Fast spectral methods for the shape identification problem of a perfectly conducting obstacle

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Fast spectral methods for the shape identification problem of a perfectly conducting obstacle

Frédérique Le Louër

To cite this version:

Frédérique Le Louër. Fast spectral methods for the shape identification problem of a perfectly con-

ducting obstacle. 2013. �hal-00780379�

Fast spectral methods for the shape identification problem of a perfectly conducting obstacle

Frédérique Le Louër

Abstract

We are concerned with fast methods for the numerical implementation of the direct and inverse scattering problems for a perfectly conducting obstacle. The scattering problem is usually reduced to a single uniquely solvable modified combined-field integral equation (M- CFIE). For the numerical solution of the M-CFIE we propose a new high-order spectral algorithm by transporting this equation on the unit sphere via the Piola transform. The inverse problem is formulated as a nonlinear least squares problem for which the iteratively regularized Gauss-Newton method is applied to recover an approximate solution. Numerical experiments are presented in the special case of star-shaped obstacles.

Keywords : Maxwell equations, perfectly conducting interface, boundary integral equation, spectral method, Fréchet derivative, regularized Newton method.

1 Introduction

In this paper we are concerned with the numerical aspects of the solution of the direct and inverse electromagnetic scattering problems for a three-dimensional bounded and perfectly conducting obstacle lit by time-harmonic incident plane waves.

We assume the perfect conductor be represented by a bounded domainΩinR³. LetΩ^cdenote the exterior domain R³\Ωand n denote the outer unit normal vector to the boundaryΓ. Let κdenote the exterior wavenumber. The propagation of electromagnetic waves are governed by the system of Maxwell equations and the time-harmonic Maxwell system can be reduced to a second order equation for the electric field only. In this case the forward problem is formulated as follows : Given an incident electric wave E^inc which is assumed to solve the second order Maxwell equation in the absence of any scatterer, find the electric scattered waveE^ssolution to the time-harmonic Maxwell equation

curl curlE^s−κ²E^s= 0 inΩ^c, (1.1a) and satisfying the boundary condition,

n×(E^s+E^inc) = 0 onΓ. (1.1b)

In addition the scattered fieldE^shas to satisfy the Silver-Müller radiation condition:

lim

|x|→+∞|x|

curlE^s(x)× x

|x|−iκE^s(x)

= 0. (1.1c)

∗f.lelouer@math.uni-goettingen.de

This scattering problem can be reduced in several different ways to a single modified combined- field boundary integral equation (M-CFIE). In section 2, we describe, via direct and indirect approaches, the single integral equations that are usually used to solve the problem (1.1a)-(1.1c).

The construction of efficient inverse obstacle scattering algorithm greatly depends on an ac- curate approximation of the solution to the forward problem. Among all the existing numerical methods to solve integral equations, we focus here on spectral methods. Ganesh and Hawkins already proposed in [8, 9, 10, 11] several spectrally accurate methods to implement the electric field integral equation (EFIE). One of them [11] consists in transporting the EFIE on the unit sphere via a normal transformation acting from the tangent plane to the boundary Γ onto the tangent plane to the unit sphere, so that one only have to seek a solution in terms of tangential vector spherical harmonics. To treat numerically the M-CFIE, we need to implement the hy- persingular part of the electromagnetic double layer boundary integral operator. This requires a transformation that move the nullspace of the surface divergence onΓ onto the nullspace of the surface divergence onS². The implementation of a hypersingular integral can then be avoided by involving integration by parts and surface derivatives of the vector spherical harmonics that can be computed analytically. Such a transformation is well known as the Piola transform. In section 3 we give the reformulation of the integral equation in spherical coordinates, based on this ap- proach, and describe an appropriate splitting of the various singularities. The fully discrete high order spectral algorithm and numerical examples are presented in section 4. The Piola transform is already used by Hohage and Le Louër [15] to implement systems of second-kind boundary integral equations in an inverse dielectric obstacle scattering algorithm. Therefore, section 3 and 4 use notations and contain various results from [15] that are essential for this paper.

The radiation condition implies that the scattered fieldE^shas an asymptotic behavior of the form

E^s(x) =e^iκ|x|

|x| E^∞(x) +b O 1

|x|

, |x| → ∞,

uniformly in all directions xb = _|x|^x . The far-field pattern E^∞ is a tangential vector function defined on the unit sphereS² ofR³and is always analytic.

Let consider the scattering ofmincident plane waves of the formE^inc_k (x) =p_ke^iκx·dk where dk,p_k ∈ S² and dk·p_k = 0. We denote by Fk the boundary to far-field operator that maps the boundary Γ onto the far-field pattern E^∞_k of the solution to the forward problem (1.1a)- (1.1c) for the incident waveE^inc_k . The inverse problem is formulated as follows: Given noisy far field measurements E^∞_1,δ, . . . ,E^∞_m,δ obtained from the scattering of the m incident plane waves characterized by the couples of directions and polarizations(d_k,p_k)_k=1,...,m, solve

F_k(Γ) =E^∞_k,δ, fork= 1, . . . , m. (1.2) Here, the noise level is measured in theL²-norm, i.e.

_m P

k=1

||E^∞_k,δ−E^∞_k ||²_L2

¹₂

< δ, and the error bound δ is assumed to be known. Kress [16] proved that a perfect conductor can be uniquely determined from the knowledge of the far-field pattern for all incoming plane waves. However, it remains an open question whether or not the boundaryΓis uniquely determined by only a finite number of incoming plane waves.

