Variational source conditions and stability estimates for inverse electromagnetic medium scattering problems
aa r X i v : . [ m a t h . NA ] D ec VARIATIONAL SOURCE CONDITIONS AND STABILITYESTIMATES FOR INVERSE ELECTROMAGNETIC MEDIUMSCATTERING PROBLEMS
FREDERIC WEIDLING AND THORSTEN HOHAGE
Institute for Numerical and Applied Mathematics, University of Goettingen,Lotzestr. 16-18, 37083 Goettingen, Germany
Abstract.
This paper is concerned with the inverse problem to recover thescalar, complex-valued refractive index of a medium from measurements ofscattered time-harmonic electromagnetic waves at a fixed frequency. The mainresults are two variational source conditions for near and far field data, whichimply logarithmic rates of convergence of regularization methods, in particularTikhonov regularization, as the noise level tends to 0. Moreover, these vari-ational source conditions imply conditional stability estimates which improveand complement known stability estimates in the literature. Introduction
In this paper we study the behavior of time-harmonic electromagnetic wavesin an inhomogeneous isotropic medium. The propagation of the time-harmonicelectric field E is described by the equation(1) − ∇ × ∇ × E + κ nE = 0where the wave number κ and the refractive index n are given by(2) κ := ω √ ǫ µ , n ( x ) := 1 ǫ (cid:18) ǫ ( x ) + i σ ( x ) ω (cid:19) . Here ǫ and σ denote the electric permittivity and the conductivity of the medium,respectively, and ω denotes the frequency of the time-dependent electric field E ( x, t ) = ℜ ( ǫ − / E ( x ) exp( − iωt )). Supposing that the inhomogeneity is compactly sup-ported, we can assume w.l.o.g. that supp( n − ⊂ B ( R ) := { x ∈ R : | x | < R } with R = π by possibly rescaling x , and sufficiently smooth a unique solution to(1) exist under a suitable radiation condition defining the behavior of the field atinfinity as will be detailed in Section 2.Corresponding inverse problems are to probe the medium with incident fields E i fulfilling (1) for n ≡ E s = E − E i with the aim of recovering the refractive index, see Section 2 for more details. Theinverse problems will be formulated as operator equations of the form F ( n ) = y with a mapping F : dom( F ) ⊂ X → Y between Hilbert spaces X , Y . Let n † denote the exact solution and y δ ∈ Y perturbed data satisfying k F ( n † ) − y δ k Y ≤ δ . To find a stable approximation to n † from such data one needs to employ E-mail address : [email protected] . regularization techniques. A prominent example is Tikhonov regularization wherethe approximation n δα is defined by(3) n δα ∈ arg min n ∈ dom( F ) ∩X (cid:20) α (cid:13)(cid:13) F ( n ) − y δ (cid:13)(cid:13) Y + 12 k n k X (cid:21) . Major issues of regularization theory are the convergence of (cid:13)(cid:13) n † − n δα (cid:13)(cid:13) X → δ → α = α ( δ, y δ ). To obtain such rates one needs additional assumptions on the true so-lution (see [8, Prop 3.11]). Starting with [17] these assumptions are now frequentlyformulated as variational source conditions (4) ∀ n ∈ dom( F ) β (cid:13)(cid:13) n † − n (cid:13)(cid:13) X ≤ k n k X − (cid:13)(cid:13) n † (cid:13)(cid:13) X + ψ (cid:16)(cid:13)(cid:13) F ( n ) − F ( n † ) (cid:13)(cid:13) Y (cid:17) , where β ∈ (0 ,
