Seismic tomography is a technique to determine the material properties of the Earth's subsurface based on the observation of seismograms. This can be stated as a PDE-constrained optimization problem governed by the elastic wave equation. We present a semismooth Newton-PCG method with a trust-region globalization for full-waveform inversion that uses a Moreau-Yosida regularization to handle additional constraints on the material parameters. We develop results on the differentiability of the parameter-to-state operator and analyze the proposed optimization method in a function space setting.
The elastic wave equation is discretized by a high-order continuous Galerkin method in space and an explicit Newmark time-stepping scheme. The matrix-free implementation relies on the adjoint-based computation of the gradient and Hessian-vector products and on a MPI-based parallelization. Randomized data reduction techniques are employed to efficiently gather information from a large number of seismic events. Especially, we utilize randomly chosen linear combinations of the seismic sources to approximate the Hessian operator and to reduce the number of required PDE solves during the CG iterations. Numerical results are shown for an application in geophysical exploration on reservoir-scale and the joint inversion for both Lamé coefficients with constraints on the Poisson's ratio.