The domain derivative in inverse obstacle scattering with nonlinear impedance boundary condition

In this paper an inverse obstacle scattering problem for the Helmholtz equation with nonlinear impedance boundary condition is considered. For a certain class of nonlinearities, well-posedness of the direct scattering problem is proven. Furthermore, differentiability of solutions with respect to the boundary is shown by the variational method. A characterization of the derivative allows for iterative regularization schemes in solving the inverse problem, which consists in reconstructing the scattering obstacle from the far field pattern of a scattered wave. An all-at-once Newton-type regularization method is developed to illustrate the use of the domain derivative by some numerical examples.


Introduction
Nonlinear phenomena in acoustic and electromagnetic obstacle scattering have gained importance in the last decade, for example in the case of thinly coated perfectly conducting objects filled with a nonlinear medium [27,29].Through an asymptotic analysis, nonlinear material properties in thin layers lead to approximate nonlinear boundary conditions, [26].E↵ective boundary conditions for thin films filled with linear media have been derived and investigated in [2,5,7,9].Scattering problems with such nonlinear boundary conditions have been studied in [25] for electromagnetic waves and in [3,4] for the acoustic wave equation, based on a boundary element method in space and convolution quadrature in time.
In this work we focus on the reconstruction of an obstacle with material properties near the boundary approximated by an impedance boundary condition with an additional nonlinear term.Generally, in inverse obstacle scattering, we are interested in determining the shape of an obstacle by illuminating the object with plane incident time-harmonic waves and considering measurements of the far-field pattern of corresponding scattered waves.In case of linear scattering models such inverse problems are extensively studied (see [6]) but much less is known in case of nonlinear media.First approaches in recovering the support of nonlinear media can be found in [23] and [8] based on the factorization method and the monotonicity method.These require, at least theoretically, measurements of the far field pattern for all incident directions.But, for the following investigations we assume access to the far field pattern for a single or just a few incident fields.
We consider time-harmonic scattering modeled by the Helmholtz equation in the exterior of an object with a nonlinear impedance boundary condition.Thus, the model may cover acoustic scattering (in R 3 ) as well as a specific polarization in electromagnetic scattering (in R 2 ).For the scattering problem with classical Robin boundary condition, i.e., without a nonlinear term, it is observed that iterative regularization schemes based on the domain derivative of the scattering problem lead to feasible reconstruction algorithms (see [22] and references cited therein).Therefore, in extending this approach to nonlinear impedance boundary conditions our task is twofold.First, we have to establish a shape derivative of the scattered field with respect to the scattering obstacle, and second, we are going to develop an iterative regularization scheme in solving the severely ill-posed reconstruction problem.
In view of the variational method for showing existence of the domain derivative we begin with a weak scattering theory for the nonlinear boundary condition.Thus, in the second section we present quite general assumptions on the nonlinear term which ensure a well-posed direct problem.It is shown that Kacurovskii's extension of the Fredholm Theory to nonlinear compact operators can be applied to the scattering problem.Furthermore, we will prove uniqueness and stability of the nonlinear scattering problem by using Rellich's Lemma.
With a well-posed scattering problem the existence of the so called material derivative is proven in the following section.The presented approach is closely related to the ideas established in [13] for semilinear boundary value problems.Similar to the case of linear boundary value problems such a derivative can be represented by its domain derivative (see [11,14,18]), which is shown to satisfy a corresponding linearized impedance boundary value problem for the Helmholtz equation assuming suitable regularity of the boundary.
The subsequent sections are devoted to the inverse problem.With the domain derivative at hand it becomes natural to consider iterative regularization schemes in solving the inverse problem.Since any iteration step requires solving a nonlinear boundary value problem we suggest an all-at-once method based on linearizing the scattering problem and applying a regularised Newton step for the reconstruction.In general, it is shown by B. Kaltenbacher in [15] that such an approach leads to regularization schemes by assuming certain mapping properties of the nonlinear operator.
