On optimality and bounds for internal solutions generated from boundary data-driven Gramians
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866916701553033216 |
|---|---|
| author | Druskin, V. Moskow, S. Zaslavsky, M. |
| author_facet | Druskin, V. Moskow, S. Zaslavsky, M. |
| contents | We consider the computation of internal solutions for a time domain plasma wave equation with unknown coefficients from the data obtained by sampling its transfer function at the boundary. The computation is performed by transforming known background snapshots using the Cholesky decomposition of the data-driven Gramian. We show that this approximation is asymptotically close to the projection of the true internal solution onto the subspace of background snapshots. This allows us to derive a generally applicable bound for the error in the approximation of internal fields from boundary data for a time domain plasma wave equation with an unknown potential $q$. For general $q\in L^\infty$, we prove convergence of these data generated internal fields in one dimension for two examples. The first is for piecewise constant initial data and sampling $τ$ equal to the pulse width. The second is piecewise linear initial data and sampling at half the pulse width. We show that in both cases the data generated solutions converge in $L^2$ at order $\sqrtτ$. We present numerical experiments validating the result and the sharpness of this convergence rate. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_15407 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On optimality and bounds for internal solutions generated from boundary data-driven Gramians Druskin, V. Moskow, S. Zaslavsky, M. Numerical Analysis 65M32, 78A46 We consider the computation of internal solutions for a time domain plasma wave equation with unknown coefficients from the data obtained by sampling its transfer function at the boundary. The computation is performed by transforming known background snapshots using the Cholesky decomposition of the data-driven Gramian. We show that this approximation is asymptotically close to the projection of the true internal solution onto the subspace of background snapshots. This allows us to derive a generally applicable bound for the error in the approximation of internal fields from boundary data for a time domain plasma wave equation with an unknown potential $q$. For general $q\in L^\infty$, we prove convergence of these data generated internal fields in one dimension for two examples. The first is for piecewise constant initial data and sampling $τ$ equal to the pulse width. The second is piecewise linear initial data and sampling at half the pulse width. We show that in both cases the data generated solutions converge in $L^2$ at order $\sqrtτ$. We present numerical experiments validating the result and the sharpness of this convergence rate. |
| title | On optimality and bounds for internal solutions generated from boundary data-driven Gramians |
| topic | Numerical Analysis 65M32, 78A46 |
| url | https://arxiv.org/abs/2504.15407 |