Numerical Analysis for a Hyperbolic PDE-Constrained Optimization Problem in Acoustic Full Waveform Inversion

TL;DR

This work develops and analyzes a fully discrete PDE-constrained optimization framework for acoustic full waveform inversion, combining a first-order state formulation with leapfrog time stepping and $P_1^h$ spatial discretization, while discretizing the control as piecewise constant. Under a CFL-type stability condition, the scheme is shown to be well-posed and stable, with the state discretization converging uniformly to the continuous forward solution. The authors establish strong convergence of discrete local minimizers to a given continuous local minimizer under a quadratic growth condition, and prove convergence of the discretized objective to the continuous one. A numerical experiment demonstrates the method's capability to reconstruct a spatially varying Slowness parameter from noisy observations, validating the theoretical results and showcasing practical applicability to acoustic FWI.

Abstract

This paper explores a fully discrete approximation for a nonlinear hyperbolic PDE-constrained optimization problem (P) with applications in acoustic full waveform inversion. The optimization problem is primarily complicated by the hyperbolic character and the second-order bilinear structure in the governing wave equation. While the control parameter is discretized using the piecewise constant elements, the state discretization is realized through an auxiliary first-order system along with the leapfrog time-stepping method and continuous piecewise linear elements. The resulting fully discrete minimization problem ($\text{P}_h$) is shown to be well-defined. Furthermore, building upon a suitable CFL-condition, we prove stability and uniform convergence of the state discretization. Our final result is the strong convergence result for ($\text{P}_h$) in the following sense: Given a local minimizer $\overline ν$ of (P) satisfying a reasonable growth condition, there exists a sequence of local minimizers of ($\text{P}_h$) converging strongly towards $\overline ν$.

Figures (1)

• Figure 1: Reconstruction in the numerical experiment.

Theorems & Definitions (15)

• Remark 2.1
• Lemma 3.1: ciarlet02
• Theorem 3.2
• proof
• Theorem 4.1
• proof
• Lemma 5.1
• proof
• Theorem 5.2
• proof