A projection method with regularization for Cauchy problem of the time-harmonic Maxwell equations

Applied Numerical Mathematics 129 (2018) 71–82

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Applied Numerical Mathematics www.elsevier.com/locate/apnum

A projection method with regularization for Cauchy problem of the time-harmonic Maxwell equations ✩ Yunyun Ma a,∗,1 , Fuming Ma b a

Guangdong Province Key Lab of Computational Science, School of Mathematics and Computational Science, Sun Yat-sen University, Guangzhou 510275, PR China b School of Mathematics, Jilin University, Changchun 130012, PR China

a r t i c l e

i n f o

Article history: Received 21 March 2017 Received in revised form 7 December 2017 Accepted 27 February 2018 Available online 6 March 2018 Keywords: Maxwell equations Cauchy problem Projection method Regularization

a b s t r a c t We develop a projection method with regularization for reconstructing the radiation electromagnetic field in the exterior of a bounded domain from the knowledge of Cauchy data. The method is divided into two parts. We first solve the complete tangential component of the electrical field on the boundary of that domain from Cauchy data. The radiation electromagnetic field is then recovered from the complete tangential component of the electrical field. For the first part, we transform the Cauchy problem into a compact operator equation by means of the electric-to-magnetic Calderón operator and propose a projection method with regularization to solve that compact operator equation. Meanwhile, we analyze the asymptotic behavior of the singular values of the corresponding compact operator. For the second part, we expend the radiation electromagnetic field to the vector spherical harmonics. Numerical examples are finally presented to demonstrate the computational efficiency of the proposed method. © 2018 IMACS. Published by Elsevier B.V. All rights reserved.

1. Introduction We consider in this paper Cauchy problem of the time-harmonic Maxwell Equations. Maxwell equations consist of two pairs of coupled partial differential equations relating six fields, two of which model sources of electromagnetism. This system can be reduced to its time-harmonic form by assuming the propagation of the electromagnetic wave at a single frequency. It arises naturally in many physical applications, such as non-destructive testing of objects by microwave interrogation [20], microwave medical imaging [11,12], design of efficient radiators [3,27], source localization [4] and mine detection [5]. For more details on physical background of electromagnetic wave, we refer the reader to [16,23]. Several prominent methods have been developed for the computational electromagnetics in the past century, such as integral equation methods [8–10,13] and the finite element methods [18,19,22]. Among these methods, the surface tangential components of the electromagnetic field are considered as the input data for computing the electromagnetic field around an object. More recently, some study in the reconstruction of the surface tangential components of the electromagnetic field from near-field measurements has been considered in the works [2,28]. The solution of these methods is not unique unless the input data is provided over a surface enclosing all the sources of the electromagnetic field. As a result, the efficiency of

✩

* 1

This research is supported by the Natural Science Foundation of China under grant No. 11371172 and 11771180. Corresponding author. E-mail addresses: [email protected] (Y. Ma), [email protected] (F. Ma). Current address: College of Computer, Dongguan University of Technology, Dongguan 523000, PR China.

https://doi.org/10.1016/j.apnum.2018.02.010 0168-9274/© 2018 IMACS. Published by Elsevier B.V. All rights reserved.

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Y. Ma, F. Ma / Applied Numerical Mathematics 129 (2018) 71–82

these methods weakens when the input data is only available over a portion of a closed surface. However, in many engineering applications the measurements over a closed surface are often infeasible or impossible, in particular for the far-field measurements. We therefore must consider the problem of the analytic continuation of the solution of Maxwell equations in a domain from its surface tangential components on a part of the boundary of this domain, i.e. a Cauchy problem of Maxwell equations. The Cauchy problem for the time-harmonic Maxwell equations is ill-posed. As that for Helmholtz equation, the problem has unique solution but is unstable (see [1,6,15]). The presence of noise in the measurements will be amplified in the solution and in most cases the solution will be useless. For that reason, there is a considerable interest in establishing reliable and fast numerical algorithms for the Cauchy problem of Maxwell equations. A recent method on approximating the solution to this problem may be found in [25,26] and the reference therein. We consider in this paper the reconstruction of the radiation solution in the exterior of a bounded domain of the time-harmonic Maxwell equations from the knowledge of Cauchy data, which is the surface tangential components of the electromagnetic field on a part of the boundary of the aforementioned domain. Our main idea to deal with this Cauchy problem is divided into two parts. We first recover the complete date of the surface tangential component of the electrical field from Cauchy data. We shall reduce the Cauchy problem to a compact operator equation using the electric-to-magnetic Calderón operator and solve the corresponding compact operator equation by a projection method with regularization strategy. We then find the radiation solution from the Dirichlet boundary condition (the tangential components of the electrical field on the boundary). This paper is organized in six sections. In Section 2, we introduce the Cauchy problem of Maxwell equations and formulate this problem into a compact operator equation. We discuss in Section 3 the properties of the corresponding compact operator and analyze the asymptotic behavior of the singular values of that operator. A projection method with regularization is proposed in Section 4 for solving the corresponding compact operator equation. In Section 5, we present three numerical examples to confirm the efficiency of the proposed method. The final section summarizes the results of this paper and describes our future works. 2. Formulation of the problems Cauchy problem of time-harmonic Maxwell equations is presented in this section. By introducing the classical electricto-magnetic Calderón operator, we reduce the problem to an operator equation, that is determining the complete surface tangential component of the electrical field from Cauchy data. We begin with describing the Cauchy problem of Maxwell equations. Let B R be a ball with its center at the origin and of radius R > 0. We assume that all the sources of the electromagnetic field are contained in B R . We consider the timeharmonic electromagnetic wave with frequency ω > 0 propagation in a homogeneous as well as an isotropic medium with space independent electric permittivity ε > 0, magnetic permeability μ > 0 and electric conductivity σ ≥ 0. Let the wave √ number k be defined by k2 = (ε + iσ /ω)μω2 , where i := −1 is the imaginary unit. The complex valued space dependent parts E = [ E 1 , E 2 , E 3 ] of the electric field and H = [ H 1 , H 2 , H 3 ] of the magnetic field satisfy Maxwell equations