Although, a complete convergence proof is not known even for the inverse object reconstruction for linear scattering problems, we elaborate such an all-at-once approach in the fourth section.We introduce an integral equation method solving for the linearized scattering problems, and develop a Newton type regularization scheme for the reconstruction of a scattering object from a far field pattern.Some examples in the last section illustrate the performance of the scheme.It turns out that the suggested method leads to reconstructions in case of nonlinear boundary conditions comparable to the known results with Robin boundary conditions.

The direct problem
Let D be a bounded domain in R N (N = 2, 3) with a Lipschitz continuous boundary @D.An incident plane wave u i (x) = e ikd•x with propagation direction d 2 S N 1 and real positive wave number k is scattered by the object D, generating a scattered wave u s .The scattering problem is to find the total wave u = u i + u s which satisfies the Helmholtz equation together with a nonlinear impedance boundary condition where ⌫ denotes the unit normal vector to @D oriented towards R N \D, 2 L 1 (@D) is a complex-valued impedance function, and g : @D ⇥ C ! C gives an additional nonlinear term with respect to u.Furthermore, the scattered field u s is assumed to satisfy the Sommerfeld radiation condition lim uniformly with respect to x = x |x| .In the given scattering problem the propagation region of the wave is unbounded.For a weak formulation we consider an equivalent representation of the problem on a bounded subdomain ⌦ := B R \D, where the radius R > 0 of the ball B R ✓ R N is chosen large enough such that D ✓ B R .We consider weak solutions in the Sobolev space H 1 (⌦).To include the Sommerfeld radiation condition, we specify a nonlocal boundary condition on the boundary @B R .Therefore, we introduce the Dirichlet-to-Neumann map , where w is the radiating solution of the Helmholtz equation in R N \B R with Dirichlet trace w = f on @B R (see [6]).Since the scattered wave u s is a solution of the Helmholtz equation satisfying the Sommerfeld radiation condition we obtain the boundary condition @u s @⌫ = ⇤u s on @B R .
Thus, a weak formulation of the scattering problem is given by with We assume throughout this work the function g(x, z) to be measurable in x, continuous in z, and satisfying a Caratheodory condition with c > 0, 2 L 2 (@D) and 1  p < 1 in case of N = 2, and 1  p  2 in case of N = 3.Then, with u 2 H 1 (⌦) we have G(u) 2 L 2 (@D) for the Nemytskii operator defined by G(u)(x) = g(x, u(x)), see [1,Theorem 5.36].Further conditions on g will be specified below.
To ensure also a well-posed direct boundary value problem in case of the nonlinear impedance condition, the function g : @D ⇥ C ! C has to fulfil some additional assumptions.The following Theorem, based on an extension of the Fredholm theory, gives a su cient condition for existence of weak solutions of the scattering problem.
Then, for any f 2 H 1 (⌦) there exists a weak solution u 2 H 1 (⌦) of the boundary value problem (4).
Proof.The variational formulation is equivalent to the operator equation The linear scattering problem is given by the operator R l .Thus, as mentioned above, it is an injective Fredholm operator with index zero and has a bounded inverse.From we see, the operator R n is continuous, since by the sublinear assumption the Nemytskii operator G is continuous as a mapping from L 2 (@D) into L 2 (@D) (see [28]).Furthermore, the trace operator from H 1 (⌦) into L 2 (@D) is compact and we observe R n : H 1 (⌦) !H 1 (⌦) to be continuous and compact.
Using the assumption on g we obtain Therefore, the Theorem of Kacurovskii, an extension of the Fredholm alternative, see [31,Theorem 29.A], can be applied and shows existence of a solution to (4) for any f 2 H 1 (⌦).
Let g be di↵erentiable in its second argument, in the sense of existence of g z (., z; w) 2 L 1 (@D) such that g(x, z + w) g(x, z) = g z (x, z; w) + o(|w|) for all z, w 2 C, a.e.x 2 @D and g z to be R-linear w.r.t.w, then a lower bound on Re( ) leads to a well-posed direct scattering problem.
Corollary 2.2 Let g : @D ⇥ C ! C be a sublinear function as in Theorem 2.1, di↵erentiable in its second argument as described above, and let kRe( ) sup z,w2C Im(g z (x, z; w)w) for any z, w 2 C and a.e.x 2 @D.
