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

TL;DR

This paper develops a fully discrete numerical method for a hyperbolic PDE-constrained optimization problem in acoustic waveform inversion, proving stability, convergence, and the existence of discrete minimizers approximating continuous solutions.

Contribution

It introduces a novel discretization approach for hyperbolic PDE constraints in waveform inversion and proves strong convergence of discrete minimizers to continuous solutions.

Findings

The discretized problem ($ ext{P}_h$) is well-defined.

Stability and convergence are established under a CFL-condition.

Discrete minimizers strongly converge to continuous minimizers.

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 $\overline \nu$ of (P) satisfying a reasonable growth condition, there exists a sequence of local minimizers of ($\text{P}_h$) converging strongly towards $\overline \nu$.