Abstract Nonlinear inverse problems are typically solved by minimizing a data-misfit functional, which is often non-convex that leads minimization algorithms to stagnate at a local minimum. A typical example is the cycle-skipping phenomenon in full waveform inversion (FWI) of seismic reflection or transmission data. To overcome this difficulty, distance functionals are constructed by training neural networks to emulate a convex distance measure. Two construction strategies are discussed: (i) a data-converter network that simplifies the forward map so that a standard quadratic loss becomes convex, and (ii) a scalar-valued distance-network based on the data residual. Training samples can be generated from measured data exploiting an approximate invariance of the forward operator. Under a set of structural assumptions, it is proven that applying the Landweber iteration to the learned functional is a well-defined regularization method that guarantees monotone error reduction and convergence to the exact solution as the noise level vanishes. Given certain restrictions on the trained network and the nonlinear forward operator, it is validated that these structural assumptions are satisfied by the learned distance. This methodology is numerically validated on the Camembert benchmark model for FWI in the acoustic regime. Replacing the conventional L 2 -misfit with the learned convexified functional considerably mitigates cycle-skipping and enlarges the domain of convergence for gradient-based optimization. Numerical experiments show that the Landweber scheme with the learned misfit reaches a lower reconstruction error than the standard least-squares approach. For comparison, in some experiments convergence has been accelerated by using a limited-memory BFGS optimizer leading to slightly larger reconstruction errors compared to the Landweber scheme. Overall, the work provides a systematic way to embed learned, convex distance measures into inverse problem solvers, supplies a solid theoretical foundation for their use, and demonstrates practical gains in a seismic imaging benchmark model.

We present a unified convergence analysis of inexact Newton regularizations with general uniformly convex penalty terms for nonlinear ill-posed problems in Banach spaces. These schemes consist of an outer (Newton) iteration and an inner iteration, which provides the update of the current outer iterate. To this end the nonlinear problem is linearized about the current iterate and the resulting linear system is inexactly solved by an inner regularization method. In our analysis we only rely on generic assumptions of the inner methods, and we show that a variety of regularization techniques satisfy these assumptions. For instance, gradient-type and iterated-Tikhonov methods are covered. Not only the technique of proof is novel, but also some results, because for the first time uniformly convex penalty terms stabilize the inner scheme in full generality. Numerical experiments based on the inverse problem of electrical impedance tomography illustrate the impact of different uniformly convex penalty terms.

We present a mathematical framework for viscoelastic full waveform inversion (FWI) in vertically transverse isotropic media. FWI can be formulated as the nonlinear inverse problem of identifying parameters in the underlying attenuating anisotropic wave equation given partial wave field measurements (seismograms). From a mathematical point of view, one has to solve an operator equation for the full waveform forward operator, which is the corresponding parameter-to-state map. We give a rigorous definition of this operator, show its Fr & eacute;chet differentiability, and explicitly characterize the adjoint operator of its Fr & eacute;chet derivative. Thus, we provide the main ingredients to implement Newton-type/gradient-based regularization schemes for FWI. Our approach can be directly applied to other concepts of anisotropy.

The term Kirchhoff migration refers to a collection of approximate linearized inversion formulas for solving the inverse problem of seismic tomography which entails reconstructing the Earth's subsurface from reflected wave fields. A number of such formulas exists, the first dating from the 1950 s. As far as we know, these formulas have not yet been mathematically compared with respect to their imaging properties. This shortcoming is to be alleviated by the present work: we systematically discuss the advantages and disadvantages of the formulas in 2D from a microlocal point of view. To this end we consider the corresponding imaging operators in an unified framework as pseudodifferential or Fourier integral operators. Numerical examples illustrate the theoretical insights and allow a visual comparison of the different formulas.

We discuss mapping properties of the parameter-to-state map of full waveform inversion and generalize the results of [M. Eller and A. Rieder, Inverse Problems, 37 (2021), 085011] from the acoustic to the viscoelastic wave equation. In particular, we establish injectivity of the Fre'\chet derivative of the parameter-to-state map for a semidiscrete seismic inverse problem in the viscoelastic regime. Here the finite-dimensional parameter space is restricted to functions having global support in the propagation medium (the nonlocal case) and that are locally linearly independent. As a consequence, we deduce local conditional well-posedness of this nonlinear inverse problem. Furthermore, we show that the tangential cone condition holds, which is an essential prerequisite in the convergence analysis of a variety of inversion algorithms for nonlinear ill-p osed problems.