To solve the inverse problem (1.2), we reformulate it as a nonlinear equation posed on an open set of parametrized boundaries. Then we apply the iteratively regularized Gauss-Newton method (IRGNM) to the nonlinear equation via first order linearization. The computation of the iterates requires the analysis and an explicit form of the first Fréchet derivative of the parametrized form of the boundary to far-field operator. The first Fréchet derivative is usually characterized as the far-field pattern of the solution to a new exterior boundary value problem. The fast spectral solver

of the direct problem is then incorporated in the IRGNM to compute the far-field pattern of the solution and its derivative at each iteration steps. In electromagnetism, Fréchet differentiability was first investigated by Potthast [21] for the perfect conductor problem, using the boundary integral equation approach. The characterization of the first derivative was then improved by Kress [16]. In section 5, we formulate the IRGNM for the inverse problem (1.2). We also recall the main result on the Fréchet differentiability of the boundary to far-field operator and give a characterization of the adjoint operator, following ideas of [14, 15], which is needed in the implementation of the utilized regularized Newton method. We present numerical experiments in the special case of star-shaped obstacles.

2 The solution of the perfect conductor problem

In this paper, we will assume that the boundaryΓofΩis a smooth and simply connected closed surface, so thatΩis diffeomorphic to a ball.

Notation: We denote byH^s(Ω),H_loc^s (Ω^c)andH^s(Γ) the standard (local in the case of the exterior domain) complex valued, Hilbertian Sobolev space of orders∈Rdefined on Ω,Ω^c and Γ respectively (with the convention H⁰ = L².) Spaces of vector functions will be denoted by boldface letters, thusH^s= (H^s)³. IfPis a differential operator, we write:

H^s(P,Ω) = {v∈H^s(Ω) : Pv ∈H^s(Ω)}, H^s_loc(P,Ω^c) = {v∈H^s_loc(Ω^c) : Pv∈H^s_loc(Ω^c)}.

The spaceH^s(P,Ω)is endowed with the graph norm||v||_Hs(P,Ω)=

||v||²_Hs(Ω)+||Pv||²_Hs(Ω)

¹₂ . This defines in particular the Hilbert spacesH^s(curl,Ω)andH^s(curl curl,Ω)and the Fréchet spacesH^s_loc(curl,Ω)and H^s_loc(curl curl,Ω). Whens= 0 we omit the upper index 0.

We use the following surface differential operators: The tangential gradient denoted bygrad_Γ, the surface divergence denoted by divΓ, the tangential vector curl denoted by curlΓ and the surface scalar curl denoted bycurl_Γ. For their definitions we refer to [15, Appendix A] or [17, pages 68-75]. Fors∈R, we introduce the Hilbert space

H^s_div(Γ) ={j∈H^s(Γ); j·n= 0and div_Γj ∈H^s(Γ)}, H^s_curl(Γ) ={j∈H^s(Γ); j·n= 0and curlΓj∈H^s(Γ)}, endowed with the norms

|| · ||H^s_div(Γ) =

|| · ||²_Hs(Γ)+||div_Γ· ||²_Hs(Γ)

^1/2 ,

|| · ||H^s_curl(Γ) =

|| · ||²_Hs(Γ)+||curl_Γ·||²_Hs(Γ)

^1/2 .

Recall that for a vector functionu ∈H(curl,Ω)∩H(curl curl,Ω), the traces n×u_|Γ and n×curlu_|Γ are inH⁻_div¹²(Γ). The dual space ofH⁻_div¹²(Γ)for theL²duality product isH⁻_curl¹² (Γ) and the exterior product with the normal vector defines a bicontinuous isomorphism between H⁻

1 2

div(Γ)and H⁻

1 2

curl(Γ); so that(n×u_|Γ)×nand(n×curlu_|Γ)×nare inH⁻

1 2

curl(Γ).

Let Φ(κ,z) = e^iκ|z|

4π|z| be the fundamental solution of the Helmholtz equation ∆u+κ²u= 0.

For any solutionE^sto the Maxwell equation (1.1a) that satisfies the radiation condition (1.1c),

it holds the Stratton-Chu representation formula:

E^s(x) = Z

Γ

curl^xn

Φ(κ,x−y) n(y)×E^s(y)o ds(y) + 1

κ² Z

Γ

curl curl^xn

Φ(κ,x−y) n(y)×curlE^s(y)o ds(y).

(2.1)

Conversely, for anyj ∈H⁻_div¹²(Γ)the potentials Sj= 1

κ Z

Γ

curl curl^xn

Φ(κ,· −y)j(y)o

ds(y)andDj = Z

Γ

curl^xn

Φ(κ,· −y)j(y)o ds(y) satisfy the Maxwell equation and the Silver-Müller radiation condition.

