Space-Time Methods for Acoustic Waves with Applications to Full Waveform Inversion

Classically, wave equations are considered as evolution equations where the derivative with respect to time is treated in a stronger way than the spatial differential operators. This results in an ordinary differential equation (ODE) with values in a function space, e.g. in a Hilbert space, with respect to the spatial variable. For instance, acoustic waves in a spatial domain $\Omega \subset \mathbb{R}^d$ for a given right-hand side $\mathbf b$ can be considered in terms of the following ODE \begin{equation*} \partial_t \mathbf y = A\mathbf y + \mathbf b\quad \text{ in }[0,T]\,,\quad \mathbf y(0) = \mathbf 0\,, \qquad A = \begin{pmatrix} 0 & \operatorname{div} \\ \nabla & 0 \end{pmatrix}, \end{equation*} where the solution $\mathbf y = (p, \mathbf v)$ is an element of the space $\mathrm C^0\big(0,T; \mathcal D(A)\big) \cap \mathrm C^1\big(0,T; \mathrm L_2(\Omega)\big)$ with $\mathcal D(A) \subset \mathrm H^1(\Omega) \times H(\operatorname{div}, \Omega)$. In order to analyze this ODE, space and time are treated separately and hence tools for partial differential equations are used in space and tools for ODEs are used in time. Typically, this separation carries over to the analysis of numerical schemes to approximate solutions of the equation. By contrast, in this work, we consider the space-time operator \begin{equation*} L (p,\mathbf v) = \begin{pmatrix} \partial_t p + \operatorname{div} \mathbf v \\ \partial_t \mathbf v + \nabla p \end{pmatrix}\,, \end{equation*} in $Q = (0,T) \times \Omega$ as a whole treating time and space dependence simultaneously in a variational manner. Using this approach, we constructed a space-time Hilbert space setting that allows for irregular solutions, e.g. with space-time discontinuities. Within this variational framework, we construct and analyzed two classes of non-conforming discretization schemes for acoustic waves, a Discontinuous Petrov-Galerkin method and a scheme of Least-Squares type. For both methods, we provide a convergence analysis exploiting tools from classical Finite Element theory for space and also time dependence. The theoretical predictions are complemented by extensive numerical experiments showing that high convergence rates are attained in practice. While considering the problem of Full Waveform Inversion (FWI), we focus on the derivation of Newton-type algorithms to tackle this inverse problem numerically. Here, we make extensive use of the space-time $\mathrm L_2(Q)$ adjoint $L^*$ that is easily accessible within our variational space-time framework. We implement a regularized inexact Newton method, CG-REGINN, and provide a numerical example for a benchmark problem

A space-time discontinuous Petrov- Galerkin method for acousticwaves

We apply the discontinuous Petrov-Galerkin (DPG) method to linear acoustic waves in space and time using the framework of first-order Friedrichs systems. Based on results for operators and semigroups of hyperbolic systems, we show that the ideal DPG method is wellposed. The main task is to avoid the explicit use of traces, which are difficult to define in Hilbert spaces with respect to the graph norm of the space-time differential operator. Then, the practical DPG method is analyzed by constructing a Fortin operator numerically. For our numerical experiments we introduce a simplified DPG method with discontinuous ansatz functions on the faces of the space-time skeleton, where the error is bounded by an equivalent conforming DPG method. Examples for a plane wave configuration confirms the numerical analysis, and the computation of a diffraction pattern illustrates a first step to applications

Space-time discontinuous Petrov-Galerkin methods for linearwave equations in heterogeneous media

We establish an abstract space-time DPG framework for the approximation of linear waves in heterogeneous media. The estimates are based on a suitable variational setting in the energy space. The analysis combines the approaches for acoustic waves in Gopalakrishnan / Sepulveda (A space-time DPG method for acoustic waves, arXiv 2017) and in Ernesti / Wieners (RICCAM proceedings, submitted 2017) and is based on the abstract definition of traces on the skeleton of the time-space sub-structuring. The method is evaluated by large-scale parallel computations motivated from applications in seismic imaging, where the computational domain can be restricted substantially to a subset of the full space-time cylinder

Visco-acoustic full waveform seismic inversion: from a DG forward solver to a Newton-CG inverse solver

In this paper we present a holistic framework for full waveform inversion (FWI) in the visco-acoustic regime. FWI entails the reconstruction of material parameters (such as density and sound speed) from measurements of reflected wave fields (seismograms). We derive a discontinuous Galerkin (DG) solver for the visco-acoustic wave equation and incorporate it into an inverse solver. For the DG discretization we provide a block diagonal preconditioner for the efficient computation of the time steps by GMRES which yields a convergence estimate in space and time. Numerical tests illustrate these results. Furthermore, we set up an inverse solver of well established Newton-CG type, and we express the required Fréchet derivative and its adjoint in the DG setting. Reconstructions from simulated cross-well seismograms highlight the challenges of FWI and demonstrate the performance of the scheme. Some of the inversion experiments use seismograms generated by an independent FDTD forward solver to avoid an inverse crime

Der Erleuchtete/ Großmütige und Gerechte Chur-Sächsische Ober-Hoff-Prediger/ Nemlich Des ... Martini Geiers/ Der Heil. Schrifft weitberühmten Doctoris ... Erleuchteter Verstand/ Großmütiges Hertz/ und Gerechter Ruhm

DER ERLEUCHTETE/ GROSSMÜTIGE UND GERECHTE CHUR-SÄCHSISCHE OBER-HOFF-PREDIGER/ NEMLICH DES ... MARTINI GEIERS/ DER HEIL. SCHRIFFT WEITBERÜHMTEN DOCTORIS ... ERLEUCHTETER VERSTAND/ GROSSMÜTIGES HERTZ/ UND GERECHTER RUHM Der Erleuchtete/ Großmütige und Gerechte Chur-Sächsische Ober-Hoff-Prediger/ Nemlich Des ... Martini Geiers/ Der Heil. Schrifft weitberühmten Doctoris ... Erleuchteter Verstand/ Großmütiges Hertz/ und Gerechter Ruhm ([1]r) Titelseite ([1]r) Widmung ([2]r) Folget der Lebens-Lauff/ Wie solcher von dem seeligen Mann selbst auffgesetzet worden. ([81]) Abdanckung/ (107