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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | https://arxiv.org/abs/2407.19273 |
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| _version_ | 1866910788812275712 |
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| author | Ammann, Luis Yousept, Irwin |
| author_facet | Ammann, Luis Yousept, Irwin |
| contents | 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 $\overline ν$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_19273 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Numerical Analysis for a Hyperbolic PDE-Constrained Optimization Problem in Acoustic Full Waveform Inversion Ammann, Luis Yousept, Irwin Numerical Analysis Optimization and Control 35L10, 35Q93, 65N30 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 $\overline ν$. |
| title | Numerical Analysis for a Hyperbolic PDE-Constrained Optimization Problem in Acoustic Full Waveform Inversion |
| topic | Numerical Analysis Optimization and Control 35L10, 35Q93, 65N30 |
| url | https://arxiv.org/abs/2407.19273 |