Then, the problem (4) has a unique solution u 2 H 1 (⌦) and it exists a constant c > 0 with Proof.Rellich's Lemma ensures that a radiating solution of the Helmholtz equation vanishes if see [6].Let u 1 and u 2 be two solutions of (4).We obtain the inequality for the scattered solution u = u 1 u 2 of the Helmholtz equation, This implies uniqueness of the solution u 2 H 1 (⌦) of problem (4).The uniform bound by a constant c > 0 follows analogously as it is known from the Riesz-Fredholm theory for an injective linear Fredholm operator.
Remark (a) As an example we may consider Obviously, the function is sublinear in z, and with and |z| 1+|z| 2  1 2 for all z 2 C we observe the assumption of Corollary 2.2.(b) Note, the restriction of a sublinear function g is quite natural, since in the general well-posed case g can be replaced by a sublinear one without changing the solution.This is seen as follows: Let us assume that the boundary value problem ( 4) is well-posed, f is defined from an incident field u i as above, and u = u s + u i denotes the solution of the scattering problem.From Green's representation theorem in R n \D for u s and in D for u i , and the boundary condition we obtain + (ik u g(., u)) ds = Du + S(iku g(., u)) on @D, where is the fundamental solution of the Helmholtz equation and D, S denote the boundary integral operators corresponding to the double and the single layer potential of the Helmholtz equation, see [20].
, see [17].Thus, the above representation holds pointwise and the Caratheodory condition implies , since first we conclude u 2 H 1 (@D) and then we can apply the Sobolev embedding H 2 (@D) ✓ C(@D) from D, S : for |z| > b + 1 .We observe g b to be sublinear in its second argument and u solves the problem (4) for all b > b 0 , if g is replaced by g b .Thus, if g b satisfies the conditions of Corollary 2.2 and u b denotes the corresponding unique solution, we conclude u = u b for all b > b 0 by uniqueness.

The domain derivative
In view of iterative regularization schemes for the inverse obstacle scattering problem, it is necessary to study di↵erentiability of the domain to far field operator.Therefore, we investigate the derivative of a solution of the scattering problem with respect to the domain D. In this chapter we assume that D is of class C 1 .A variation of a domain D ✓ B R = {x 2 R 3 : |x| < R} is described by a su ciently small vector field We denote a perturbed domain by Throughout, we consider and g on @D and on perturbed boundaries to be traces of functions : R N !C and g : R N ⇥ C ! C and assume there exist unique solutions u 2 H 1 (⌦), u h 2 H 1 (⌦ h ) of ( 4) and (8), respectively.For instance Corollary 2.2 gives su cient conditions.
A transformation of variables with Here, we denote by J ' the Jacobian matrix of the transformation ' and e u h (x) = (u h ')(x).Furthermore, the Jacobian with respect to the surface @D is given by for local parametrizations and b = + h of @D and @D h , respectively.Additionally, note that for any perturbation h we have the same f in (9) as in (4) given by the incident field on @B R .
For the tangential and the normal component of a vector on the boundary @D we introduce the notations h ⌧ = ⌫ ⇥ (h ⇥ ⌫) and h ⌫ = h • ⌫ and thus h = h ⌧ +h ⌫ ⌫ holds on @D. Furthermore, we introduce the tangential gradient, given by r ⌧ v = rv @v @⌫ ⌫, and the tangential divergence, DivV = divV ⌫ > J V ⌫, for smooth functions v or V in a neighborhood of @D, which both can be extended to linear bounded operators r ⌧ , Div : Elementary calculations show the following linearizations of the Jacobians.
, C) and  be the mean curvature of @D. Then Proof.See [12].
