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

Abstract

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