⎧ ⎪ ⎨curlE − ik H = 0, curlH + ik E = 0, ⎪ ⎩ divE = 0, and divH = 0,

in R3 \ B R ,

(2.1)

with one of the Silver–Müller radiation conditions

lim r ( H × xˆ − E ) = 0, or lim r ( E × xˆ − H ) = 0,

r →∞

r →∞

(2.2)

where x := [x1 , x2 , x3 ] ∈ R3 , r := |x| and xˆ := x/r. In this paper, we shall simply call E = [ E 1 , E 2 , E 3 ] and H = [ H 1 , H 2 , H 3 ] the electric field and magnetic field, respectively. The Cauchy data is the tangential components of the electrical field and magnetic field on   ∂ B R given by

ν × E = f , ν × H = g , on ,

(2.3)

where ν is the unit outward normal of the boundary ∂ B R , the vector functions f = [ f 1 , f 2 , f 3 ] and g = [ g 1 , g 2 , g 3 ] belong to the space of the surface tangential vector field. We shall consider the reconstruction of the electromagnetic field in R3 \ B R from Cauchy data, that is to find a radiation solution E and H of the system (2.1)–(2.2) from the Cauchy condition (2.3). We assume that the parameters ε , μ and σ are constants and the wave number k is a positive constant in the following parts of this paper. We next present some notations and functional spaces to formulate an operator equation. For n ∈ N := {1, 2, . . .}, let Zn := {1, 2, . . . , n} and Znc := N \ Zn . The dot product of two vectors a = [a1 , a2 , a3 ] ∈ C3 and b = [b1 , b2 , b3 ] ∈ C3 is  defined by a · b := j ∈Z3 a j b j . For a continuously differentiable function ϕ defined on the unit sphere S, we use Gradϕ to

Y. Ma, F. Ma / Applied Numerical Mathematics 129 (2018) 71–82

73

denote its surface gradient, the definition of which can be found in [10,22]. The vector spherical harmonics of order n ∈ N are defined by

1 ˆ ) := √ ˆ ) = xˆ × U m ˆ ), for xˆ ∈ S and |m| ∈ Zn , Um GradY nm (xˆ ) and V m n (x n (x n (x n(n + 1) where Y nm are spherical harmonics. Let H s (∂ B R ) for s ≥ 0, denote the Sobolev space defined on ∂ B R . A function u =   3 m m s s |m|∈Zn {an,m U n + bn,m V n } ∈ [H (∂ B R )] is said to belong to the space H t (∂ B R ) of the surface tangential vector field n∈N m of order s ∈ N0 := N ∪ {0}, if the coefficients an,m := ∂ B u · U n dS and bn,m := ∂ B u · V m n dS for n ∈ N, |m| ∈ Zn satisfy



n∈N



|m|∈Zn (n(n + 1))



s

R

R

(|an,m |2 + |bn,m |2 ) < +∞. We note that H t0 (∂ B R ) coincides with L t2 (∂ B R ) defined by



3

L t2 (∂ B R ) := u ∈ L 2 (∂ B R )

 : ν · u = 0 a.e. on ∂ B R .

By using Div to indicate the surface divergence [10,22], we define the space H s (Div; ∂ B R ) with order s ∈ N of tangential field possessing a surface divergence by



H s (Div; ∂ B R ) := u ∈ H ts (∂ B R ) :

 {(n(n + 1))s+1 |an,m |2 + (n(n + 1))s |bn,m |2 } < +∞ ,



n∈N |m|∈Zn

equipped with the inner product

u , v s,Div :=

  (n(n + 1))s+1 an,m cn,m + (n(n + 1))s bn,m dn,m

n∈N |m|∈Zn







m for u = n∈N |m|∈Zn an,m U m and v = n + bn,m V n   ∂ B R , we define the space





3

L t2 () := u ∈ L 2 ()



n∈N



|m|∈Zn {cn,m U n

m

1/ 2

+ dn,m V m n }, and the norm · s,Div := ·, ·s,Div . For

 : ν · u = 0 a.e. on  



3

equipped with the inner product u , v  :=  u · vdS for u = [u 1 , u 2 , u 3 ] , v = [ v 1 , v 2 , v 3 ] ∈ L 2 () , and the norm  2 2  · := ·, ·1/2 . For F = [ f 1 ; f 2 ] and G = [ g 1 ; g 2 ] ∈ L t2 () we define the inner product in the space L t2 () as F , G 2 := 1/ 2

f 1 , g 1  + f 2 , g 2  and the norm as · 2 := ·, ·2 . For more details on the definitions of these Sobolev spaces, we refer

the reader to [22]. We now formulate an operator equation to determine the complete tangential component of the electrical field from Cauchy data, where the tangential component of the electrical field is denoted by

λ := ν × E on ∂ B R .

