The aim of full waveform inversion is the reconstruction of subsurface material parameters (e.g., the velocity of compression and shear wave and the attenuation of these waves) from measurements of the full elastic wave field which has been excited locally by controlled sources or globally by earthquakes. From a mathematical point of view, we have to deal with a nonlinear parameter identification problem for the elastic wave equation. The least squares approach for the full nonlinear problem is called *full waveform inversion* (FWI). It needs to be regularized due to its illposedness. The resulting optimization problem with PDE constraints is mostly solved by gradient-type schemes which require repeated solutions of the direct and the adjoint time-dependent PDE. This approach became reasonable just recently due to the tremendously increased performance of modern computers. Nevertheless, the approach suffers twofold:

- Its practicality is still restricted by the numerical effort needed, for instance, by explicit time integrators, by the storage requirements for the full wave propagation in space and time, by non-linearities of the inverse problem, and by the slow convergence of gradient-type schemes.
- A comprehensive mathematical analysis of the inverse problem justifying the numerics is missing. However, such an analysis is needed to achieve the best possible reconstruction within the uncertainty and incompleteness of the measurement data.

We will work on both issues under the following guiding principle: Develop a sound mathematical theory driven by realistic scenarios. Furthermore, we will set up regularization schemes adapted to the underlying physics (for instance, preserving discontinuities of the material parameters and taking into account strongly anisotropic and heterogeneous porous media properties).

Some of the analytical questions we will address are:

- Between which spaces is the parameter-to-solution mapping injective? In other words: Which subsurface information is uniquely determined by the measurements? What is the sensitivity of different combinations of elastic material parameters and which combination of parameters is best resolvable?
- Between which spaces is the parameter-to-solution map Fréchet differentiable?
- Is the inverse problem illposed in the strict mathematical sense? Which analytical setting is appropriate to describe a noise level and to analyze a discrepancy principle?
- Which kind of regularization fits the physics of the problem?

Further, we will construct implicit hierarchical space-time discretizations leading to efficient parallel schemes for solving the direct problem and for evaluating the Fréchet derivative of the parameter-to-solution map and its adjoint.

In the long run, we want to incorporate dispersion, attenuation and anisotropy in a 3D setting, thus reducing the gap between mathematical idealization and practical needs by extending the model step by step to a realistic physical scenario. Our vision is to realize highly reliable reconstruction algorithms complying with industrial standards. To this end, we will combine up-to-date techniques for inverse problems with massively parallel explicit and implicit direct solvers for the coarse level of adaptive hierarchy space-time discretizations which allow for locally very exact resolutions and provide efficient coarse approximation for preconditioning.