The eikonal equation is instrumental in many applications in several fields ranging from computer vision to geoscience. This equation can be efficiently solved using the iterative Fast Sweeping (FS) methods and the direct Fast Marching (FM) methods. However, when used for a point source, the original eikonal equation is known to yield inaccurate numerical solutions, because of a singularity at the source. In this case, the factored eikonal equation is often preferred, and is known to yield a more accurate numerical solution. One application that requires the solution of the eikonal equation for point sources is travel time tomography. This inverse problem may be formulated using the eikonal equation as a forward problem. While this problem has been solved using FS in the past, the more recent choice for applying it involves FM methods because of the efficiency in which sensitivities can be obtained using them. However, while several FS methods are available for solving the factored equation, the FM method is available only for the original eikonal equation.In this paper we develop a Fast Marching algorithm for the factored eikonal equation, using both first and second order finite-difference schemes. Our algorithm follows the same lines as the original FM algorithm and requires the same computational effort. In addition, we show how to obtain sensitivities using this FM method and apply travel time tomography, formulated as an inverse factored eikonal equation. Numerical results in two and three dimensions show that our algorithm solves the factored eikonal equation efficiently, and demonstrate the achieved accuracy for computing the travel time. We also demonstrate a recovery of a 2D and 3D heterogeneous medium by travel time tomography using the eikonal equation for forward modeling and inversion by Gauss-Newton.

SUMMARYA novel multigrid algorithm for computing the principal eigenvector of column-stochastic matrices is developed. The method is based on an approach originally introduced by Horton and Leutenegger (Perform. Eval. Rev. 1994; 22:191-200) whereby the coarse-grid problem is adapted to yield a better and better coarse representation of the original problem. A special feature of the present approach is the squaring of the stochastic matrix-followed by a stretching of its spectrum-just prior to the coarse-grid correction process. This procedure is shown to yield good convergence properties, even though a cheap and simple aggregation is used for the restriction and prolongation matrices, which is important for maintaining competitive computational costs. A second special feature is a bottom-up procedure for defining coarse-grid aggregates.

Estimating parameters of Partial Differential Equations (PDEs) from noisy and indirect measurements often requires solving ill-posed inverse problems. These so called parameter estimation or inverse medium problems arise in a variety of applications such as geophysical, medical imaging, and nondestructive testing. Their solution is computationally intense since the underlying PDEs need to be solved numerous times until the reconstruction of the parameters is sufficiently accurate. Typically, the computational demand grows significantly when more measurements are available, which poses severe challenges to inversion algorithms as measurement devices become more powerful.In this paper we present jInv, a flexible framework and open source software that provides parallel algorithms for solving parameter estimation problems with many measurements. Being written in the expressive programming language Julia, jInv is portable, easy to understand and extend, cross-platform tested, and well-documented. It provides novel parallelization schemes that exploit the inherent structure of many parameter estimation problems and can be used to solve multiphysics inversion problems as is demonstrated using numerical experiments motivated by geophysical imaging.

No abstract

Full waveform inversion (FWI) is a process in which seismic numerical simulations are fit to observed data by changing the wave velocity model of the medium under investigation. The problem is non-linear, and therefore optimization techniques have been used to find a reasonable solution to the problem. The main problem in fitting the data is the lack of low spatial frequencies. This deficiency often leads to a local minimum and to non-plausible solutions. In this work we explore how to obtain low frequency information for FWI. Our approach involves augmenting FWI with travel time tomography, which has low-frequency features. By jointly inverting these two problems we enrich FWI with information that can replace low frequency data. In addition, we use high order regularization, in a preliminary inversion stage, to prevent high frequency features from polluting our model in the initial stages of the reconstruction. This regularization also promotes the non-dominant low-frequency modes that exist in the FWI sensitivity. By applying a joint FWI and travel time inversion we are able to obtain a smooth model than can later be used to recover a good approximation for the true model. A second contribution of this paper involves the acceleration of the main computational bottleneck in FWI-the solution of the Helmholtz equation. We show that the solution time can be reduced by solving the equation for multiple right hand sides using block multigrid preconditioned Krylov methods.