An inverse problem for a semi-linear wave equation: A numerical study

Figure 1.  The set $ W $, where we can recover the potential $ q(x, t) $. In the 1+1 dimensional case this is the set, where we can make two approximate plane waves (one traveling to the left and the other to the right) collide

Figure 2.  The example of Section 2.1. The potential function $ q(x, t) $ (left) and the corresponding solution $ u $ of (1) (right)

Figure 3.  Left: The relative difference $ (| u_{h+1} |_2 - | u_{h} |_2)/| u_{h} |_2 $ between $ L^2 $-energies of solutions $ u_h $ to (19) at successively finer grids in log-scale. Right: the absolute differences $ \Vert u_{h+1}-u_h\Vert_{L^\infty} $ evaluated at the common grid points between the successively finer grids in log-scale. We see that the $ L^2 $-energy stays bounded along with finer grids and the values of the numerical solutions converge at the common grid points

Figure 4.  Demonstration of regularization of numerical differentiation of noisy measurements. Example 1, the function (28), on top row and example 2, the function in (29), on bottom row. (a) The function $ g $ to be differentiated. (b) The exact derivative $ \partial_x \partial_yg $ of the function $ f $. (c) Regularized approximation $ {\bf y}_\mathrm{reg} $ to the derivative $ {\bf y} $ of the noisy measurement $ {\bf g}_{i, j} $. (d) Finite difference approximation of the derivative of the noisy measurement $ g_{i, j}+ \varepsilon_{i, j} $. (e) The absolute difference $ | \partial_x \partial_y g - {\bf y}_\mathrm{reg}| $. Experimentally chosen regularization parameters $ \lambda = 0.00001 $ and $ \lambda = 0.01 $ have been used in example 1 and 2 respectively

Figure 5.  Left: Convergence rate for the integral in Lemma 4.2 with respect to $ \tau\in [0, 1500] $. We compare the absolute errors between the integral approximations in sup-norm. Here the red curve depicts the theoretical upper bound $ O(\tau^{-1/2}) $ given by Lemma 4.2, the blue curve corresponds to the precise integral of Lemma 4.2 and the black dot-dash curve depicts the integral of (11). The difference in convergence rates in both pictures is explained by the test function $ \chi $ included in the functions $ H_1^\tau $ and $ H_2^\tau $ in the integral (11). Right: The L-curve obtained by comparing the DN maps against the finite differences (30) in $ \log\log $-scale. In this case, the "corner" of the curve is located approximatively at $ \varepsilon = 0.03 $

Figure 6.  Example 1: Comparison of the true target, supported on the time-interval, and its numerical reconstruction

Figure 7.  Example 2: Comparison of the true target, which changes sign on the time-interval, and its numerical reconstruction

Figure 8.  Example 3: Comparison of the true target and its numerical reconstruction in the presence of discontinuities. The jump discontinuity of the target is blurred in the reconstruction

Figure 9.  Example 4: Comparison of the true non-compactly supported target target and its numerical reconstruction. The true target is a smooth bump-function, whose location moves in space along time

Figure 10.  Example 5: Comparison of the true target and its numerical reconstruction in case of time-independent sign-changing target. Top: reconstruction of the potential function in time. Bottom: a cross-section of the true target and its reconstruction at $ t_0 = 1.5 $ in black and red dot-dash, respectively

Figure 11.  Example 6: To demonstrate the resolution of our numerical method, we consider time-independent potentials $ q(x) $ given by the characteristic functions of two small intervals of width $ 0.1 $ at distances $ 0.4 $, $ 0.3 $, $ 0.2 $, $ 0.14 $, $ 0.12 $ and $ 0.1 $ measured from the center points of the intervals. The cross-sections of the true targets and their reconstructions at fixed time $ t_0 = 1.5 $ are given in black and red dot-dash, respectively. Two sufficiently far apart target intervals are clearly distinguishable. However, we lose details when the targets are close to each other. There are two reasons for this. Firstly, our approximate plane waves have a finite width of approximately $ 0.1 $ units, which causes loss of small details. Secondly, the potential functions considered are discontinuous and thus the jump discontinuity gets blurred (which is similar to Example 3 in Figure 8)