On optimality and bounds for internal solutions generated from boundary data-driven Gramians

TL;DR

The paper develops a data-driven ROM approach for the time-domain plasma wave equation with unknown potential $q$, constructing data-generated internal fields via a Cholesky factorization of the boundary-derived Gramian. It proves that these fields asymptotically approximate the projection of the true internal solution onto the background snapshot space, yielding a general error bound that ties the data-driven error to best-approximation errors and a causality residual. In 1D, for two pulse-types, the data-generated solutions converge at rate $O(\sqrt{\tau})$ as the sampling interval $\tau$ shrinks, with numerical experiments confirming the rate and sharpness. The framework clarifies when data-driven internal data faithfully represent the true fields, discusses extensions to higher dimensions via MIMO setups, and outlines future connections to related inverse problems and redatuming methods.

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 $τ$ 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.

Figures (3)

• Figure 1: Reconstructed solution ${\bf{U}}$, true solution $U$, background solution $U_0$ and causal projection $\hat{U}$ of $U$ onto the the subspace of background snapshots for the terminal time $T=100$ and the samplings $n=75$ (a), $n=150$ (b), $n=300$ (c), $n=600$ (d).
• Figure 2: Errors of the reconstructed solutions for the terminal time $T=100$. As one can expect, the error is mostly concentrated near the singularity.
• Figure 3: Convergence rates of the approximants ${\bf U}$ and causal projection $\hat{U}$. As one can observe, both are aligned pretty well with the $O(\sqrt{\tau})$ error estimate.

Theorems & Definitions (35)

• Lemma 3.1
• Proof 1
• Remark 3.2
• Definition 3.3
• Lemma 4.1
• Proof 2
• Theorem 4.2
• Proposition 4.3
• Proof 3
• Lemma 4.4