Reconstruction of a compactly supported sound profile in the presence of a random background medium

Abstract

In this paper, we present algorithms for reconstructing an unknown compact scatterer embedded in a random noisy background medium, given measurements of the scattered field and information about the background medium and the sound profile. We present six different methods for the solution of this inverse problem using different amounts of scattered data and prior information about the random background medium and the scatterer. The different inversion algorithms are defined by a combination of stochastic programming methods and Bayesian formulation. Our basic results show that if we have data for just one instance of the random background medium the best strategy is to invert for both random medium and unknown scatterer with appropriate regularization. However, if we have data for multiple instances of the medium it may be worth solving a coupled set of multiple inverse problems. We present several numerical results for inverting for various scatterer geometries under different inversion scenarios. The main take-away of our study is that one should invert for both unknown scatterer and random medium, with appropriate, prior-information based regularization. Furthermore, if data from multiple realizations of the background medium is available, then it may be beneficial to combine results from multiple inversions.

An Adaptive High Order Direct Solution Technique for Elliptic Boundary Value Problems

“…The recursive linearization procedure requires solving a sequence of linear least squares problems at successively higher frequencies to reconstruct an unknown sound speed. Next, in [6], the HPS method was utilized for inverse scattering problems with a random noisy background medium. In each of these inverse scattering solvers, the least squares solve requires solving the same variable coefficient elliptic partial differential equation many times to apply the forward and adjoint operators.…”

“…Since there are n c − 2 points on each panel, the interpolation operators are n c − 3 order. Then the solution and DtN matrices on Ω τ = Ω α ∪ Ω β are given by inserting the interpolation operators into the appropriate locations in equations (6) and Figure 6. Notation for the merge operation when boxes are on different levels as described in Section 3.2.…”

“…Compute the y-directional Chebychev coefficients B j of the approximate solution on τ . (5) end for (6) Let S y = max j=2,...,nc−1 (|B j (n c − 1)| + |B j (n c ) − B j (n c − 2)|) (7) for j = 2, . .…”

“…Let σ 1 and σ 2 be the children of τ . (6) Split I σ 1 b and I σ 2 b into vectors I 1 , I 2 , and I 3 as shown in Figure 3. ( 7)…”

Multifrequency inverse obstacle scattering with unknown impedance boundary conditions using recursive linearization

“…The combination of small number iterates required for each optimization problem, and the objective function requiring the solution of only 1 boundary value problem with N d boundary conditions allows for the efficient reconstruction of the unknown sound speed. The RLA has been subsequently coupled with fast algorithms for recovering more complicated sound speed profiles in [10,11], for shape recovery of sound-soft scatters in [12], and for shape recovery of axisymmetric sound-soft scatters in [13].…”