We present a novel full-waveform inversion approach which can reduce the computational cost by up to an order of magnitude compared to conventional approaches, provided that variations in medium properties are sufficiently smooth. Our method is based on the usage of wavefield-adapted meshes which accelerate the forward and adjoint wavefield simulations. By adapting the mesh to the expected complexity and smoothness of the wavefield, the number of elements needed to discretise the wave equation can be greatly reduced. This leads to spectral-element meshes which are optimally tailored to source locations and medium complexity. We demonstrate a workflow which opens up the possibility to use these meshes in full-waveform inversion and show the computational advantages of the approach. We provide examples in 2-D and 3-D to illustrate the concept, describe how the new workflow deviates from the standard full-waveform inversion workflow, and explain the additional steps in detail.

An order of magnitude speed-up in finite-element modelling of wave propagation can be achieved by adapting the mesh to the anticipated space-dependent complexity and smoothness of the waves. This can be achieved by designing the mesh not only to respect the local wavelengths, but also the propagation direction of the waves depending on the source location, hence by anisotropic adaptive mesh refinement. Discrete gradients with respect to material properties as needed in full waveform inversion can still be computed exactly, but at greatly reduced computational cost. In order to do this, we explicitly distinguish the discretization of the model space from the discretization of the wavefield and derive the necessary expressions to map the discrete gradient into the model space. While the idea is applicable to any wave propagation problem that retains predictable smoothness in the solution, we highlight the idea of this approach with instructive 2D examples of forward as well as inverse elastic wave propagation.Furthermore, we apply the method to 3D global seismic wave simulations and demonstrate how meshes can be constructed that take advantage of high-order mappings from the reference coordinates of the finite elements to physical coordinates. Error level and speed-ups are estimated based on convergence tests with 1D and 3D models.