Using these results, the forward problem (1.1a)-(1.1c) can be reduced, in several different ways, to a single uniquely solvable modified combined-field integral equation (M-CFIE). We will consider the following two different approaches. The first one can be used to solve electromagnetic scattering problem with general Dirichlet conditions of the formn×E^s=f wheref ∈H⁻_div¹²(Γ) is given. It is based on the layer ansatz :

E^s=Dj+iηSΛj, (2.2)

where Λ is a bounded operator fromH⁻_div¹²(Γ) to itself, self-adjoint and elliptic for the bilinear form

(j,m)7→

Z

Γ

j·(n×m)ds (2.3)

andη is a non vanishing real constant. By the jump relations, the fieldE^s given by (2.2) solves the Dirichlet boundary value problem (1.1a)-(1.1c) if the densityj solves the following integral equation

(I−Mκ−iηCκΛ)j= 2f onΓ. (2.4)

Here the single layer potentialC_κ and the double layer potentialM_κare defined by M_κj(x) = −

Z

Γ

n(x)×curl^x{2Φ(κ,x−y)j(y)}ds(y), Cκj(x) = −1

κ Z

Γ

n(x)×curl curl^x{2Φ(κ,x−y)j(y)}ds(y)

= −κn(x)× Z

Γ

2Φ(κ,x−y)j(y)ds(y) +1 κcurlΓ

Z

Γ

2Φ(κ,x−y) divΓj(y)ds(y).

The operator Mκ : H⁻

1 2

div(Γ) → H⁻

1 2

div(Γ) is compact and the operator Cκ has a hypersingular kernel but is bounded onH⁻

1 2

div(Γ). The operator Λis then chosen such that(I−Mκ−CκΛ)is a Fredholm operator of index zero. Kress first proposed a compact regularizationΛj=n×S²₀j where S₀ is the single layer boundary integral operator of the Laplace equation [3], thus (I + M_κ+C_κΛ)is a Fredhom operator of the second kind. One can also use the elliptic and invertible operatorΛj=C₀j which is a variant of the operatorC_κ [5, 22] defined onH⁻_div¹²(Γ)by

C₀j=n×S₀j+curl_ΓS₀div_Γj.

The far-field pattern can be computed via the integral representation formulaE^∞=F∞j where the far-field operatorF∞:H⁻

1 2

div(Γ)→L²_t(S²)is defined forxb∈S² by : F_∞j(bx) = iκ

4π Z

Γ

e^−iκx·yb bx×j(y)

ds(y) +iη κ 4π

Z

Γ

e^−iκx·yb

bx×Λj(y)

×xb ds(y).

The alternative boundary integral equation approach is based on the Stratton-Chu represen- tation formula (2.1) and the following equality forx∈Ω^c

0 = 1 κ²

Z

Γ

curl curl^xn

Φ(κ,x−y) n(y)×curlE^inc(y)o ds(y) +

Z

Γ

curl^xn

Φ(κ,x−y) n(y)×E^inc(y)o ds(y).

Adding the last equation to (2.1) and using the boundary condition ofE^s, it yields E^s(x) = 1

κ² Z

Γ

curl curl^xn

Φ(κ,x−y) n(y)×curl E^s(y) +E^inc(y)o

ds(y), x∈Ω^c. Taking the Neumann and Dirichlet trace of the above equation we obtain

Cκ

1

κn×curl E^s+E^inc

= 2n×E^inc, and

(I +Mκ) 1

κn×curl E^s+E^inc

= 2 1

κn×curlE^inc

. Setting m =κ⁻¹n×curl E^s+E^inc

, by a linear combination of the two above equalities we obtain the integral equation

(I +Mκ+iηΛCκ)m= 2 1

κn×curlE^inc

+iηΛ n×E^inc

onΓ, (2.5) whereΛis still an elliptic operator for the bilinear form (2.3).

The far-field pattern can then be computed via the integral representation formula E^∞(bx) = κ

4π Z

Γ

e^−iκbx·y

bx×m(y)

×bx ds(y).

3 Spherical reformulation of the integral equation

We transport the surface integral equations (2.4) and (2.5) on a spherical reference domain and work in the spherical coordinate system. We denote byθ, φthe spherical coordinates of any point bx∈S², that means we can write

bx=ψ(θ, φ) = (sinθcosφ,sinθsinφ,cosθ)^T, with(θ, φ)∈]0;π[×[0; 2π[∪{(0,0); (0, π)}.

The tangent and the cotangent planes at any point bx =ψ(θ, φ) ∈ S² is generated by the unit vectorse_θ = ^∂ψ_∂θ(θ, φ)ande_φ = _sin¹_θ^∂ψ_∂φ(θ, φ). The triplet(bx,e_θ,e_φ) forms a direct orthonormal system. The determinant of the Jacobian of the change of variable isJψ(θ, φ) = sinθ.

Letq:S² →Γ be a parametrization of classC¹ at least. The total derivative[Dq(bx)]maps the tangent planeT

bxtoS²at the pointbxonto the tangent planeT_q(

x)b toΓat the pointq(x). Theb latter is generated by the vectorst1(bx) = [Dq(bx)]eθ andt2(x) = [Dq(b x)]eb φ. The determinant Jq of the Jacobian of the change of variable q : S² 7→∈Γ and the normal vector n◦q can be computed via the formulasJq =

t1×t2

and τq(n) = ^t1_J^×t2

q . The parametrizationq : S² →Γ being a diffeomorphism, we set[Dq(bx)]⁻¹ = [Dq⁻¹]◦q(bx). The transposed matrix[Dq(bx)^*]⁻¹ maps the cotangent planeT^∗

bx to S² at the pointxb onto the cotangent plane T^∗_q(

bx) to Γ at the

pointq(bx). The latter is generated by the vectorst¹(x) =b ^{t2(q(x))×n(q(b}_J ^bx))

q(x)b = [Dq(bx)^*]⁻¹e_θ and t²(bx) =^n(q(bx))×t_J ^1(q(bx))

q(bx) = [Dq(bx)^*]⁻¹eφ.