1] and ψ : [0 , ∞ ) → [0 , ∞ ) is an index function, that is ψ is a con-tinuous, monotonically increasing function satisfying ψ (0) = 0. We only refer to[10, 28] for overviews.Variational source conditions have many advantages over classical spectral sourcecondition which we discussed in detail in [18]. Let us mention only two of them here:Firstly, if the index function ψ is concave, it can be shown by a simple argument[12] (see also [30, Thm. 3.3]) that they lead to the convergence rate(5) β (cid:13)(cid:13) n † − n δα (cid:13)(cid:13) X ≤ ψ ( δ ) , for Tikhonov regularization with an optimal choice of the regularization parameter α . Secondly, if a variational source condition with the some function ψ holds forall n † ∈ K , where K is a (usually compact) subset of X , they imply the stabilityestimate(6) ∀ n , n ∈ K β k n − n k X ≤ ψ (cid:16) k F ( n ) − F ( n ) k Y (cid:17) , while it is not clear whether every stability estimate can be sharpened to a varia-tional source condition.There are however only few verifications of variational source conditions so far.They can under certain assumptions be derived from spectral source conditions, butthen they do not yield any new information. For linear operators F and l q penaltieswith respect to certain bases in the range of F ∗ , characterizations of variationalsource condition have been derived in [2, 3, 11]. Reformulations of (4) have beenproven for phase retrieval and an option pricing problem in [17]. Moreover weshowed that the acoustic inverse medium scattering problem fulfills such a conditionrecently [18].The purpose of this paper is to demonstrate how similar techniques can be ap-plied to the electromagnetic inverse medium scattering problem to derive a varia-tional source condition and hence also a stability estimate via (6). We are consid-ering refractive indices fulfilling(7) n ∈ D := (cid:8) n ∈ C ,α ( R ) : supp(1 − n ) ⊂ B ( π ) , ℜ ( n ) ≥ b, ℑ ( n ) ≥ (cid:9) , b > C ( π ) = ( − π, π ) .As a first kind of measurement operator consider the near field operator F n mapping an refractive index to the corresponding Green’s tensor w n ( x, y ) measuredon R S × R S with the unit sphere S := { x ∈ R : | x | = 1 } and R > π . Theorem 1.1.
Let s > m > / , s = 2 m + 3 / and R > π . Suppose that the truerefractive index n † satisfies n † ∈ D ∩ H s with k n † k H s ≤ C s . Then a variational SC AND STABILITY OF INVERSE EM SCATTERING 3 source condition (4) holds true for the operator F n with dom( F n ) := D ∩ H m ( C ( π )) , Y = ( L ( R S × R S )) × , β = 1 / , and ψ given by ψ n ( t ) := A (cid:0) ln(3 + t − ) (cid:1) − ν , ν := min (cid:26) s − mm + 5 / , s − ms − m + 1 (cid:27) , where the constant A > depends only on m, s, C s , κ, b and R . As in [18] the choice of β ∈ (0 ,
1) is actually arbitrary, but A depends on thechoice of β . For a discussion of the exceptional case s = 2 m + 3 / Corollary 1.2.
Let s > m > / , s = 2 m + 3 / and R > π . Assume that therefractive indices n † , n and n satisfy n † , n , n ∈ D ∩ H s with k n † k H s , k n j k H s ≤ C s for some C s ≥ , then • Convergence rate: the error bound (cid:13)(cid:13) n δα − n † (cid:13)(cid:13) H m ≤ √ A (cid:0) ln(3 + δ − ) (cid:1) − ν in terms of the noise level δ with respect to the operator F n holds true forthe regularization scheme (3) if α = (2 A ∂ ln(3+ t − ) − ν ∂t (cid:12)(cid:12) t =4 δ ) − . • Stability estimate: one obtains the estimate k n − n k H m ≤ √ A (cid:16) ln (cid:16) k F n ( n ) − F n ( n ) k − L ( R S × R S )) × (cid:17)(cid:17) − ν . As R ≫ R → ∞ in a sense explained in Section 2. Then incident fields are plane wavesinstead of point sources, and it is assumed that only the so-called far field patternsof the scattered fields can be measured. In this case we will show essentially thesame results, but with slightly smaller exponents: Theorem 1.3.
Under the assumptions of Theorem 1.1 and for all < θ < avariational source condition (4) holds true for the operator F f with dom( F f ) := D ∩ H m ( C ( π )) , Y = ( L ( S × S )) × , β = 1 / , and ψ given by ψ n ( t ) := B (cid:0) ln(3 + t − ) (cid:1) − νθ where ν is given as in Theorem 1.1 the constant B > depends only on m, s, C s , κ, b ,and R . Again one obtains as a corollary results via (5) and (6).