Therefore, from now on we assume 2 C 1 (R N , C).As in [13], we also have to specify assumptions on the nonlinear function g which ensure existence of the domain derivative.We collect su cient conditions which will be applied in the following considerations.
where the partial derivatives g x satisfy a growth condition as in (5), i.e.
for a.e.x 2 @D and all z 2 C. (ii) the derivatives g z , g xz exist in the sense as above, i.e. , g z (., z; w), g xz (., z; w) 2 L 1 (@D h ) for any admissible variation h, are R-linear functions with respect to w 2 C with for all z, w 2 C and a.e.x 2 @D h .Additionally, these derivatives also satisfy growth conditions With Lemma 3.1 and Assumption (A), we conclude continuous dependence on variations of the domain D. Theorem 3.2 Let g satisfies (A) and let u 2 H 1 (⌦) be the solution of (4) and u h 2 H 1 (⌦ h ) the solutions of the scattering problem with respect to perturbed domains.Furthermore, let |g z (x, z; w)|  ⌘|w| with a su ciently small constant ⌘ > 0 locally for a.e.x 2 U and for all z 2 V in open sets @D ✓ U ✓ R 3  and {u(x) : x 2 @D} ✓ V ✓ C.Then, the solution of the boundary value problem depends continuously on the domain D, i.e. lim where the notation e u h = u h ' with ' = I + h is used.
Proof.According to well-posed scattering problems, the operators If we consider the di↵erence between these two operators, we obtain With the growth conditions ( 5) and (10), respectively, the expression in the last integral can be estimated by ).Thus, for w 2 H 1 (⌦) according to Lemma 3.1 we get Furthermore, for solutions of T (u) = f and T h (e u h ) = f we obtain the identity ⇤wv ds denotes the linear part and the nonlinear part of the operator T .Analogous operator splitting holds for T h .As mentioned in chapter 2 we know that the operator T l has a bounded inverse . Using Neumann's series, we obtain the existence of the inverse operator T 1 h,l and the inequality for a constant e C > 0, see [21,Theorem 10.1].With these considerations, the error estimates Overall, with the assumption on g z we obtain Thus, the solution of the boundary value problem depends continuously on the domain Note from the presented proof that assumptions on the second derivatives of g are not required for the above continuity result.
In the following we also have to consider the R-linear boundary value problem with g replaced by g z .Thus, we introduce the problem With the above considerations we now can formulate the main result on existence of a derivative of the solution of the scattering problem with respect to the shape of D.
Theorem 3.3 We assume the boundary value problems (4) and ( 13) are well-posed and the solution u of ( 4) is continuous with respect to the domain D in the sense of (12).With g satisfying assumption (A) the solution u is di↵erentiable with respect to variations h 2 C 1 c (B R ) of the domain, i.e., it exists w 2 H 1 (⌦), linearly depending on h with The material derivative w is given by the unique weak solution of the R linear boundary value problem Proof.Since the problem ( 13) is well posed, w 2 H 1 (⌦) exists and according to a support of h close to the boundary @D the function w can be extended to a radiating solution of the Helmholtz equation in the exterior of B R .

Theorem 3.4
The material derivative of the previous theorem is of the form w = u 0 + ru • h, where u denotes the solution of ( 4) and the domain derivative u 0 2 H 1 (⌦) is the unique weak solution of the R linear boundary value problem with @u 0 @⌫ Here, the last boundary condition means that u 0 can be uniquely extended to a weak radiating solution of the Helmholtz equation in R N \D.
Proof.From Green's representation theorem and the radiation condition we obtain Thus, the mapping properties of the potentials DL : [24, p.210 ↵.], leads to u 2 H 2 (⌦).Therefore, the traces of h • ru can be read in H 1 2 (@D) and H 1 2 (@D), respectively.We insert h • ru w in (14), where h⇤(h • ru), vi @B R = 0 vanishes since h has compact support in B R .The product rule leads to the relation and we get For the material derivative w we use equation ( 15) and thus it holds Using Gauss' divergence theorem and the compactness of the support of Applying the product rule to the surface divergence operator leads to Thus u 0 can be extended in R N \D to the weak scattered solution of ( 16)-( 17).

Shape Reconstruction
The Sommerfeld radiation condition guarantees the asymptotic behaviour of the scattered field uniformly in all directions, where the function u 1 defined on the unit sphere S N 1 is called the far field pattern of u s .Thus, the inverse problem under consideration consists in determining the shape of D from measurements of the far field pattern u 1 .We consider a fixed incident field u i and define the domain-to-far-field operator F , which maps an admissible boundary onto the far field pattern of the scattered wave.
Then, the inverse problem can be read as solving the nonlinear equation Since the far field pattern is analytic we expect this nonlinear operator equation to be severely ill-posed.