(2.4)

To this end, we introduce two operators. The first operator is the classical electric-to-magnetic Calderón operator defined as

Ge λ := ν × H on ∂ B R ,

(2.5)

which maps the tangential component of the electrical field onto that of the magnetic field on ∂ B R . It is well known that the exterior Maxwell problem (2.1), (2.2) and (2.4) has at most one solution [10]. The operator Ge is therefore well defined. For 



3





2

u ∈ L 2 (∂ B R ) , let u  (x) := u (x) = [u 1 (x), u 2 (x), u 3 (x)] with x ∈ . We define the operator K : H 2 (Div; ∂ B R ) → L t2 ()

by



      Kλ := λ , (Ge λ) for λ ∈ H 2 (Div; ∂ B R ), 



(2.6)

which maps the tangential component of the electrical field defined on ∂ B R onto Cauchy data defined on . We then formulate an operator equation for the Cauchy problem as follows. Question 2.1. Given f , g ∈ L t2 (), find a vector function λ ∈ H 2 (Div; ∂ B R ) satisfying

Kλ = ( f , g ).

(2.7)

To close this section, we show the main idea to solve the problem (2.1)–(2.3). Since the vector spherical harmonics U m n 2 and V m n for n ∈ N, |m| ∈ Zn form a complete orthonormal system in the space L t (S) [10,22], we assume that any function λ ∈ H 2 (Div; ∂ B R ) can be expanded as

λ=



m an,m U m n + bn,m V n

n∈N |m|∈Zn



(2.8)

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Y. Ma, F. Ma / Applied Numerical Mathematics 129 (2018) 71–82

for an,m , bn,m ∈ C. We have an explicit representation for the operator Ge [22] given by

Geλ =

 −ikR

σn

n∈N |m|∈Zn

bn,m U m n +

σn ikR



an,m V m n ,

(2.9)

with







σn := kR hn(1) (kR ) hn(1) (kR ) + 1,

(2.10)

(1)

(1)

where hn denotes the spherical Hankel function of the first kind with order n. From [7], we conclude that hn (kR ) = 0 (1) for any n ∈ N and R > 0. It is therefore reasonable that hn (kR ) for n ∈ N are in the denominator. We find the radiation solution E and H to the problem (2.1)–(2.3) as follows. We first determine the coefficients an,m and bn,m for n ∈ N, |m| ∈ Zn of λ from the equation (2.7). Notice that the operator K is represented by (2.8) and (2.9). We then recover the radiation solution E and H to the exterior Maxwell problem (2.1), (2.2) and (2.4) with the expansion (see [10,22])

E=



⎧ ⎪ ⎨a

1

√

m n,m M n (1 ) ⎪ ⎩ hn (kR )

+

ikRbn,m N m n



 (1 )

⎫ ⎪ ⎬ (2.11)

, ⎪ (kR ) ⎭ ⎫ ⎧ ⎪ ⎪ ⎬ ⎨ m m 1 an,m N n ikRbn,m M n − H= , √  

(1 ) n(n + 1) ⎪ ⎭ ⎩ hn (kR ) hn(1) (kR ) + kR hn(1) (kR ) ⎪ n∈N |m|∈Zn n∈N |m|∈Zn

n(n + 1)



(1 )

hn (kR ) + kR hn

(2.12)



(1)

m m where for n ∈ N, |m| ∈ Zn , M m x) and N m n (x) = curl xhn (k|x|) Y n (ˆ n (x) = curlM n (x)/(ik) for x ∈ R \ B R are the vector wave

functions. In the following sections, we turn our attention to analyzing the compactness of the operator K and propose a projection method with regularization to solve the operator equation (2.7). 3. The compactness of the operator K



2

In this section we shall show that K : H 2 (Div; ∂ B R ) → L t2 () defined as in (2.6) is a compact operator and estimate the asymptotic behavior of the singular values of K. We recall that k > 0 is the wave number and R > 0 is the radius of the ball B R . We first present two properties of the operator Ge described in [22] as follows. Lemma 3.1. If k and R are two positive constants, then there exist two positive constants c 1 and c 2 such that for all n ∈ N, σn defined as in (2.10) satisfy c 1n ≤ |σn | ≤ c 2 n. Lemma 3.2. If k and R are two positive constants, then Ge : H s (Div; ∂ B R ) → H s (Div; ∂ B R ) with s ∈ N is a bounded operator. The proof of Lemmas 3.1–3.2 can be found in [22]. We omit the proof of these lemmas here. We next show the compactness of K.



2

Theorem 3.3. If k and R are two positive constants, then K : H 2 (Div; ∂ B R ) → L t2 ()

is a compact operator.

Proof. The result of this theorem follows from Lemma 3.2 with s = 2 and the compactness of the imbedding operator

3



mapping the space H 2 (∂ B R )

3



2

onto L 2 (∂ B R ) .

