The inverse source problem for the wave equation revisited: A new approach

The inverse problem of reconstructing a source term from boundary measurements, for the wave equation, is revisited. We propose a novel approach to recover the unknown source through measuring the wave fields after injecting small particles, enjoying a high contrast, into the medium. For this purpose, we first derive the asymptotic expansion of the wave field, based on the time-domain Lippmann-Schwinger equation. The dominant term in the asymptotic expansion is expressed as an infinite series in terms of the eigenvalues $\{\lambda_n\}_{n\in \mathbb{N}}$ of the Newtonian operator (for the pure Laplacian). Such expansions are useful under a certain scale between the size of the particles and their contrast. Second, we observe that the relevant eigenvalues appearing in the expansion have non-zero averaged eigenfunctions. We prove that the family $\{\sin(\frac{c_1}{\sqrt{\lambda_n}}\, t),\, \cos(\frac{c_1}{\sqrt{\lambda_n}}\, t)\}$, for those relevant eigenvalues, with $c_1$ as the contrast of the small particle, defines a Riesz basis (contrary to the family corresponding to the whole sequence of eigenvalues). Then, using the Riesz theory, we reconstruct the wave field, generated before injecting the particles, on the center of the particles. Finally, from these (internal values of these) last fields, we reconstruct the source term (by numerical differentiation for instance). A significant advantage of our approach is that we only need the measurements on $\{x\}\times(0, T)$ for a single point $x$ away from $\Omega$, i.e., the support of the source, and large enough $T$.