreover, we provide explicit formulae for recovering , k.