Gunther Uhlmann

Maarten V. de Hoop, Gunther Uhlmann, Andras Vasy et al.

We develop an algorithm for the computation of general Fourier integral operators associated with canonical graphs. The algorithm is based on dyadic parabolic decomposition using wave packets and enables the discrete approximate evaluation of the action of such operators on data in the presence of caustics. The procedure consists in the construction of a universal operator representation through the introduction of locally singularity-resolving diffeomorphisms, enabling the application of wave packet driven computation, and in the construction of the associated pseudo-differential joint-partition of unity on the canonical graphs. We apply the method to a parametrix of the wave equation in the vicinity of a cusp singularity.

Jianliang Qian, Plamen Stefanov, Gunther Uhlmann et al.

We present a new 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 might be also discontinuous across some surface. The latter problem arises in brain imaging. The new algorithm is based on an explicit formula in the form of a Neumann series. We present numerical examples with non-trapping, 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 new algorithm.

Tak Shing Au Yeung, Eric T. Chung, Gunther Uhlmann

In this paper, we consider the inverse problem of determining an unknown function defined in three space dimensions from its geodesic X-ray transform. The standard X-ray transform is defined on the Euclidean metric and is given by the integration of a function along straight lines. The geodesic X-ray transform is the generalization of the standard X-ray transform in Riemannian manifolds and is defined by integration of a function along geodesics. This paper is motivated by Uhlmann and Vasy's theoretical reconstruction algorithm for geodesic X-ray transform and mathematical formulation for traveltime tomography to develop a novel numerical algorithm for the stated goal. Our numerical scheme is based on a Neumann series approximation and a layer stripping approach. In particular, we will first reconstruct the unknown function by using a convergent Neumann series for each small neighborhood near the boundary. Once the solution is constructed on a layer near the boundary, we repeat the same procedure for the next layer, and continue this process until the unknown function is recovered on the whole domain. One main advantage of our approach is that the reconstruction is localized, and is therefore very efficient, compared with other global approaches for which the reconstructions are performed on the whole domain. We illustrate the performance of our method by showing some test cases including the Marmousi model. Finally, we apply this method to a travel time tomography in 3D, in which the inversion of the geodesic X-ray transform is one important step, and present several numerical results to validate the scheme.

Gunther Uhlmann, Yiran Wang

We study inverse problems consisting on determining medium properties using the responses to probing waves from the machine learning point of view. Based on the understanding of propagation of waves and their nonlinear interactions, we construct a deep convolutional neural network in which the parameters are used to classify and reconstruct the coefficients of nonlinear wave equations that model the medium properties. Furthermore, for given approximation accuracy, we obtain the depth and number of units of the network and their quantitative dependence on the complexity of the medium.