Theory of the linear sampling method for time-dependent fields

Abstract

The linear sampling method seeks to localize the unknown source of an observed, time-dependent field. The unknown source could be, for example, a scatterer embedded within a medium, or an impulsive excitation such as an earthquake or explosion. The source of the observed field is localized by means of solving the so-called near-field equation and mapping the obtained solutions through an indicator functional over a test region assumed to contain the source. In its current formulation, however, the linear sampling method suffers from an ambiguous time parameter that strongly influences its ability to localize the unknown source. Our paper consists of two fundamental results central to the theoretical understanding of the linear sampling method and its numerical implementation. First, we prove the so-called blowup behavior of solutions to the near-field equation for a general source function that is separable in space and time. Second, we show that the linear sampling method can be formulated such that the ambiguous time parameter is irrelevant. We demonstrate that a dependence of the linear sampling method on the time parameter arises from an incorrect implementation of a convolution-type operator found in the near-field equation. When the operator is implemented correctly, the dependence on the time parameter vanishes. We provide detailed algorithms for efficient and proper implementations of the convolutional operator in both the time and frequency domains. The crucial result of the improved implementations is that they allow the linear sampling method to be completely automated, as one does not need to know the space-time dependence of the unknown source. We demonstrate the effectiveness of the improved time-and frequency-domain implementations using several numerical examples applied to imaging scatterers.

The Time Domain Linear Sampling Method for determining the shape of multiple scatterers using electromagnetic waves

“…Moreover, p z = p is typically fixed to be a unit vector (|p| = 1). Note that in their analysis of the TD-LSM for the wave equation [30], the authors argue that one may choose τ z = τ = 0.…”

An acoustic Lippmann-Schwinger inversion method: applications and comparison with the linear sampling method

“…Of course, this is physically possible only when the prescribed point lies inside the scatterer. It can be shown that the focusing function j is proportional to the inverse of the contrast source function χ [13]. In particular, the inverse of χ exists and is bounded in the L 2 norm only for points zäD (where χ is strictly nonzero).…”

“…The linear sampling method has been interpreted as a focusing technique [13,19]. As we show in this paper, the solution to the ill-posed integral equation in the linear sampling method corresponds to a focusing function, which is a type of distributed source for focusing wave fields in a medium.…”

“…The linear sampling method is another imaging technique that has garnered considerable attention [8][9][10][11][12][13]. To demarcate the boundary of a scatterer, the method relies on a characteristic blowup behavior of the solution to an ill-posed integral equation.…”

“…Such a process has a physical solution with finite energy only when the given point lies inside the scatterer. Clearly, the inverse source cannot exist for any point outside the scatterer, and numerically this 'law of causality' manifests as a blowup in the norm of the solution [13]. Consequently, an image of the scatterer can be obtained by noting where the norm of the solution becomes arbitrarily large.…”