Preprint
On optimality and bounds for internal solutions generated from boundary data-driven Gramians
21 Apr 2025
Abstract
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 $\tau$ 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 $\sqrt{\tau}$. We
present numerical experiments validating the result and the sharpness of this
convergence rate.
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Details
- Title
- On optimality and bounds for internal solutions generated from boundary data-driven Gramians
- Creators
- V DruskinS MoskowM Zaslavsky
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991022048905404721