Corollary 1.4.
Suppose the assumptions of Corollary 1.2 hold true, then • Convergence rate: the error bound (cid:13)(cid:13) n δα − n † (cid:13)(cid:13) H m ≤ √ B (cid:0) ln(3 + δ − ) (cid:1) − νθ in terms of the noise level δ with respect to the operator F f holds true forthe regularization scheme (3) if α = (2 B ∂ ln(3+ t − ) − ν ∂t (cid:12)(cid:12) t =4 δ ) − . • Stability estimate: one obtains the estimate k n − n k H m ≤ √ B (cid:16) ln (cid:16) k F f ( n ) − F f ( n ) k − L ( S × S )) × (cid:17)(cid:17) − νθ . Compared to our results for the acoustic medium scattering problem [18] weobtain slower rates of convergence. The reason for this will be discussed afterLemma 3.2. However, it can be shown by entropy techniques [22, 23] that thelogarithmic rates of convergence are optimal up to the value of the exponent ν .While we prove the first variational source condition for this problem, stabilityestimate exists for similar problems. Table 1 gives a short overview over the differentresults known to the authors. For the problem with measured far field data as VSC AND STABILITY OF INVERSE EM SCATTERING described by (10) a stability estimate was proven in [15] under the assumption that n j ∈ C ,γ for a γ >
0. The stability estimate is in the L ∞ -norm and the obtainedexponent of the logarithmic factor is ν = 1 /
15, but a very strong norm has to beapplied in the image space.Furthermore there are stability estimates using Cauchy data [5, 21]. The Cauchydata in this setup is defined as the set of the tangential parts of the electric and mag-netic field on a sphere R S for all possible solutions of the time-harmonic Maxwellsystem in the ball B ( R ) and the distance between different Cauchy sets is measuredby a Hausdorff like distance. The stability estimate in [5] is in the H norm andholds also for non constant magnetic permeability µ but the obtained exponent isunknown and bounded by 1 / κ and for κ → ∞ the logarithmic stability turns intoa H¨older stability. The obtained exponent is linearly increasing with the assumedsmoothness up to ν = 1. new H¨ahner [15] Caro [5]
Lai et al [21] data near/far field far field Cauchy Cauchy validity global local anywhere global local around 0 stability of σ, ǫ σ, ǫ σ, ǫ, µ σ norm H m L ∞ H H − s exponent < /
15 unknown, < ≤ special strong norm inimage space H¨older-logarithmic Table 1.
Comparison between different stability estimates.In the following section we will give a precise formulation of the consideredproblems. In Section 3 we discuss how to use complex geometrical optics solutionsto estimate differences of Fourier coefficients of refractive indices. The proof ofTheorem 1.1 is given in Section 4. The connection of near and far field datais discussed in Section 5 which will provide an easy way to prove Theorem 1.3in Section 6. The construction of the complex geometric optics solution will bedetailed in Appendix A.2.
