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We develop a deep learning approach to extract ray directions at discrete locations by analyzing highly oscillatory wave fields. A deep neural network is trained on a set of local plane-wave fields to predict ray directions at discrete locations. The resulting deep neural network is then applied to a reduced-frequency Helmholtz solution to extract the directions, which are further incorporated into a ray-based interior-penalty discontinuous Galerkin (IPDG) method to solve the Helmholtz equations at higher frequencies. In this way, we observe no apparent pollution effects in the resulting Helmholtz solutions in inhomogeneous media. Our 2D and 3D numerical results show that the proposed scheme is very efficient and yields highly accurate solutions., Comment: 30 pages

We will develop a nonlinear upscaling method for nonlinear transport equation. The proposed scheme gives a coarse scale equation for the cell average of the solution. In order to compute the parameters in the coarse scale equation, a local downscaling operator is constructed. This downscaling operation recovers fine scale properties using cell averages. This is achieved by solving the equation on an oversampling region with the given cell average as constraint. Due to the nonlinearity, one needs to compute these downscaling operations on the fly and cannot pre-compute these quantities. In order to give an efficient downscaling operation, we apply a deep learning approach. We will use a deep neural network to approximate the downscaling operation. Our numerical results show that the proposed scheme can achieve a good accuracy and efficiency., Comment: 18 pages

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., Comment: 31 pages