Throughout the last two chapters, we consider the scattering problem only in R 2 and specify the set of admissible domains to be starlike with respect to the origin.A parametrization of the boundary is given by with positive, 2⇡-periodic radial function r 2 C 2 (0, 2⇡).For the sake of simplicity, we write F (r) instead of F (@D) without changing notation.From Section 3, we know that F is di↵erentiable, with the derivative given by the far field pattern of the domain derivative, i.e., F 0 [@D]h = u 0 1 .We suggest an all-at-once Newton-type method based on linearization of the forward problem and of the domain-to-far-field operator in the sense of [15].A linearization of the nonlinear operator equation ( 19) leads to with noise level , i.e. ku 1 u 1 k L 2  .Since the evaluation of the operators F (r) and F 0 [r] require the far field u 1 and the Cauchy-data u| @D , @u @⌫ | @D of the solution u of the nonlinear scattering problem for the domain represented by r, we replace in any iteration step u by u n = u i + u s n and r by r n = r n 1 + h n 1 .Choosing an initial guess @D 0 , we iteratively compute the solution u n of the scattering problem with boundary condition on @D n and u s n = u n u i satisfying the radiation condition.Note that the boundary condition is obtained by linearizing the nonlinear term g(•, u n ) in the sense of (11), i.e.
The domain-to-far-field operator is then defined by F n (r n ) = u n,1 and the derivative of F n is given by , where u is replaced by u n on the right hand side of the boundary condition (17).
Since the ill-posed linear operator equation ( 20) requires a regularization, we apply as for the regularized Levenberg-Marquardt method a Tikhonov regularization in any iteration step, which finally leads to the update h n given as the solution of where ↵ > 0 denotes a positive regularization parameter, see [10].
For such an iterative regularisation method a stopping rule is required because the approximations will deteriorate for noisy data after a certain number of iterations.The most commonly used stopping rule is the discrepancy principle, where the iterations are terminated at an index n for which is valid for the first time.If the operator F satisfies the tangent cone condition kF (r) F (r) F 0 [r](r r)k  ckr rkkF (r) F (r)k locally in a neighbourhood of the exact solution in suitable Hilbert spaces, then the approach (23) with stopping condition (24) yields a regularisation method (see [16]).So far it has not been confirmed that the cone condition is fulfilled in case of inverse obstacle scattering problems.
It is known that boundary integral equations solved by Nyström's method are suitable for solving the scattering problem ( 21)- (22).Therefore we consider a single layer potential ansatz for the scattered wave with a density ' n 2 C(@D) and the fundamental solution of the Helmholtz equation 0 (k|x y|), x 6 = y, in R 2 .Using the jump relations for the single-layer potential ( [6]), the scattered wave u s n = u n u i defined by ( 25) is a radiating solution of the Helmholtz equation satisfying the impedance boundary condition (22) on @D n if and only if the density ' n satisfies the integral equation 1 2 with Here, S and K 0 denote the boundary integral operators corresponding to the single and the adjoint double layer potential of the Helmholtz equation given by S'(x) = Z @D (x, y)'(y) ds y , x 2 @D, For @D 2 C 2 the operators S, K 0 : C(@D) !C(@D) are compact, see [6].For the implementation of the linearized operator equation ( 23), we need the evaluation of F and F 0 as well as its adjoint operator F 0⇤ .To avoid computing the adjoint of the derivative, we have considered the adjoint of the discretised operator F 0 in our implementation.Overall, the shape reconstruction algorithm for the scattering problem ( 21)-( 22) is given as follows.
(i) First, an initial guess is made for the shape @D 0 of the scattering obstacle.In order to solve the direct problem we start with the incident wave u 0 = u i .(ii) By solving the integral equation ( 26) using the initial guess u 0 , the Cauchy data u n | @Dn , @un @⌫ @Dn for the approximate boundary curve @D n can be determined.The far field pattern F n (r n ) = u n,1 is computed from the density ' n by (iii) The radial component of the starlike boundary and the variations are approximated by trigonometric polynomials For each basis function h(t) = cos(jt) and h(t) = sin(jt) the domain derivative from Theorem 3.4 is determined by solving the same integral equation ( 26) with (iv) Again with (27) the far field pattern of the domain derivatives is calculated for all basic functions related to the current boundary curve.The collection of these far field patterns leads to a Jacobian matrix J at the points chosen for the discretisation.The application of Tikhonov regularisation yields where h denotes the vector of 2m + 1 Fourier coe cients of h.