To transport the boundary integral equations on the unit sphere, we use the Piola transform of q, which is an invertible operator from H⁻

1 2

div(Γ) to H⁻

1 2

div(S²) defined as follows [15, Lemma 3.1]:

Pq : H⁻_div¹²(Γ) −→ H⁻_div¹²(S²)

j 7→ j_s=J_q[Dq]⁻¹(j◦q). (3.1)

This allows us to use the following identities stated in [15]:

(grad_Γu)◦q= [Dq^*]⁻¹grad_S2(u◦q), (curl_Γu)◦q= 1

J_q[Dq]curl_S2(u◦q), (div_Γv)◦q= 1

Jq

div_S2 J_q[Dq]⁻¹(v◦q)

, (curl_Γw)◦q= 1 Jq

curl_S2 [Dq^*](w◦q) .

(3.2)

The construction of the spectral method is based on the combination of the boundary integral operators Mκ andCκ with the Piola transform (3.1) and its inverse. In practice, we apply the operator (3.1) to both sides in the boundary integral equations (2.4) and (2.5) and we insert the identity I

H⁻

1 2

div(Γ) =PqP_q⁻¹ between the integral operators and the unknown. We thus replace the operatorsΛ,Mκ andCκin (2.4) and (2.5) by the following operators defined for any density j_s∈H⁻

1 2

div(S²)by:

Λ_sj_s =P_q⁻¹ΛP_qj_s,

Mκj_s =P_q⁻¹MκPqj_s=−Jq[Dq]⁻¹ Z

S²

τq(n)×curl{2Φ(κ,|q(·)−q(y)|)[Dq(b y)]jb _s(y)}ds(b y),b C_κj_s =P_qC_κP_qj_s=−κ J_q[Dq]⁻¹

Z

S²

τ_q(n)× {2Φ(κ,|q(·)−q(by)|)[Dq(by)]j_s(by)}ds(by) +1

κcurl_S² Z

S²

{2Φ(κ,|q(·)−q(y)|) divb _S²j_s(y)}ds(b y).b

The new unknown will be a tangential vector density inH⁻_div¹²(S²)obtained by applying the Piola transform (3.1) to the unknown in (2.4) or (2.5).

Lemma 3.1 The operatorΛis an elliptic operator onH⁻

1 2

div(Γ)for the bilinear form (2.3)if and only ifΛ_s is an elliptic operator on H⁻_div¹²(S²) for the bilinear form

H⁻

1 2

div(S²)×H⁻

1 2

div(S²) → C

(j_s,ms) 7→

Z

S²

j_s·(bx×ms)ds (3.3)

Proof. Let j_s ∈ H⁻_div¹²(S²). Notice that we have (P_q⁻¹j_s)◦q = _J¹

q[Dq]j_s. In view of the definition of the vectorst¹ andt², we obtain the following equality :

n× P_q⁻¹j_s

◦q= [Dq^*]⁻¹(bx×j_s). (3.4)

Now, we setj=P_q⁻¹j_s. We haveΛj=P_q⁻¹Λ_sj_s and Z

Γ

Λj·(n×j)ds = Z

S²

Jq((Λj)◦q)·((n×j)◦q)ds

= Z

S²

Jq

P_q⁻¹Λsj_s

· n× P_q⁻¹j_s

◦qds

= Z

S²

[Dq]Λsj_s

·

[Dq^*]⁻¹(xb×j_s) ds

= Z

S²

Λsj_s·(xb×j_s)ds.

It follows that the ellipticity ofΛ_s for the bilinear form (3.3) results from the ellipticity ofΛfor

the bilinear form (2.3), and conversely.

Remark 3.2 A suitable choice is to defineΛ_s such thatj_s7→(Λ_sj_s)×bxis the duality operator fromH⁻_div¹²(S²)toH⁻_curl¹²(S²)[17, pp. 208], that is

Λ_sj_s=−∇_S²(−∆_S²)⁻³²curl_S2−curl_S2(−∆_S²)⁻¹² div_S2. In practice we will use the operator defined byΛ_sj_s=xb×j_s.

The implementation of the operator Mκ is already discussed in [15]. Here, we focus on the operatorCκ. We introduce the functions :

S1(q;κ,bx,by) = 1

2πcos(κ|q(x)b −q(by)|), S2(q;κ,x,b y) =b 1

2π







sin(κ|q(bx)−q(y)|)b

|q(x)b −q(by)| bx6=y,b

κ bx=y.b

and

R(q;bx,y) =b |bx−y|b

|q(bx)−q(by)|. The operatorCκcan then be rewritten as

Cκj_s(bx) = Z

S²

R(q;bx,by)

|bx−by| W1(q;κ,x,b y)jb _s(by)ds(y) +b i Z

S²

W2(q;κ,bx,y)jb _s(by)ds(by) +1

κcurl_S2

Z

S²

R(q;bx,by)

|bx−y|b S1(q;κ,bx,by) div_S2j_s(y)ds(b by) +i

κcurl_S2

Z

S²

S2(q;κ,bx,by) div_S2j_s(y)ds(b by) whereW1(q;κ,bx,by)andW2(q;κ,bx,y)b are3×3 matrices given by

W1(q;κ,x,b by) = κS1(q;κ,x,b by)V(q;bx,y),b W2(q;κ,x,b by) = κS2(q;κ,x,b by)V(q;bx,y),b with

V(q;bx,y)b = t2(bx)·t1(y)b eθ(bx)⊗eθ(by) +t2(bx)·t2(y)b eθ(x)b ⊗eφ(by)

−t1(x)b ·t1(by)eφ(x)b ⊗eθ(y)b −t1(x)b ·t2(by)eφ(x)b ⊗eφ(by).

