Reconstructing the wave speed and the source (original) (raw)
2021, Mathematical Methods in the Applied Sciences
We are concerned with the inverse problem of recovering the unknown wave speed and also the source in a multidimensional wave equation. We show that the wave speed coefficient can be reconstructed from the observations of the solution taken at a single point. For the source, we may need a sequence of observation points due to the presence of multiple spectrum and nodal lines. This new method, based on spectral estimation techniques, leads to a simple procedure that delivers both uniqueness and reconstruction of the coefficients at the same time.
Related papers
Inverse problems for wave equation with under-determined data
arXiv: Analysis of PDEs, 2017
We consider the inverse problems of determining the potential or the damping coefficient appearing in the wave equation. We will prove the unique determination of these coefficients from the one point measurement. Since our problem is under-determined, so some extra assumption on the coefficients is required to prove the uniqueness.
The inverse source problem for the wave equation revisited: A new approach
2021
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 {λn}n∈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( c1 √ λn t), cos( c1 √ λn t)}, for those relevant eigenvalues, with c1 as the contrast of the small particle, defines a Riesz basis (contrary to the family corresp...
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.