### Nonlinear inverse (boundary value) problems: Direct methods

**Direct methods – Time-harmonic waves**

We used complex geometrical optics (CGO) solutions in a pilot study of a direct nonlinear method for the inverse boundary value problem for the Helmholtz equation. The underlying ideas are based on techniques introduced in proofs of the uniqueness of this inverse problem. In two dimensions, we developed a computational approach for generating the *nonlinear scattering transform *from boundary (Dirichlet-to-Neumann map) data via solving a boundary integral equation (step 1). Once the scattering transform is obtained one can compute the CGO solution in the interior (subsurface) by solving another integral equation (step 2). From an expansion of the CGO solution we then extract the wavespeed. We introduced a regularization corresponding with a truncation of the scattering transform to mitigate the intrinsic instability associated with step 1, which enabled us to recover coarse scale variations in the wavespeed [preprint 4]. The nonlinear lowpass filtering also provides insight in possible smooth approximations of complex models with corresponding approximations of the data.

**Nonlinear, coupled physics and hybrid inverse problems**

We analyzed the *time-harmonic diffuse electromagnetic inverse problem with internal data *obtained from *electroseis**mic *conversion [preprint 6]. We obtained a Lipschitz stability estimate for the recovery of the electrokinetic mobility parameter, conductivity and relative permittivity. We derived a reconstruction of the electrokinetic mobility parameter via solving a transport equation. Typically, hybrid inverse problems arising in nonlinear physics yield improved stability estimates with the promise of fundamentally improved resolution.

**Direct geometric methods**

Subject to sufficient regularity of and structure in the coefficients, the seismic inverse boundary value problem is natu- rally related to certain geometric inverse problems. We have been conncerned with partial data and the reconstruction of part of the structure and coarse-scale variations. To this end, we have been exploiting techniques from Riemannian geometry, so far, restricting ourselves to the isotropic case. The Riemannian metric corresponds with the wavespeed.

- We extended techniques known as time migration to estimate the so-called time-migration velocity matrix [97]. This matrix determines the so-called shape operator which we consider now as the data.
*Generalization of Dix’ method*. We recently succeeded in generalizing Dix’ original method to the direct recon- struction of wavespeeds in three dimensions along neighborhoods of rays admitting the formation of caustics. The key idea is to introduce Riemannian normal coordinates which generalize ‘time’ coordinates in reflection seismology. In the first step we reconstruct the Riemannian metric in the these coordinates via the reconstruction of directional Riemannian curvature [111]. In the second step we exploit a relationship between the Riemannian curvature and the Ricci tensor to obtain the metric in Cartesian (or depth) coordinates [112].- The generalized Dix’ method is, under certain conditions, applicable to the joint mapping of microseismic activity and recovery of wavespeed, connected with aftershocks of earthquakes or the illumination of open fractures.