Next, we introduce a change of coordinate system in order to move all the singularities in the weakly singular integrals to only one point that is chosen to be the North pole. Forxb∈ S² we consider an orthogonal transformation T

xb which maps bxonto the North pole denoted by η. Ifb bx=bx(θ, φ)then T

bx:=P(φ)Q(−θ)P(−φ)whereP andQare defined by P(φ) =





cosφ −sinφ 0 sinφ cosφ 0

0 0 1



 and Q(θ) =





cosθ 0 sinφ

0 1 0

−sinφ 0 cosφ



. We also introduce an induced linear tranformationT

bx defined byT

bxu(y) =b u(T⁻¹

bx y)b and we still denote byT

bx its bivariate analogue T

bxv(yb₁,by₂) =v(T⁻¹

bx yb₁, T⁻¹

bx yb₂). If we write bz=T

bxyb then we have the identity

|xb−by|=|T⁻¹

bx (bη−bz)|=|ηb−bz|.

The boundary integral operatorCκ can be rewritten in the form:

Cκj_s(bx) = Z

S²

T

xbR(q;bη,bz)

|bη−bz| T

xbW1(q;κ,η,b bz) +iT

bxW2(q;κ,η,b bz)

Tbxj_s(bz)ds(bz) +1

κcurl_S2

Z

S²

T

bxR(q;η,b bz)

|ηb−bz| T

bxS1(q;κ,η,b bz) +iT

bxS2(q;κ,η,b bz)

Tbx div_S2j_s

(bz)ds(bz),

(3.5)

and it can be shown that the mappings (θ⁰, φ⁰)7→ T

bxR(q; ˆη,bz(θ⁰, φ⁰))T

bxS1(q;κ,η,ˆ bz(θ⁰, φ⁰))and (θ⁰, φ⁰)7→ T

bxR(q; ˆη,bz(θ⁰, φ⁰))T

bxC1(q;κ,η,ˆ bz(θ⁰, φ⁰)) are smooth. An important point is that the singularity _|ˆη− ¹

bz(θ⁰,φ⁰)| = ¹

2 sinθ⁰ 2

is cancelled out by the surface elementds(bz) = sinθ⁰dθ⁰dφ⁰. Finally, settingf_s=Pqf, the parametrized form of the equation (2.4) is

I

H⁻

1 2

div(S²)− Mκ−iηCκΛs

j_s= 2f_s, onS², (3.6)

and setting u^inc₁ = Pq n×E^inc

and u^inc₂ = Pq n×curlE^inc

the parametrized form of the equation (2.5) is

I

H⁻

1 2

div(S²)+Mκ+iηΛsCκ

j_s= 2

1 κu^inc₂

+iηΛsu^inc₁

, onS². (3.7)

4 A high order spectral algorithm

The parametrized boundary integral equations (3.6) and (3.7) are now propicious to obtain spec- trally accurate approximations of the solution by combining the spectral method of Ganesh and Graham [7] and the hybrid spectral method of Ganesh and Hawkins [10, 11]. The numerical scheme is based on the numerical integration formula over the unit sphere for a continuous func- tion

Z

S²

u(bx)ds(x)b ≈

2n+1

X

ρ=0 n+1

X

τ=1

µ_ρν_τu(x(θb _τ, φ_ρ)), (4.1) where θ_τ = arccosζ_τ where ζ_τ, for τ = 1, . . . , n+ 1, are the zeros of the Legendre polynomial P_n+1⁰ of degree n+ 1andν_τ, forτ= 1, . . . , n+ 1, are the corresponding Gauss-Legendre weights and

µρ= π

n+ 1, φρ= ρπ

n+ 1, forρ= 0, . . . ,2n+ 1.

Forl∈Nand0≤j≤l, letP_l^j denote thej-th associated Legendre function of orderl. When j= 0we writeP_l⁰=P_l. The spherical harmonics defined by

Y_l,j(bx) = (−1)^|j|+j2 s

2l+ 1 4π

(l− |j|!)

(l+|j|!)P_l^|j|(cosθ)e^ijφ

forj=−l, . . . , l andl= 0,1,2, . . .form a complete orthonormal system in L²(S²). The formula (4.1) is exact for the spherical polynomials of order less than or equal to2n+ 1(see [20]). This induces the discrete inner product(·,·)n

(ϕ1, ϕ2)n=

2n+1

X

ρ=0 n+1

X

τ=1

µρντϕ1(bxρτ)ϕ2(bxρτ),

on the space Pn of all scalar spherical polynomials of degree less than or equal to n. We have (ϕ1, ϕ2)_S2 = (ϕ1, ϕ2)n for all ϕ1, ϕ2 ∈Pn. We introduce a projection operator`n onPn defined by