We then establish two technical lemmas to study the asymptotic behavior of the singular values of the operator K. To this end, we define that m,m









m,m





m m m A n,n := U m n , U n , B n,n := V n , V n , m,m

m C n,n := U m n , V n

m,m





m and D n,n := V m n , U n     for n, n ∈ N, |m| ∈ Zn and m  ∈ Zn . By using K∗ we denote the adjoint operator of K. For n, n ∈ N, |m| ∈ Zn and m  ∈ Zn

let

Y. Ma, F. Ma / Applied Numerical Mathematics 129 (2018) 71–82

m ,m

m ,m

T 1,n ,n := A n ,n + m ,m

m ,m

T 3,n ,n := C n ,n +

σn σn

75

σn m ,m C , σn n ,n

(3.1)

σn m ,m m ,m k2 R 2 m ,m m ,m D n ,n , T 4,n ,n := B n ,n + A . σn σn σn n ,n

(3.2)

k2 R 2

m ,m

m ,m

m ,m

B n ,n , T 2,n ,n := D n ,n +

For λ ∈ H 2 (Div; ∂ B R ) expanded as in (2.8), we define that



 λ=

⎧ ⎨ 

Um n

m ,m an ,m T 1,n ,n

(n(n + 1))3 ⎩ n ∈N |m |∈Zn ⎧ ⎨  Vm

+

m ,m bn ,m T 2,n ,n

+

m ,m bn ,m T 4,n ,n

⎫ ⎬ ⎭

n∈N |m|∈Zn

m ,m an ,m T 3,n ,n

n

+

n∈N |m|∈Zn

(n(n + 1))2 ⎩

n ∈N |m |∈Zn



⎫ ⎬ ⎭

(3.3)

. 



m,m m,m m,m m,m Lemma 3.4. The upper bound of A n,n , B n,n , C n,n and D n,n for n, n ∈ N, |m| ∈ Zn and m  ∈ Zn is meas(), where meas() denotes the measure of the domain . m,m

m,m

m,m



m,m



Proof. Note that A n,n , B n,n , C n,n and D n,n for n, n ∈ N, |m| ∈ Zn and m  ∈ Zn have a similar structure. We shall only   m,m m,m m,m m,m estimate the bound of A n,n for n, n ∈ N, |m| ∈ Zn and m  ∈ Zn . The analogous assertions for B n,n , C n,n and D n,n can   be proved in the same way. For n, n ∈ N, |m| ∈ Zn and m  ∈ Zn , by using the Cauchy–Schwarz inequality we obtain that

⎞1/2 ⎛ ⎞1/2 ⎛ ⎞1/2 ⎛ ⎞1/2 ⎛          2  m 2

 m,m  ⎝  m 2 ⎠ ⎝  m 2 ⎠   U n dS ≤ ⎝ Y n  dS ⎠ ⎝ Y nm  dS ⎠ ≤ meas().  An,n  ≤ U n  dS 







2

This yields the desired result. The proof is finished.



Lemma 3.5. If λ ∈ H 2 (Div; ∂ B R ) can be expanded as in (2.8), then  λ, h

 2,Div

= K∗ Kλ, h2,Div for all h ∈ H 2 (Div; ∂ B R ), where  λ is

defined in (3.3). Proof. The proof of this lemma is done by the definitions of the adjoint operator K∗ and the inner product in the space H 2 (Div; ∂ B R ). For h ∈ H 2 (Div; ∂ B R ) with the expansion



m cn,m U m n + dn,m V n

h=



n∈N |m|∈Zn

and the definition of  λ in (3.3), we have that

!

Kλ, Kh2 = λ, h + Ge λ, Ge h

an,m U m n

=

+ bn,m V m n

n∈N |m|∈Zn

! +

 −ikRbn,m

n∈N |m|∈Zn

=



cn,m

+

n∈N |m|∈Zn

n∈N |m|∈Zn

Um n

dn,m

σn an,m

+

ikR

⎧ ⎨  ⎩

n ∈N |m |∈Zn

n∈N |m|∈Zn



σn

n ∈N |m |∈Zn

Vm n

  " −ikRdn,m m σn cn,m m , Un + Vn n∈N |m|∈Zn

m ,m

m ,m

an ,m T 1,n ,n + bn ,m T 2,n ,n

⎧ ⎨  ⎩

"

  m , cn,m U m n + dn,m V n

m ,m an ,m T 3,n ,n

+

σn

ikR

⎫ ⎬

m ,m bn ,m T 4,n ,n

⎭ ⎫ ⎬ ⎭

  λ, h = 

2,Div

.

This together with Kλ, Kh2 = K∗ Kλ, h2,Div yields the conclusion. The proof is complete.

2

76

Y. Ma, F. Ma / Applied Numerical Mathematics 129 (2018) 71–82

We now show the main theorem in this section. For this purpose, we define an operator A N for N ∈ N by

A N λ :=



Um n

⎧ ⎨ 

m ,m an ,m T 1,n ,n

(n(n + 1))3 ⎩ n ∈N |m |∈Zn ⎧ ⎨  Vm

+ bn ,m T 2m,n, m,n

n∈Z N |m|∈Zn

+

n

(n(n

n∈Z N |m|∈Zn

+ 1))2

⎩

n ∈N |m |∈Zn



m ,m

m ,m

an ,m T 3,n ,n + bn ,m T 4,n ,n

⎫ ⎬ ⎭ ⎫ ⎬ ⎭

,



m ,m where T j ,n ,n for j ∈ Z4 , n, n ∈ N, |m| ∈ Zn and m  ∈ Zn are given by (3.1) and (3.2), and λ ∈ H 2 (Div; ∂ B R ) is expanded as in (2.8).

