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 Locking-Free hp DPG Method for Linear Elasticity with Symmetric Stresses

We present two new methods for linear elasticity that simultaneously yield stress and displacement approximations of optimal accuracy in both the mesh size h and polynomial degree p. This is achieved within the recently developed discontinuous Petrov- Galerkin (DPG) framework. In this framework, both the stress and the displacement ap- proximations are discontinuous across element interfaces. We study locking-free convergence properties and the interrelationships between the two DPG methods

Efficient Resolution of Anisotropic Structures

We highlight some recent new delevelopments concerning the sparse representation of possibly high-dimensional functions exhibiting strong anisotropic features and low regularity in isotropic Sobolev or Besov scales. Specifically, we focus on the solution of transport equations which exhibit propagation of singularities where, additionally, high-dimensionality enters when the convection field, and hence the solutions, depend on parameters varying over some compact set. Important constituents of our approach are directionally adaptive discretization concepts motivated by compactly supported shearlet systems, and well-conditioned stable variational formulations that support trial spaces with anisotropic refinements with arbitrary directionalities. We prove that they provide tight error-residual relations which are used to contrive rigorously founded adaptive refinement schemes which converge in $L_2$. Moreover, in the context of parameter dependent problems we discuss two approaches serving different purposes and working under different regularity assumptions. For frequent query problems, making essential use of the novel well-conditioned variational formulations, a new Reduced Basis Method is outlined which exhibits a certain rate-optimal performance for indefinite, unsymmetric or singularly perturbed problems. For the radiative transfer problem with scattering a sparse tensor method is presented which mitigates or even overcomes the curse of dimensionality under suitable (so far still isotropic) regularity assumptions. Numerical examples for both methods illustrate the theoretical findings

Wavenumber Explicit Analysis of a DPG Method for the Multidimensional Helmholtz Equation

We study the properties of a novel discontinuous Petrov Galerkin (DPG) method for acoustic wave propagation. The method yields Hermitian positive definite matrices and has good pre-asymptotic stability properties. Numerically, we find that the method exhibits negligible phase errors (otherwise known as pollution errors) even in the lowest order case. Theoretically, we are able to prove error estimates that explicitly show the dependencies with respect to the wavenumber ω, the mesh size h, and the polynomial degree p. But the current state of the theory does not fully explain the remarkably good numerical phase errors. Theoretically, comparisons are made with several other recent works that gave wave number explicit estimates. Numerically, comparisons are made with the standard finite element method and its recent modification for wave propagation with clever quadratures. The new DPG method is designed following the previously established principles of optimal test functions. In addition to the nonstandard test functions, in this work, we also use a nonstandard wave number dependent norm on both the test and trial space to obtain our error estimates