`nu=

n

X

l=0 l

X

j=−l

(u, Yl,j)nYl,j, and we set

Lnu=





`nu1

`nu2

`nu3



 whereu= (u1, u2, u3)^T. The tangential vector spherical harmonics defined by

Y⁽¹⁾_l,j = 1

pl(l+ 1)grad_S2Y_l,j andY⁽²⁾_l,j = 1

pl(l+ 1)curl_S2Y_l,j

for j =−l, . . . , l and l = 1,2, . . . form a complete orthonormal system in L²_t(S²). The formula (4.1) is exact for the vector spherical polynomials of order less than or equal to2nand we have Y^(k)_l,j ·Y^(k_l0,j⁰⁾⁰ ∈ P2n+1 for l, l⁰ ≤n, |j| ≤ l, |j⁰| ≤ l⁰ and k, k⁰ = 1,2 (see [20]). This induces the discrete inner product

(v1|v2)n=

2n+1

X

ρ=0 n+1

X

τ=1

µρντv1(xbρτ)·v2(xbρτ), on the subspaceTn ⊂H⁻

1 2

div(S²)of finite dimension2(n+ 1)²−2generated by the orthonormal basis of tangential vector spherical harmonics of degree less than or equal ton∈N. We introduce a projection operatorLn onTn defined by

Lnv=

2

X

i=1 n

X

l=1 l

X

j=−l

(v|Y⁽ⁱ⁾_lj)nY⁽ⁱ⁾_lj. In a first step, the operatorCκ is approached by

Cκ,n⁰j_s(x) =b Z

S²

1

|ηˆ−bz|Ln⁰{T

bxR(q; ˆη,·)T

bxW1(q;κ,η,ˆ ·)T

bxj_s(·)}(bz)ds(bz) + i

Z

S²

Ln⁰{T

bxW2(q;κ,η,ˆ ·)T

bxj_s(·)}(bz)ds(bz) + 1

κcurl_S2

Z

S²

1

|ˆη−bz|`n⁰{T

bxR(q;bx,·)T

bxS1(q;κ,bx,·)T

bx div_S2j_s(·)

}(bz)ds(bz)

+ i κcurl_S2

Z

S²

`n⁰{T

bxS2(q;κ,bx,·)T

bx div_S2j_s(·)

}(bz)ds(bz).

for somen⁰=an+ 1 with fixeda >1andn⁰−n >3 (see [11] and [8, Appendix A]). By the use of the fact that the scalar spherical harmonics are eigenfunctions of the single layer potential on the sphere and additional identities [4] we obtain

Cκ,n⁰j_s(bx) =Wκ,n⁰j_s(x) +b 1

κcurl_S2Sκ,n⁰div_S2j_s(bx), (4.2) with

W_κ,n⁰j_s(bx) =

2n⁰+1

X

ρ⁰=0 n⁰+1

X

τ⁰=1

µ_ρ⁰ν_τ⁰α_τ⁰T

bxR(q; ˆη,bz_ρ⁰_τ⁰)T

bxW₁(q;κ,η,ˆ bz_ρ⁰_τ⁰)T

bxj_s(bz_ρ⁰_τ⁰) + i

2n⁰+1

X

ρ⁰=0 n⁰+1

X

τ⁰=1

µ_ρ0ν_τ0T

xbW2(q;κ,η,ˆ bz_ρ0τ⁰)T

bxj_s(bz_ρ0τ⁰), and

Sκ,n⁰ϕs(bx) =

2n⁰+1

X

ρ⁰=0 n⁰+1

X

τ⁰=1

µρ⁰ντ⁰ατ⁰T

bxR(q; ˆη,bzρ⁰τ⁰)T

bxS1(q;κ,η,ˆ bzρ⁰τ⁰)T

bxϕs(bzρ⁰τ⁰) + i

2n⁰+1

X

ρ⁰=0 n⁰+1

X

τ⁰=1

µρ⁰ντ⁰T

xbS2(q;κ,η,ˆ bzρ⁰τ⁰)T

bxϕs(bzρ⁰τ⁰), where ατ⁰ =Pn⁰

l=0Pl(ζτ⁰).We proceed in the same way for the operator Mκ as for the weakly singular part of Cκ and we denote by Mκ,n⁰ its approximation. Our fully discrete scheme for (3.6) is as follows: computej_n ∈Tn such that

j_n−LnMκ,n⁰j_n−iηLnCκ,n⁰Λsj_n= 2Lnf_s, (4.3) whereΛs is defined as in remark 3.2. Since any densityj_n∈Tn can be written

j_n=

n

X

l=1 l

X

j=−l

α_l,jY⁽¹⁾_l,j +β_l,jY⁽²⁾_l,j,

the equation (4.3) is equivalent to a system of2× (n+ 1)²−1

equations for the2× (n+ 1)²−1 unknown coefficients(α_l,j)and (β_l,j)by applying the scalar product(· |Y⁽¹⁾_l,j)_n and (· |Y⁽²⁾_l,j)_n, for l = 1, . . . , n and j = −l, . . . , l to the equation (4.3). The same remarks hold true for the equation (3.7).

In the remaining of the section we briefly describe the fully discrete scheme for the equation (4.3) and give numerical results. The discrete approximationWκ of the weakly singular part of the operator ofCκis of the form

Wκ=

W1,1 W1,2

W2,1 W2,2

, whereW_a,b, fora, b= 1,2 is a (n+ 1)²−1

× (n+ 1)²−1

matrix. The coefficients ofW_a,b, for1≤l, l⁰ ≤n,|j| ≤land|j⁰| ≤l⁰ are given by

W_a,b^ljl0j0 = (LnWκ,n⁰Y^(b)_l,j|Y^(a)_l0j⁰)n.