Theorem 3.6. If k and R are two positive constants, then there exists a positive constant c such that for all ρ ∈ (0, 1) and integers n > 3, μn (K∗ K) ≤ cn−ρ /3 meas(), where μn (K∗ K) denotes the n-th eigenvalue of K∗ K listed in decreasing order. Proof. We prove this result by using the min–max principle on self-adjoint operator (see [24]) to the operator K∗ K. According to the definition of A N for N ∈ N, we have that

 m A N ( H 2 (Div; ∂ B R )) ⊂ span U m n , V n : |m| ∈ Zn , n ∈ Z N ,   and dim A N ( H 2 (Div; ∂ B R )) ≤ 2N ( N + 1). It means that A N has at most 2N ( N + 1) non-vanishing eigenvalues. Applying the min–max principle on self-adjoint operator, we get the relation

#

#

μ2N (N +1)+1 (K∗ K) ≤ μ2N (N +1)+1 (AN ) + μ1 (K∗ K − AN ) ≤ 0 + #K∗ K − AN #2,Div . We then estimate K∗ K − A N 2,Div as follows. From the definition of A N and Cauchy inequality, we obtain that for λ ∈ H 2 (Div; ∂ B R ) expanded as (2.8),

# ∗ # #(K K − A N )λ#2

2,Div

 2    

m ,m m ,m   = an ,m T 1,n ,n + bn ,m T 2,n ,n  3  ( n ( n + 1 )) n ∈N |m |∈Z  n∈ZcN |m|∈Zn n  2    

1 m ,m m ,m  



+ a T + b T n ,m 3,n ,n n ,m 4,n ,n   2 (n(n + 1))   n ∈N |m |∈Zn n∈ZcN |m|∈Zn  2  2  m ,m   m ,m   T 2,n ,n   T 1,n ,n  λ 22,Div ≤ + (n(n + 1))3 (n (n + 1))3 (n (n + 1))2 c



1

n∈Z N |m|∈Zn

+

n∈ZcN |m|∈Zn

n ∈N |m |∈Zn



λ 22,Div (n(n + 1))2

n ∈N |m |∈Zn

 2  m ,m   T 3,n ,n  (n (n + 1))3

+

 2  m ,m   T 4,n ,n  (n (n + 1))2

. 



m ,m Applying the definitions (3.1) and (3.2) of T j ,n ,n for j ∈ Z4 , n, n ∈ N, |m| ∈ Zn and m  ∈ Zn with Lemmas 3.1–3.2, we  

   m ,m  have that there exists a positive constant c such that for all n, n ∈ N, |m| ∈ Zn and m  ∈ Zn ,  T 1,n ,n  ≤ cn nmeas(),        m ,m   m ,m   m ,m   T 2,n ,n  ≤ c (1 + n/n )meas(),  T 3,n ,n  ≤ c (1 + n /n)meas() and  T 4,n ,n  ≤ cmeas(). We obtain that there exists a positive constant c such that for all ρ ∈ (0, 1),

# # ∗ #(K K − A N )λ#2

2,Div

≤ c λ 22,Div (meas())2



n∈ZcN |m|∈Zn



n2 n 2

(n (1 + n ))3

+

1

(n (1 + n ))2

+

1

(n(n

+ 1))3

 n ∈N |m |∈Zn

n2

(n (1 + n ))2n 2



+

1

(n(n + 1))2

Y. Ma, F. Ma / Applied Numerical Mathematics 129 (2018) 71–82

  n ∈N |m |∈Zn

1

(n (1 + n ))3

≤ c λ 22,Div (meas())2



+

n 2

(n (1 + n ))3n2

+

77



1

(n (1 + n ))2

n−3 ≤ c λ 22,Div (meas())2 N −2ρ



n −3 +2 ρ

n∈N

n∈ZcN

≤ c λ 22,Div (meas())2 N −2ρ . We conclude that there exists a positive constant c such that for all

ρ ∈ (0, 1) and integers n > 3,

μN 3 (K∗ K) ≤ μ2N (N +1)+1 (K∗ K) ≤ cN −ρ meas(), 2

which finishes the proof.

We finally show two remarks as follows. Remark 3.7. The Cauchy problem of Maxwell equations (2.1)–(2.3) is ill-posed since K is a compact operator. Remark 3.8. The measure of  has influence on the ill-posedness of the Cauchy problem of Maxwell equations. 4. The projection method with regularization We develop in this section a numerical method for solving the Cauchy problem (2.1)–(2.3). The method has two steps. An approximation to the solution λ of the compact operator equation (2.7) is first obtained via a suitable projection method in conjunction with Tikhonov regularization. We then reconstruct the radiation electromagnetic field from the approximation of λ by (2.11) and (2.12). We now describe a projection method with regularization for solving the compact operator equation (2.7). According to



2

Tikhonov regularization [10,17], for y := [ f ; g ] ∈ L t2 () , we determine v α ∈ H 2 (Div; ∂ B R ) satisfying the normal equation

α v α + K∗ K v α = K∗ y , where

(4.1)

α > 0 is the regularization parameter, and v α minimizes the Tikhonov functional J α ( v ) := K v − y 22 + α v 22,Div , for v ∈ H 2 (Div; ∂ B R ).

