Real-time reconstruction of time-varying point sources in a three-dimensional scalar wave equation

Abstract

This paper discusses a reconstruction of point sources in a three-dimensional scalar wave equation from boundary measurements. We assume that the number, locations and magnitudes of point sources are unknown. Under these assumptions, we propose a real-time reconstruction method of these point sources based on the concept of the reciprocity gap functional. In our method, the number, locations and magnitudes of point sources can be identified within small delay. The effectiveness of the proposed method is shown by numerical examples.

Real-time reconstruction of moving point/dipole wave sources from boundary measurements

“…Derivations of (26)-(30) are given in Appendix A. We note that equations (26)-(28) are similar to the results for fixed point sources [16], but some supplemental terms arise due to the effect of moving velocities of sources, e.g. ξ k (τ ) and d τ (p k,xy (t k (τ ))).…”

“…(6) for moving dipole sources (3). Reconstruction methods of unknown sources can be categorised into two types, one is based on an optimisation technique like the least squares method [5,[9][10][11][12], and the other is based on an algebraic idea [13][14][15][16]. The former method has high versatility, but it usually requires large computational cost because we need to solve partial differential equation iteratively.…”

“…The key ideas of their method are the reciprocity gap functional [17] with respect to the space variable r and the Fourier transform with respect to the time variable t. Since their method needs to compute the inverse Fourier transform in the reconstruction procedure, we can not reconstruct unknown source is simultaneously on observing data. In 2011, Inui, Ohnaka and the author extended the idea of [14], and propose a new algebraic procedure in which one can reconstruct the locations and magnitudes of several point sources almost simultaneously on observations [16]. We can also find some researches which apply the reciprocity gap functional to localise a point wave source [18,19].…”

Inverse moving source problem for fractional diffusion(-wave) equations: Determination of orbits

“…In a series of works [20][21][22], numerical algorithms were examined for reconstructing the moving orbit from boundary data of solutions of the scalar wave equation.…”

Identification of the Point Sources in Some Stochastic Wave Equations

“…The total number and each location of point sources are estimated from ( ( , ), ∈ , 0 ≤ ≤ ). We refer to, for example, [1] and in particular the references therein for some recent study in this area. This theory has found substantial applications in determining the heat sources in heat conduction, the magnetic sources in brain, the earthquake sources of seismic waves, and so on.…”