Generalized Radon transforms (GRT) serve, for instance, as linear models for seismic imaging in the acoustic regime. They occur when the corresponding inverse problem is linearized about a known background compression wave speed (Born approximation). The resulting GRT is completely determined by this background velocity. In this work, we present an implementation of approximate inversion formulas for this class of GRTs proposed and analyzed in [Grathwohl et al., Inverse Problems, 34 (2018), 014002] and [Grathwohl et al., Inverse Problems, 34 (2018), 114001], where we restrict ourselves to layered background velocities in two dimensions. In a series of numerical experiments, we intensively test our implementation, reproducing theoretical predictions. Further, we drive the validity of the linearization to its limits.

SUMMARY Full-waveform inversion (FWI) has been proven to be an effective tool for high-resolution multiparameter imaging of the shallow subsurface. It has been shown that the Gauss–Newton (GN) optimization method uses the off-diagonal information contained in the Hessian matrix and can increase resolution and mitigate crosstalk in multiparameter viscoelastic FWI. In this work, we demonstrate the advantages of GN viscoelastic FWI over the conventional FWI with a conjugate gradient optimization method by using synthetic examples. We also investigate the potential of shallow seismic-wave 2-D viscoelastic FWI as a method for high-resolution hydrogeological characterization. The GN viscoelastic FWI is applied to two orthogonal profiles acquired at the Krauthausen natural laboratory (Germany). The groundwater table is located at around 2 m, which nicely agrees with an abrupt increase of P-wave velocity in the inverted results. FWI also reveals a low S-wave velocity layer at the depth of 4–6 m with high Poisson’s ratio values close to 0.5, which corresponds to a saturated sand layer known from previous studies. A K-mean cluster analysis is used to further analyse the multiparameter FWI results. By considering the derived Poisson’s ratio, P- and S-wave velocities, we convert the complex relationship between the multivariate data into a lithological meaningful zonation of the shallow subsurface. By comparing the lithological units in the alluvial aquifer with the cone penetration tests clusters, we conclude that the divided facies describe valuable characterization information about the heterogeneity and connectivity of the aquifer. This experiment indicates that the multiparameter models derived by viscoelastic FWI contain useful information for high-resolution aquifer characterization, and the potential of multiparameter FWI combined with cluster analysis in shallow subsurface characterization is encouraging.

Parameter identification tasks for partial differential equations are non-linear illposed problems where the parameters are typically assumed to be in L^∞ . This Banach space is non-smooth, non-reflexive and non-separable and requires therefore a more sophisticated regularization treatment than the more regular L^p -spaces with 1<p<∞ . We propose a novel inexact Newton-like iterative solver where the Newton update is an approximate minimizer of a smoothed Tikhonov functional over a finite-dimensional space whose dimension increases as the iteration progresses. In this way, all iterates stay bounded in L^∞ and the regularizer, delivered by a discrepancy principle, converges weakly- ⋆ to a solution when the noise level decreases to zero. Our theoretical results are demonstrated by numerical experiments based on the acoustic wave equation in one spatial dimension. This model problem satisfies all assumptions from our theoretical analysis.

Generalized Radon transforms are Fourier integral operators which are used, for instance, as imaging models in geophysical exploration. They appear naturally when linearizing about a known background compression wave speed. In this work we first consider a linearly increasing background velocity in two spatial dimensions. We verify the Bolker condition for the zero-offset scanning geometry and provide meaningful arguments for it to hold even if the common offset is positive. Based on this result we suggest an imaging operator for which we calculate the top order symbol in the zero-offset case to study how it maps singularities. Second, to support the usage of background models obtained from linear regression we present a stability result for the Bolker condition under perturbations of the background velocity and of the offset.

This book presents the notes from the seminar on wave phenomena given in 2019 at the Mathematical Research Center in Oberwolfach.

Mathematical modeling of physical processes yields a system of partial differential equations that describes the behavior of a system physically correct and allows for analytical and numerical predictions of the system behavior. Here we start by shortly summarizing modeling principles which are illustrated for simple linear models in one space dimension. Then this is specified for different types of wave equations.

In this chapter we extend the results from the previous one to linear and quasilinear Maxwell systems on a spatial domain G, endowed with boundary conditions. The general theory of symmetric hyperbolic systems is much more sophisticated in this case.

The linear wave equation can be analyzed in the framework of symmetric Friedrichs systems as a special case of linear hyperbolic conservation laws. Here, we introduce a general framework for the existence and uniqueness of strong and weak solutions in space and time which applies to general linear wave equations.

In an inverse problem we draw conclusions about the cause from its observed (measured) effect. This kind of task is typically ill-posed, that is, small changes (noisy measurements) in the effects lead to dramatic changes in the corresponding causes. Yet to obtain meaningful results the inversion or reconstruction process needs stabilization which is commonly referred to as regularization. All these basic concepts will be introduced first in some detail before we study advanced inverse wave problems. We start by giving two examples of inverse problems which are also ill-posed as we will learn later.

In this chapter, we analyze the full discretization error caused by the four schemes presented in Chap.