A Formula for Time-to-Frequency Wave Boundary Data Conversion by the Boundary Control Method

Abstract

Given the wave equation on a compact Riemannian manifold with boundary, we derive an explicit reconstruction procedure to represent the frequency-domain Neumann-to-Dirichlet map in terms of the time-domain Neumann-to-Dirichlet map at any non-eigenfrequency. If the wave equation is exactly controllable, we derive an explicit formula to compute the former from the latter. The derivation is based on the boundary control method and requires only knowledge on the boundary of the manifold. The formula is stable when the level of regularization is fixed. The numerical feasibility is validated using one-dimensional examples in both Euclidean and non-Euclidean geometries.

Figures (5)

• Figure 1: First 50 singular values of $[K], [S^*S]$ and $[A^*_0 A_0]$
• Figure 2: Left: Elliptic solution $\mathfrak{u}_0$ versus wave snapshot $u^f(T)$ in the Euclidean geometry with $0\%$$1\%$$2\%$ and $5\%$ random Gaussian noise. Right: Absolute difference $|[\mathfrak{u}_0] - [u^f(T)]|$.
• Figure 3: Left: Relative Frobenius error of the reconstructed elliptic ND map $[\tilde{\mathfrak{L}}_0]$ versus $\alpha = 10^{-1}, 10^{-2}, \dots, 10^{-10}$. Right: Relative 2-norm error of the reconstructed elliptic solution $[\mathfrak{u}_0]$ versus $\alpha = 10^{-1}, 10^{-2}, \dots, 10^{-10}$.
• Figure 4: Left: Elliptic solution $\mathfrak{u}$ versus wave snapshot $u^f(T)$ in a non-Euclidean geometry with $0\%$$1\%$$2\%$ and $5\%$ random Gaussian noise. Right: Absolute error $|[\mathfrak{u}_0] - [u^f(T)]|$.
• Figure 5: Left: Error in the wave snapshot $u^f(T)$ (relative to the elliptic solution $\mathfrak{u}$) for real $\lambda\in (-8,8)$. Right: Error in the wave snapshot $u^f(T)$ for purely imaginary $\lambda\in (-8i,8i)$.

Theorems & Definitions (28)

• Remark 2.1
• Lemma 2.2
• proof
• Remark 2.3
• Lemma 2.4
• proof
• Lemma 4.1
• Lemma 4.2
• proof
• Proposition 4.3