For the numerical treatment to (4.1), let T N ⊂ H 2 (Div; ∂ B R ) with N ∈ N denote a finite dimension space given by

T N :=

⎧ ⎨ ⎩

{an,m U m n

+ bn,m V m n}

n∈Z N |m|∈Zn

⎫ ⎬

: an,m , bn,m ∈ C . ⎭

We define a projection operator P N : H 2 (Div; ∂ B R ) → T N by

PN w =



m {an,m U m n + bn,m V n },

n∈Z N |m|∈Zn





m 2 for w = n∈N |m|∈Zn {an,m U m n + bn,m V n } ∈ H (Div; ∂ B R ). Instead of using the infinite series expansion, we find an approximation to the solution of the normal equation (4.1) formed by

v αN =



N ,α m {anN,,mα U m n + bn,m V n } ∈ T N

(4.2)

n∈Z N |m|∈Zn

satisfying the projection equation

P N (α I + K∗ K) v αN = P N K∗ y .

(4.3)

That is the projection method with regularization for solving the compact operator equation (2.7). The existence and uniqueness of v αN ∈ T N follow easily since T N is finite dimensional and (α I + K∗ K) is one to one. The convergence of the projection method is the same as the projection method with regularization for the Cauchy problem of Helmholtz equation, which can be found in [21]. The parameter α > 0 is selected by using Hansen’s L-curve criterion [14].

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Y. Ma, F. Ma / Applied Numerical Mathematics 129 (2018) 71–82

m We nextshow the finite system of linear algebraic equations for solving (4.3). By choosing the basis U m n , V n : |m| ∈ Zn α and n ∈ Z N in T N , the solution v N of the projection equation (4.3) is characterized by

$

%

$

%

$

%

$

%

$

%

$

%

α v αN , U ls 2,Div + K v αN , KU ls 2 = y , KU ls 2 , α v αN , V ls 2,Div + K v αN , K V ls 2 = y , K V ls 2 , for |s| ∈ Zl and l ∈ Z N . Applying (4.2) with the definitions (3.1) and (3.2) in the equations above, we obtain that the coefficients of v nα satisfy



α (l(l + 1))3alN,s,α +

m,s

m,s



$

anN,,mα T 1,n,l + bnN,,mα T 2,n,l = y , K U ls

n∈Z N |m|∈Zn



α (l(l + 1))2 blN,s,α +

m,s

m,s



$

anN,,mα T 3,n,l + bnN,,mα T 4,n,l = y , K V ls

n∈Z N |m|∈Zn

for |s| ∈ Zl and l ∈ Z N , where

K U ls

=





 U ls 

,

&

 

σl

 ikR

 V ls 



, and

K V ls

=



 V ls 



,

−ikR

σl

%

2

% 2

, ,

 '

 U ls 

.



Using the coefficients of v nα , we finally recover the radiation electromagnetic field with the form

E=



H=

1

√

n,m

n

(1 ) n(n + 1) ⎪ ⎩ hn (kR )

n∈Z N |m|∈Zn



⎧ ⎪ ⎨ a N ,α M m

√

n∈Z N |m|∈Zn

+

⎧ ⎪ ⎨ a N ,α N m

1

n(n + 1)

n,m n (1 ) ⎪ ⎩ hn (kR )



(1 )

 (1 )

hn (kR ) + kR hn N ,α

−

⎫ ⎪ ⎬

N ,α ikRbn,m N m n

ikRbn,m M m n



(1 )

 (1 )

hn (kR ) + kR hn

⎪ (kR ) ⎭ ⎫ ⎪ ⎬

,

⎪ (kR ) ⎭

.

5. Numerical examples We present in this section numerical examples to demonstrate the performance of the algorithm developed in Section 4. The numerical results are all obtained by using Matlab in a modest desktop. We begin with introducing the parameters used in the following examples. The Cauchy data is generated by a magnetic dipole located at a point y = [ y 1 , y 2 , y 3 ] with A = [1, 1, 1] , i.e.

(

)

E (x) = curlx A (x, y ) and H (x) =

1 ik

(

)

curlx curlx A (x, y ) ,

(5.1)

where is the fundamental solution of Helmholtz equation defined by

(x, y ) =

eik|x− y | 4π | x − y |

, x = y .

