The monotonicity method for the inverse crack
scattering problem
Takashi FURUYA (Nagoya University)
Let Γ ⊂ R2 be a smooth non-intersecting open arc (crack), and we assume that Γ can beextended to an arbitrary smooth, simply connected, closed curve ∂Ω enclosing a bounded domainΩ in R2 . Let k > 0 be the wave number, and let θ ∈ S1 be incident direction. We consider thefollowing direct scattering problem: For θ ∈ S1 determine us such that∆us + k 2 us = 0 in R2 \ Γ,(0.1)

us = −eikθ·x on Γ√ ∂us− ikus = 0,lim rr→∞∂r(0.2)(0.3)

where r = |x|, and (0.3) is the Sommerfeld radiation condition. It is well known that there exists
a unique solution us and it has the following asymptotic behaviour:oeikr nx.
us (x) = √ u∞ (x̂, θ) + O 1/r , r → ∞, x̂ :=|x|r(0.4)

The function u∞ is called the far field pattern of us . With the far field pattern u∞ , we define thefar field operator F : L2 (S1 ) → L2 (S1 ) byZF g(x̂) :=u∞ (x̂, θ)g(θ)ds(θ), x̂ ∈ S1 .
(0.5)
S1

The inverse scattering problem we consider is to reconstruct the unknown arc Γ from the far field
pattern u∞ (x̂, θ) for all x̂ ∈ S1 , all x̂ ∈ S1 with one k > 0. In other words, given the far field
operator F , reconstruct Γ.
In order to solve such a problem, we use the idea of the monotonicity method. The featureof this method is to understand the inclusion relation of an unknown target and artificial object
by comparing the data operator with some operator corresponding to an artificial one. For recent
developments of the monotonicity method, we refer to [2, 3, 4]. The following theorem is our main
result for solving the inverse crack scattering problem.
Theorem 0.1 (Theorem 1.1 in [1]). Let σ ⊂ R2 be a smooth non-intersecting open arc. Then,Hσ∗ Hσ ≤fin −ReF,(0.6)

where the Herglotz operator Hσ : L2 (S1 ) → L2 (σ) is given byZHσ g(x) :=eikθ·x g(θ)ds(θ), x ∈ σ,(0.7)σ⊂Γ⇐⇒

S1and the inequality on the right-hand side in (0.6) denotes that −ReF − Hσ∗ Hσ has only finitely
many negative eigenvalues, and the real part of an operator F is self-adjoint operators given by
1
ReF := (F + F ∗ ).
2
Based on Theorem 0.1, we give numerical examples. The indicator function in our examples isgiven by
I(σ) := # {negative eigenvalues of − ReF − Hσ∗ Hσ } .
(0.8)
The idea to reconstruct Γ is to plot the value of I(σ) for many of small σ in the sampling region.
Then, we expect from Theorem 0.1 that the value of the function I(σ) is low if σ is close to Γ.
1RRverHere, σ is chosen in two ways; One is the vertical line segment σi,j:= zi,j + {0} × [− 2M, 2M]
Ri Rj
Rver
where zi,j := ( M , M ) (i, j = −M, −M + 1, ..., M ) denote the center of σi,j , and M is the length
verof σi,j, and R > 0 is length of sampling square region [−R, R]2 , and M ∈ N is large to take aRRhorsmall segment. The other is horizontal one σi,j:= zi,j + [− 2M, 2M] × {0}.
The far field operator F is approximated by the matrix

2π ∞(0.9)
u (x̂l , θm ) 1≤l,m≤N ∈ CN ×N ,N2πl2πm2πm∗where x̂l = cos( 2πlN ), sin( N ) and θm = cos( N ), sin( N ) . The operator Hσ Hσ is approximated byZ

2πHσ∗ Hσ ≈eiky·(θm −x̂l ) dy
∈ CN ×N .
(0.10)
N
σ1≤l,m≤NF ≈In our examples, we fix R = 1.5, M = 100, N = 30, and wavenumber k = 1, and consider thetrue shape of Γ as a blue curve in Figure 1. Figures 2 and 3 is given by plotting the values of thevertical and horizontal indicator functions
verIver (zi,j ) := I(σi,j),hor
Ihor (zi,j ) := I(σi,j),(0.11)

for each i, j = −100, 99, ..., 100, respectively.

Figure 1: true shape of ΓFigure 2: verticalFigure 3: horizontal

References[1] T. Furuya, T. Daimon, R. Saiin, The monotonicity method for the inverse crack scattering
problem,, to appear in Inverse Probl. Sci. Eng., (2020).
[2] R. Griesmaier, B. Harrach, Monotonicity in inverse medium scattering on unbounded domains,
SIAM J. Math. Anal. 78, (2018), no. 5, 2533–2557.
[3] B. Harrach, V. Pohjola, M. Salo, Monotonicity and local uniqueness for the Helmholtz equation,
Anal. PDE, 12, (2019), no. 7, 1741–1771.
[4] B. Harrach, M. Ullrich, Monotonicity based shape reconstruction in electrical impedance tomography, SIAM J. Anal. Math. 45, (2013), no. 6, 3382–3403.
Graduate School of Mathematics, Nagoya University, Furocho, Chikusa-ku, Nagoya, 464-8602,
Japane-mail: takashi.furuya0101@gmail.com
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