The discrete approximation of the operatorM_κis obtained in the same way and is denoted byM_κ. We refer to the paper of Hohage and Le Louër [15] for details on their numerical implementation.

The discrete approximation of the hypersingular part of the operatorCκis of the form Hκ=

0 0 H2,1 0

, whereT2,1 is a (n+ 1)²−1

× (n+ 1)²−1

matrix. The coefficients ofH2,1, for1≤l, l⁰≤n,

|j| ≤l and|j⁰| ≤l⁰ are given by

H^ljl_2,1^0j0 =κ⁻¹(Lncurl_S2Sκ,n⁰div_S2Y⁽¹⁾_l,j|Y⁽²⁾_l0j⁰)_n

=−κ⁻¹(curl_S2`_nSκ,n⁰

pl(l+ 1)Y_l,j,Y⁽²⁾_l0j⁰)_n

=−κ⁻¹(`nSκ,n⁰

pl(l+ 1)Yl,j,p

l⁰(l⁰+ 1)Yl⁰j⁰)n.

where Sκ,n⁰ is the discrete approximation of the acoustic single layer potential and we refer to the paper of Ganesh and Graham [7] for details on its numerical implementation. The discrete approximation of the operatorC_κisC_κ=W_κ+H_κ. The discrete approximation of Λ_sis of the form

Λ_n=

0 −I

C^(n+1)2−1

IC^(n+1)2−1 0

.

Finaly, the discrete approximation of the integral operator associated to (2.4) isI−Mκ−iηCκΛn, and the discrete approximation of the integral operator associated to (2.5) isI +Mκ+iηΛnCκ. The discrete approximation of the right-hand side 2f_s = 2Pqf of one of the boundary integral equations is the vector2f, wheref = (f₁,f₂)^T, whose coefficients are given fork= 1,2,l= 1, . . . , n andj=−l, . . . , l byf_k^lj= (LnPqf|Y^(k)_lj )n.

The parametrized form of the far-field operatorF_∞ isF_∞j_s=F1j_s+iηF2Λsj_s,with (F1j_s) (x) =b iκ

4π Z

S²

e^{−iκbx·q(y)b}

eθ(by)·j_s(y)b

xb×t1(y) +b eφ(by)·j_s(y)b

bx×t2(y)b ds(by), and

(F2ms) (bx) = κ 4π

Z

S²

e^{−iκbx·q(by)}

eθ(y)b ·ms(by)

bx×t1(by) + eφ(by)·ms(by)

bx×t2(by)

×bxds(y).b The discrete approximation ofF∞ evaluated at the2(n_∞+ 1)² Gauss-quadrature points on the unit far sphere is

F_∞= F₁ F₂ I

C^2(n+1)2−2 0

0 Λ_n

with

F1= F1,1 F1,2

, F2= F2,1 F2,2

where F_a,b, for a, b= 1,2 is a 6(n_∞+ 1)²×((n+ 1)²−1) matrix. The coefficients of F_a,b, for a, b = 1,2, 1 ≤ l⁰ ≤ n, |j⁰| ≤ l⁰ and ρ = 0, . . . ,2n_∞+ 1 and τ = 1, . . . , n_∞+ 1 are given by F^{ρτ l}_a,b^0j0 =

FaY^(b)_l0j⁰

(bxρτ).

Table 2 exhibits emphasized fast convergence for the far-field patternE^∞for boundaries with parametric representations given in Table 1. As a first test we compute the electric far-field denoted,E^∞_ps, created by an off center point source located inside the perfect conductor :

E^inc(x) =gradΦ(κ,|x−s|)×p, s∈Ωandp∈S².

Table 1: Parametric representation of the perfectly conducting obstacles [1, 18]

surface parametric representation

sphere q◦ψ(θ, φ) =Rψ(θ, φ),R∈R^∗+;

great stellated dodecahedron q◦ψ(θ, φ) = 0.33r(θ, φ)ψ(θ, φ), wherer(θ, φ)>0solves p−|r(θ,φ)f(δ,ξ,0)|^p+p−|r(θ,φ)f(δ,−ξ,0)|^p+p−|r(θ,φ)f(0,δ,ξ)|^p

+p^{−|r(θ,φ)f(0,δ,−ξ)|p}+p^{−|r(θ,φ)f(ξ,0,δ)|p}+p−|r(θ,φ)f(−ξ,0,δ)|^p= 2.5, withf(a, b, c) =asinθcosφ+bsinθsinφ+ccosθ,

δ= q5−√

5 10 ,ξ=

q

5+√ 5

10 ,p∈N,p >2.

Table 2: Numerical examples for the perfect conductor problem

surface n ||[E^∞_ps]n−E^∞_exact||∞ Re[E^∞_pw(d)]n·p Im[E^∞_pw(d)]n·p

sphere 5 3.0071E−04 0.153 874 899 2.392 810 874

R= 1 10 2.2449E−12 0.156 527 151 2.383 319 427

κ= 4.49340945 15 1.0175E−14 0.156 527 163 2.383 319 428 20 9.5447E−15 0.156 527 163 2.383 319 428 great stellated dodecahedron 25 1.1905E−03 −0.292 692 794 2.229 490 540

p= 4 35 2.1864E−04 −0.294 669 310 2.234 334 948

κ=π 45 5.0831E−05 −0.294 839 844 2.232 714 273

55 1.4777E−05 −0.294 712 464 2.233 078 537

In this case the total exterior wave has to vanish so that the far-field pattern of the scattered waveE^s is the opposite of the far field pattern of the incident wave:

E^∞_exact(bx) =−iκ

4πe^−iκbx·s(bx×p).

