Explicit tight bounds on the stably recoverable information for the inverse source problem

Abstract

For the inverse source problem with the two-dimensional Helmholtz equation, the singular values of the source-to-near-field operator reveal a sharp frequency cut-off in the stably recoverable information on the source. We prove and numerically validate an explicit, tight lower bound Bfor the spectral location B of this cut-off. We also conjecture, justify and support numerically a tight upper bound B + for the cut-off. The bounds are expressed in terms of zeros of Bessel functions of the first and second kind. Finally, we propose our near-field upper bound B + as an improvement of a commonly used upper bound on the spectral cutoff for the source-to-far-field operator.

Stable source reconstruction from a finite number of measurements in the multi-frequency inverse source problem

“…Building on these results, Section 4 shows how distribution of the zeros of Bessel functions allows us to substantially reduce the number of frequencies initially needed to reconstruct the finite dimensional sources. By relating our analysis with results in [16], we also show that, with an additional requirement on the frequency set, reconstructions will be robust towards noise in the measurements. In Section 5, we conduct a set of numerical experiments to validate our findings.…”

“…Hence, if we have measurements at all frequencies {k m,n } we could use (16) to reconstruct any s ∈ L 2 (D 0 ). This is similar to what was proposed in [11].…”

“…We see that reconstruction of s amounts to reconstructing the vectorŜ fromÛ , and hence depends on the invertibility of K. The properties of K depend on the choice of frequency set Q r . From (16), it is evident that if Q r = Q M,N , we have that K = I sincê…”

“…This problem has been investigated by several authors in recent years, for example in [4,5,6,16,26]. It has a wide range of applications, for example in optical tomography [3], antenna control [18] and in the acoustic reconstruction part of several hybrid imaging methods, most notably photo-and thermoacoustic tomography [2,19], see also [1,6].…”

Stability of the inverse source problem for the Helmholtz equation in R3

“…The main result there is Theorem 2. The analysis in sections 2 and 3 follows our earlier work [14] in R 2 rather closely. Section 4 concerns the multi-frequency ISP, and here we identify certain finite-dimensional spaces of sources that can be stably reconstructed from the boundary measurements.…”