For r > 0, let



r := r (sin θ cos ϕ , sin θ sin ϕ , cos θ) : θ ∈ I θ , ϕ ∈ I ϕ . The surface   ∂# B R is denoted by  :=  R , where I θ  [0, π ] and I ϕ  [0, 2π ]. We use δ to denote the relative noise # # # # # level defined by # f δ − f # / f ≤ δ and # g δ − g # / g ≤ δ , where f δ and g δ are the surface tangential components of the electrical and magnetic field with random noise on . In the following examples, we recover the electromagnetic field on a sphere ∂ B r with r ≥ R. The approximations to the electrical field and magnetic field are denoted by N um E [ N , α ] = [ Num E1 , Num E2 , Num E3 ] and N um H [ N , α ] = [ Num H1 , Num H2 , Num H3 ] , respectively. The results on the curve

γ := r (sin θ cos ϕ , sin θ sin ϕ , cos θ) : θ = 1.03, ϕ ∈ I ϕ



are shown in the figures below. Example 5.1. This example is to verify the performance of the algorithm (4.3) without regularization for Cauchy data with random noise. We set k = 3, y = [1, 0, 0] , I θ = [0, π ] and I ϕ = [0, 3π /2]. We list in Table 5.1 the relative errors of N um E [ N , α ] and N um H [ N , α ] on ∂ B R with α = 0, for different choices of R and N, where

RE( E ) := N um E [ N , α ] − E / E and RE( H ) := N um H [ N , α ] − H / H .

Y. Ma, F. Ma / Applied Numerical Mathematics 129 (2018) 71–82

79

Table 5.1 Relative errors of N um E [ N , α ] and N um H [ N , α ] of Example 5.1. N

5 6 7 8 9

R = 10

R = 20

R = 30

R = 40

RE(E)

RE(H)

RE(E)

RE(H)

RE(E)

RE(H)

RE(E)

RE(H)

0.2115 0.1314 0.1273 0.1230 0.1126

0.2691 0.2076 0.2075 0.2057 0.2046

0.1884 0.0786 0.0717 0.0634 0.0617

0.2033 0.1129 0.1095 0.1040 0.1023

0.1794 0.0641 0.0558 0.0444 0.0416

0.1882 0.0840 0.0786 0.0706 0.0685

0.1775 0.0582 0.0490 0.0353 0.0371

0.1826 0.0711 0.0643 0.0542 0.0517

Fig. 5.1. Electrical field Num E1 .

Fig. 5.2. Magnetic field Num H1 .

Fig. 5.3. Electrical field Num E2 .

Fig. 5.4. Magnetic field Num H2 .

We then choose the relative noise level δ = 0.005 and δ = 0.05. For R = 30 and N = 6, we plot the real part of the cartesian components of N um E [ N , α ], N um H [ N , α ], E and H on γc in Figs. 5.1–5.6, where

γc := {30(sin θ cos ϕ , sin θ sin ϕ , cos θ) : θ = 1.03, ϕ ∈ [0, 2π ]} . We find in Table 5.1 that for a fixed R the approximation accuracy of the projection method increases as N grows and for a fixed N it slightly increase as R grows. We confirm that the method is convergence for a fixed R, and R (R > 10) has little influence on the accuracy of the method for a fixed N. From Table 5.1 and Figs. 5.1–5.6, we conclude that the approximations to the tangential components of the electromagnetic field is accurate on the remainder of the surface ∂ B R . We note that the algorithm (4.3) with α = 0 is the least squares method for solving (2.7). We obtain that the accuracy of the least squares method is acceptable for N = 6 and meas( R ) = 3meas(∂ B R )/4. Example 5.2. In this example, we confirm the stability of the algorithm (4.3) with regularization for Cauchy data. Let δ = 0. We set k = 5, y = [1, 2, 0] , R = 30 and I θ = I ϕ = [0, π ], where meas( R ) = meas(∂ B R )/2. For different N = 6 and N = 10,

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Y. Ma, F. Ma / Applied Numerical Mathematics 129 (2018) 71–82

Fig. 5.5. Electrical field Num E3 .

Fig. 5.7. Electrical field Num E1 .

Fig. 5.9. Electrical field Num E2 .

Fig. 5.6. Magnetic field Num H3 .

Fig. 5.8. Magnetic field Num H1 .

Fig. 5.10. Magnetic field Num H2 .

we choose α = 0 and α = 0.0002, we present the real part of the cartesian components of N um E [ N , α ], N um H [ N , α ], E and H on γ with r = 32 in Figs. 5.7–5.12. We compare the results for N = 6 and N = 10 with α = 0 in Figs. 5.7–5.12, we see that the accuracy of the least squares method is acceptable for N = 6, but the errors of the least squares method for N = 10 are large. The approximation for N = 10 is useless. The reason is that the eigenvalues of the matrix corresponding to the least squares method decrease more rapidly as N tends to infinity for the ill-posed problems. The least squares method is unstable. But we see in Figs. 5.7–5.12

Y. Ma, F. Ma / Applied Numerical Mathematics 129 (2018) 71–82

Fig. 5.11. Electrical field Num E3 .

Fig. 5.13. Electrical field Num E1 .

Fig. 5.15. Electrical field Num E2 .

that the accuracy of algorithm (4.3) for N = 10 with regularization is stable.

81

Fig. 5.12. Magnetic field Num H3 .

Fig. 5.14. Magnetic field Num H1 .

Fig. 5.16. Magnetic field Num H2 .

α = 0.0002 is acceptable. We conclude that the projection method with

Example 5.3. The purpose of this example is to verify the efficiency of the algorithm (4.3) for Cauchy data with random noise. We set δ = 0.005. Let k = 1, y = [1, 2, 3] , R = 30 and I θ = I ϕ = [0, π ]. For N = 8 with α = 0.00002, the absolute values of the cartesian components of N um E [ N , α ], N um H [ N , α ], E and H on γ with r = 32 are reported in Figs. 5.13–5.18. We conclude that the projection method with regularization is stable and effective.