(v) The iteration is stopped at index n for which , is valid for the first time.
To avoid an inverse crime, we compute synthetic data by a di↵erent integral equation approach than in (25).We consider the direct approach based on the Green's representation theorem, which is given by u s (x) = DLu(x) SL @u(x) @⌫ = Z @D @ (x, y) @⌫ y u(y) (x, y) @u(y) @⌫ y ds y for x 2 ⌦.
The jump relations of the single layer and double layer potential at the boundary @D and the linearized boundary condition (22) lead to the integral equation 1 2 where denotes the boundary integral operator corresponding to the double layer potential.Note that for the direct approach, the far field pattern is given by

Numerical Results
For numerical examples, we choose an apple-shaped and a peanut-shaped obstacle as in [22].The parameterizations are given by z(t) = 0.5 + 0. Numerically, we solve the integral equations ( 26) and ( 28) both by Nyström's method, see [6, chap. 3.5].For the computation of the synthetic data we use 128 discretisation points on the boundary curve, while for the inverse scheme we evaluate 64 discretisation points on the boundary with equidistant angles.We consider the wave number k = 2 and choose trigonometric polynomials of degree m = 4 for the approximation of the boundary curve.Furthermore, we use the plane wave u i (x) = e ikx•d with direction d = cos( ⇡ 6 ), sin( ⇡ 6 ) as an incident field.First we consider the reconstruction from the synthetic data without noise using the regularised Newton scheme, see figure 1.The angle of the incident plane wave is indicated by an arrow, the original boundary curve by a dotted line and the initial guess, a circle with radius r = 0.3, by the dashed line.One can observe that choosing the radius of the circle between 0.1 and 1.1 gives similarly results, possibly with slightly more iteration steps.On the right hand side of the figure the relative discrete L 2 errors are plotted.The residual error kF n (r n ) u 1 k/ku 1 k by the full line and the reconstruction error kr n r exact k/kr exact k by the dashed line.For this example we have chosen g(x, u) = u + sin(u) as nonlinear function on the boundary.A nonlinearity of this form was already examined for the Laplace problem in [30] by Wendland and Ruotsalainen.This function does not satisfy condition ( 6), but g is Lipschitz continuous and the assumptions in (A) are guaranteed.In the case of noise-free synthetic data, we observe a stable performance for the apple-shaped object, as well as for other test objects.We have set the regularisation parameter here to ↵ = 0.05, whereas a wide range of regularisation parameters leads to comparable results.For a di↵erent incident direction d of u i we obtain similar results, while adding more incident fields gives us a small improvement in our reconstructions, as known from linear scattering problems.Now we consider reconstructions of the peanut-shape obstacle from running 100 experiments with 5% random noise added.The best and the worst result are shown in figure 2. Comparable results are obtained using the apple-shaped object.In the case of noisy data the regularization parameter has to be increased.In the example considered, we fixed the regularization parameter with ↵ = 2, which turned out to be su cient in all tests.The discrepancy principle (24) with ⌧ = 1.3 was used as stopping condition.
Very similar results are achieved using the nonlinearity g(x, u) = u/(1 + |u| 2 ) from chapter 2, which satisfies the required assumptions to ensure a well-posed direct problem.If we choose the Kerr-type nonlinearity g(x, u) = |u| 2 u a small deterioration in the results is visible, but the reconstructions are still promising.In conclusion, we can state that the performance of the presented all-at-once regularized Newton method for acoustic scattering problems with nonlinear impedance boundary condition is satisfactory in all tested examples and comparable to the well-known linear cases.

Figure 1 .
Figure 1.First and 10th iteration in the case of noise-free data as well as the relative discrete L 2 errors.

Figure 2 .
Figure 2. The best and the worst result after 12 iteration and 5% equally distributed random noise.