The direct and inverse problems
In this paper we study electromagnetic waves ( E , H )( x, t ) in an inhomogeneousisotropic medium. The propagation of the electric field E and the magnetic field H is described by Maxwell’s equations ∇ × E + µ ∂ H ∂t = 0 , ∇ × H − ǫ ∂ E ∂t = J. We will assume that the magnetic permeability µ = µ is constant, which is a goodapproximation for most materials, while the electric permittivity ǫ = ǫ ( x ) > J ( x, t ) = σ ( x ) E ( x, t ) where the conductivity satisfies σ ( x ) ≥
0. We will consider the case of acompactly supported inhomogeneity, that is ǫ ( x ) = ǫ and σ ( x ) = 0 for all x largerthen a certain radius R and assume without loss of generality that R = π .Supposing a time-harmonic dependence of the electromagnetic wave of the form E ( x, t ) = ℜ (cid:18) √ ǫ E ( x )e − i ωt (cid:19) , H ( x, t ) = ℜ (cid:18) √ µ H ( x )e − i ωt (cid:19) SC AND STABILITY OF INVERSE EM SCATTERING 5 with frequency ω > E and H must satisfy the equations(8a) ∇ × E − i κH = 0 ∇ × H + i κnE = 0where the wave number κ and the refractive index n are given as in (2). In thefollowing we will assume that the total field ( E, H ) is given as the superposition ofan incident field ( E i , H i ) that solves the Maxwell equations (8a) for n ≡ E s , H s ) = ( E, H ) − ( E i , H i ) . To guarantee that the scattered field is unique and models outgoing waves weassume that the Silver-M¨uller radiation condition(8c) lim | x |→∞ ( H s ( x ) × x − | x | E s ( x )) = 0is satisfied. One can take the first equation in (8a) as an equation defining H andthus obtain (1).For a known incident wave E i and a known refractive index n in the admissibleset D defined in (7) the system (8) has a unique solution E ∈ C ( R ). E is theunique solution of the electromagnetic Lippmann-Schwinger equation E ( x ) = E i ( x ) − κ Z B ( π ) Φ κ ( x − y ) (1 − n ( y )) E ( y ) d y + ∇ Z B ( π ) Φ κ ( x − y ) 1 n ( y ) ∇ n ( y ) · E ( y ) d y, and H = ( iκ ) − ∇ × E where Φ κ ( x ) = e i κ | x | / (4 π | x | ) is the fundamental solution tothe Helmholtz equation, see [7]. The regularity assumptions on n can be relaxed ifthose on the regularity of the total field are relaxed.For our regularization scheme (3) we will choose X = H m ( C ( π )) for m > / k f k H m = X γ ∈ Z (1 + | γ | ) m (cid:12)(cid:12)(cid:12) b f ( γ ) (cid:12)(cid:12)(cid:12) where b f ( γ ) are the Fourier coefficients of f in C ( π ). Our choice of m implies thatthere exists some constant L m such that(9) L m k f k H m ( C ( π )) ≥ k f k C ( C ( π )) := max | α |≤ sup x ∈ C ( π ) | ∂ α f ( x ) | , Our requirements on the smoothness of n and the additional lower bound b onthe electric permittivity will be needed in our analysis of the problem and arerequirements for the construction of our main tool the complex geometric opticssolutions, which we will present in detail in the appendix.The two inverse problems we consider differ in the type of incident fields andthe type of measurements of the solution to (8). For the first inverse problem weconsider incident fields generated by a time harmonic electromagnetic dipole withmoment a ∈ R located at y ∈ R : E i y,a ( x ) = − κ ∇ × ∇ × a Φ( x, y ) , H i y,a = ∇ × a Φ( x, y )Here Φ( x, y ) = Φ κ ( x − y ). We assume that we can place electric dipoles withmoments given by the Cartesian unit vectors d , d and d at all points y on asphere R S , R > π and measure the corresponding electric fields on the samesphere. Then due to linearity of (8a) we know the total electric field E y,a ( x ) onthe sphere R S for any incident electric dipole field E i y,a with moment a ∈ R and VSC AND STABILITY OF INVERSE EM SCATTERING source point y ∈ R S , and the measurements can be arranged in a matrix w n ( x, y )called the near field scattering data such that E y,a ( x ) = w n ( x, y ) a. Obviously, w n is the Green’s tensor of the problem. We will further denote by w s n ( x, y ) = w n ( x, y ) − w ( x, y ) the contribution of the scattered field to the matrix,where w ( x, y ) is the near field corresponding to the homogeneous medium case n ≡