We choosed s = 0,^0.1√

2,−^0.1√

2

^T

and p = (1,0,0)^T. In the tabulated results we indicate the uniform-norm error (by taking the maximum of errors obtained over 1300 observed directions, i.e n_∞= 25) :

||[E^∞_ps]n−E^∞_exact||_∞= max

x∈b S²

[E^∞_ps]n−E^∞_exact .

As a second task we compute the electric far field denoted,E^∞_pw, created by the scattering of an incident plane wave :

E^inc(x) =pe^iκx·d, where d,p∈S² andd·p= 0.

In the tabulated results we indicate the real part and the imaginary part of the polarization component of the electric far field evaluated at the incident direction : [E^∞_pw(d)]n·p. We choosed d= 0,0,1^T

andp= (1,0,0)^T. In each of the examples we takeη= 1.

5 Numerical solution of the inverse problem and IRGNM

With the objective of using a regularized Newton-type method to solve numerically the inverse problem (1.2) we first reformulate it as a nonlinear equation posed on an open subset of admissible parametrizations of a Hilbert space. More precisely, we choose some reference domainΩref with a simply connected closed boundaryΓref and consider mappingsq: Γref→Γ belonging to

Q={q∈H^s(Γ_ref,R³) :q injective, det(Dq(bx))6= 0for allxb∈Γ_ref}.

Fors >2the setQis open inH^s(Γ_ref,R³)andH^s(Γ_ref,R³)⊂C¹(Γ_ref,R³). Since now, forq∈ Q we defineΓ_q:=q(Γ_ref)and denote byn_qthe exterior unit normal vector toΓ_q. More generally we will label all quantities and operators related to the perfect conductor problem for the boundary Γq by the index q. We denote by Fk : Q → L²_t(S²) the boundary to far field operator that maps the parametrizationqof the boundaryΓq onto the far-field patternE^∞_q,k corresponding to the scattering of the incident wave E^inc_k by the perfect conductor Γq. These operators may be combined into one operator F : Q →L²_t(S²)^m, F(q) := (F₁(q), . . . , F_m(q))^T. We also combine the measured far-field patterns into a vector E^∞_δ := (E^∞_1,δ, . . . ,E^∞_m,δ)^T∈L²_t(S²)^m such that the inverse problem (1.2) can be written as the following nonlinear equation

F(q) =E^∞_δ . (5.1)

A regularized iterative method can then be applied to the linearized equationF⁰[q]ξ=E^∞_δ −F[q].

Here, we use the Iteratively Regularized Gaus-Newton Method (IRGNM). This requires to prove the Fréchet differentiability of the boundary to far-field operatorF and an explicit form of the first Fréchet dérivativeF⁰[q]and its adjointF⁰[q]^*. Then the iterates of the IRGNM can be computed by

q^δ_N+1:= argmin_q∈Hs(S²,R)

||F⁰[q^δ_N](q−q^δ_N) +F(q^δ_N)−E^∞_δ ||²_L2+α_N||q−q₀||²_Hs

, (5.2) Here q₀ = q^δ₀ is some initial guess and the regularization parameters are chosen of the form αN =α0 2

3

^N

. The updates(∂q)N :=q^δ_N+1−q^δ_N are the unique solutions to the linear equations

αNI +F⁰[q^δ_N]^*F⁰[q^δ_N]

(∂q)^δ_N =F⁰[q^δ_N]^* E^∞_κ,δ−F(q^δ_N)

+αN q^δ₀−q^δ_N

. (5.3) For the analysis of the derivative and its adjoint, we restrict ourselves to the case m = 1 since the general case can be reduced to this special case by the obvious formulas F⁰[q]ξ = (F₁⁰[q]ξ, . . . , F_m⁰ [q]ξ)^T and F⁰[q]^*h = Pm

k=1F_k⁰[q]^*h. The following theorem is a rewriting, for the electric field only, of the theorem established in [16] by Kress.

Theorem 5.1 (characterization ofF⁰[q]). The mapping F :Q →L²_t(S²)with s >2 is Fréchet differentiable at allq∈ Qfor whichΓ_q is of classC², and the first derivative atqin the direction ξ∈H^s(Γ_ref,R³)is given by

F⁰[q]ξ=E^∞_q,ξ,

whereE^∞_q,ξ is the far-field pattern of the solution E^s_q,ξ to the Maxwell equation (1.1a)inΩ^c_q that satisfies the Silver-Müller radiation condition and the following Dirichlet boundary condition

nq×E^s_q,ξ=− ξ◦q⁻¹·nq

uq×nq− 1

κ²curlΓ_q

(ξ◦q⁻¹·nq) divΓ_quq

. (5.4) on Γq where uq = nq ×curl E^s_q +E^inc

and E^s_q is the solution of the scattering problem (1.1a)-(1.1c) satisfying the Dirichlet boundary condition

n_q× E^s_q+E^inc

= 0 onΓ_q.

To define the adjoint ofF⁰[q] :H^s(Γ_ref;R³)→L²_t(S²), we interpret the naturally complex Hilbert spaceL²_t(S²)as a real Hilbert space with the real-valued inner productReh·,·i_L²

t(S²). For bounded linear operator between complex Hilbert spaces such a reinterpretation of the spaces as real Hilbert spaces does not change the adjoint.