Inverse obstacle scattering for acoustic waves in the time domain

Abstract

<p style='text-indent:20px;'>This paper concerns an inverse acoustic scattering problem which is to determine the location and shape of a rigid obstacle from time domain scattered field data. An efficient convolution quadrature method combined with nonlinear integral equation method is proposed to solve the inverse problem. In particular, replacing the classic Fourier transform with the convolution quadrature method for time discretization, the boundary integral equations for the Helmholtz equation with complex wave numbers can be obtained to guarantee the numerically approximate causality property of the scattered field under some condition. Numerical experiments are presented to demonstrate the effectiveness and robustness of the proposed method.</p>

Quadrature method and extrapolation algorithm for solving the boundary integral equations of the acoustic wave scattering problems

“…To x obs are on the circle whose radius is 4 and the center is at the origin. The scattered fields u(x, t) at time t = 8 are recorded (blue asterisk dashed line is our method and black square dashed line is the trigonometric interpolation method due to [38], respectively) in Figure 4, which implies that our method is valid again.…”

“…Moreover, due to the inverse scaled discrete Fourier transform (see [7], [38]), then the time domain scattered field u n (z) (z = (z 1 , z 2 ) ∈ R 2 \Γ) of (2.1) shall be computed by the formula…”

“…Moreover, the scattered field u(x, t) are also compared with the results by using the trigonometric interpolation method in space (see [11], [38]). To x obs are on the circle whose radius is 4 and the center is at the origin.…”

Inverse obstacle scattering for elastic waves in the time domain

“…Many numerical methods fall into this category, such as the point source method [30], the probe method [3], the factorization method [4], the enclosure method [18], the strengthened total focusing method [12] and the linear sampling method [6,7,[13][14][15]. In order to obtain finer reconstruction, a quantitative method, namely convolution quadrature based nonlinear integral equation method, is proposed for the inverse acoustic obstacle scattering problem in [34]. The convolution quadrature method for time discretization was proposed by Lubich [24,[27][28][29], and was extended to solve the acoustic scattered field by combining various numerical methods for space discretization [1,8,31].…”

“…The obstacle is assumed to be embedded in a homogeneous and isotropic medium, and the scattered field data is measured on a circle which has a finite distance away from the obstacle. Specifically, motivated by the recent works [2,34], we convert the model problem into a coupled initial-boundary value problem for wave equations by using the Helmholtz decomposition, and we prove the uniqueness of the solution for this coupled problem by employing an energy method. Then, we establish coupled boundary integral equations with the help of retarded layer potentials and prove the uniqueness of the solution for these equations.…”