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Y. Ma, F. Ma / Applied Numerical Mathematics 129 (2018) 71–82

Fig. 5.17. Electrical field Num E3 .

Fig. 5.18. Magnetic field Num H3 .

6. Conclusions In this paper, we propose a projection method with regularization to solve Cauchy problem of Maxwell equations in the exterior of a ball. Utilizing the electric-to-magnetic Calderón operator, the Cauchy problem is reduced to a compact operator equation, which implies the ill-posedness of that problem. An approximation to the complete tangential component of the electrical field on the boundary is obtained by using the projection method with regularization to solve the corresponding compact operator equation. The radiation electromagnetic field is then reconstructed from the complete tangential component of the electrical field from the vector spherical harmonics by (2.11) and (2.12). In the future work, we will consider the Cauchy problem of Maxwell equations on more general geometries, and that problem with large wavenumber. References [1] T. Abboud, J.C. Nédélec, Electromagnetic waves in an inhomogeneous medium, J. Math. Anal. Appl. 164 (1992) 40–58. [2] H.F. Alqadah, N.P. Valdivia, E.G. Williams, A super-resolving near-field electromagnetic holographic method, IEEE Trans. Antennas Propag. 62 (2014) 3679–3692. [3] J.S. Asvestas, R.E. Kleinman, Electromagnetic scattering by indented screens, IEEE Trans. Antennas Propag. 42 (1994) 22–30. [4] G. Bao, H. Ammari, J. Fleming, An inverse source problem for Maxwell’s equations in magnetoencephalography, SIAM J. Appl. Math. 62 (2002) 1369–1382. [5] C.E. Baum, Detection and Identification of Visually Obscured Targets, Taylor & Francis, 1999. [6] L. Bers, F. John, M. Schechter, Partial Differential Equations, Wiley, New York, 1964. [7] Z. Chen, X. Liu, An adaptive perfectly matched layer technique for time-harmonic scattering problem, SIAM J. Numer. Anal. 43 (2006) 645–671. [8] W.C. Chew, J.M. Jin, E. Michielssen, J.M. Song, Fast and Efficient Algorithms in Computational Electromagnetics Waves: Numerical Analysis, Artech House, Boston–London, 2001. [9] D. Colton, R. Kress, Integral Equation Methods in Scattering Theory, John Wiley & Sons, New York, 1983. [10] D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, second ed., Springer-Verlag, New York, 1998. [11] D. Colton, P. Monk, The detection and monitoring of leukemia using electromagnetic waves: mathematical theory, Inverse Probl. 10 (1994) 1235–1251. [12] D. Colton, P. Monk, The detection and monitoring of leukemia using electromagnetic waves: numerical analysis, Inverse Probl. 11 (1995) 329–342. [13] M. Ganesh, S.C. Hawkins, A spectrally accurate algorithm for electromagnetic scattering in three dimensions, Numer. Algorithms 43 (2006) 25–60. [14] P.C. Hansen, The L-curve and its use in the numerical treatment of inverse problems, in: P. Johnston (Ed.), Computational Inverse Problems in Electrocardiology, in: Advances in Computational Bioengineering, WIT Press, Southampton, 2000, pp. 119–142. [15] V. Isakov, Inverse Problems for Partial Differential Equations, Springer-Verlag, New York, 1998. [16] D.S. Jones, Acoustic and Electromagnetic Waves, Clarendon Press, Oxford, 1989. [17] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problem, Springer-Verlag, New York, 1996. [18] A. Kirsch, P. Monk, A finite element/spectral method for approximating the time-harmonic Maxwell system in R3 , SIAM J. Appl. Math. 55 (1995) 1324–1344. [19] A. Kirsch, P. Monk, A finite element method for approximating electromagnetic scattering from a conducting object, Numer. Math. 92 (2002) 501–534. [20] K.J. Langenberg, Applied inverse problems for acoustic, electromagnetic and elastic wave scattering, in: P. Sabatier (Ed.), Basic Methods for Tomography and Inverse Problems, Adam Hilger, Bristol, 1987. [21] Y. Ma, F. Ma, H. Dong, A projection method with regularization for the Cauchy problem of Helmholtz equation, J. Comput. Math. 30 (2012) 157–176. [22] P. Monk, Finite Element Methods for Maxwell’s Equations, Oxford University Press, New York, 2003. [23] C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves, Springer-Verlag, New York, 1969. [24] M. Reed, B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, New York, 1978. [25] È.N. Sattorov, Regularization of the solution of the Cauchy problem of the system of Maxwell equations in an unbounded domain, Math. Notes 86 (2009) 422–431. [26] È.N. Sattorov, D.A. Mardanov, The Cauchy problem for the system of Maxwell equations, Sib. Math. J. 44 (2003) 671–679. [27] T.B.A. Senior, K. Sarabandi, J.R. Natzke, Scattering by a narrow gap, IEEE Trans. Antennas Propag. 38 (1990) 1102–1110. [28] N.P. Valdivia, E.G. Williams, The reconstruction of surface tangential components of electromagnetic field from near-field measurements, Inverse Probl. 23 (2007) 785–798.