1. The problem to reconstruct the refractive index from such measurementscan be posed as an operator equation F n ( n ) = w n where the near field operator isdefined as F n : D ∩ H m ( C ( π )) → ( L ( R S × R S )) × , n w n . To describe the second inverse problem, recall that the Silver-M¨uller radiationcondition (8c) implies for the scattered field the asymptotic behavior E s d,p ( x ) = e i κr r (cid:0) E ∞ d,p (ˆ x ) + o (1) (cid:1) , r → ∞ where r = | x | and ˆ x = x/r . If we choose as the incident wave a plane wave of theform E i d,p ( x ) = d × ( p × d )e i κd · x , H i d,p ( x ) = (i κ ) − ∇ × E i d,p ( x ) , x ∈ R traveling in direction d ∈ S and having polarization p ∈ C we can associatewith each refractive index a far field E ∞ that maps (ˆ x, d, p ) ∈ S × S × C to E ∞ d,p (ˆ x ) ∈ C . As this mapping is linear in p , there exists a matrix valued function e ∞ n : S × S → C × such that E ∞ d,p (ˆ x ) = e ∞ n (ˆ x, d ) p. Thus we can define the far field operator(10) F f : D ∩ H m ( C ( π )) → ( L ( S × S )) × , n e ∞ n . Estimation of low order Fourier coefficients
A main tool for our proof are complex geometrical optics (CGO) solutions, whichare solutions to (8a) having exponential growth in one direction. They are impor-tant tools in the analysis of scattering problems, for example to prove uniqueness[4, 24, 27] or stability estimates [1, 16, 19, 26]. Recently we employed them in [18] toprove a variational source condition for acoustic inverse medium scattering problem.Concerning their usage for the electromagnetic case we refer to the constructionsfor a uniqueness proof in [6] and for a stability proof in [15], which we will mostlyfollow. We point out that other constructions exist which can also cover the case ofnon constant magnetic permeability, see [5, 25]. For further references about thesesolutions introduced in [9] we recommend the review [29].Since the time harmonic Maxwell equations are not of Helmholtz type, the con-struction of CGO solutions is more complicated. Mainly one has to find a trans-formation of (8a) such that the resulting equations are of Helmholtz type. Thecorresponding transformation will be discussed in Appendix A, and the followingresult will be derived:
Theorem 3.1.
Let π < R , n ∈ D ∩ H m for m > / with k n k H m ≤ C m , κ > and ζ, η ∈ C such that ζ · ζ = κ , ζ · η = 0 and |ℑ ( ζ ) | ≥ t , where (11) t := 60 Rπ (1 + κ ) b − ( L m C m ) with the embedding constant L m given in (9) . Then there exists a solution to (8a) in the ball B (2 R ) of the form (12) E ( x, ζ, η ) = e i ζ · x [ η + f ( x, ζ, η ) ζ + V ( x, ζ, η )] , for x ∈ B (3 R/ and H = (i κ ) − ∇ × E such that k f ( · , ζ, η ) k L ( B (3 R/ + k V ( · , ζ, η ) k L ( B (3 R/ ≤ M | η ||ℑ ( ζ ) | . with a constant M depending on C m , κ, b and R . This result is an analog of [15, Theorem 4], but we have made the dependenceof the constant t on the parameters explicit. Lemma 3.2.
Let
R > π , m > / and assume that n and n are two refractiveindices satisfying n j ∈ D ∩ H m with k n j k H m ≤ C m for some C m ≥ . Let E j , H j ∈ C ( B (3 R/ ∩ L ( B (3 R/ be solutions to (8a) in B (3 R/ for n = n j for j = 1 , .Then the estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B ( π ) ( n − n ) E E d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ M k w − w k ( L ( R S × R S )) × k E k L ( B (3 R/ k E k L ( B (3 R/ holds true, where w j is the near field scattering data for n = n j for j = 1 , and M depends on κ, R, b and C m .Proof. The proof follows along the lines of [15, Lemma 5] but other norms areused there therefore we sketch the proof here. For j = 1 , E j , H j ) toradiating solutions ( V j , W j ) fulfilling (8c), with ( V j , W j ) | B (3 R/ = ( E j , H j ) and ν × E − j = ν × V + j on 3 R/ S , where ν denotes the outer normal vector on 3 R/ S and E − j and V + j denote the inner and outer Dirichlet trace on 3 R/ S of E j and V j respectively. By [15, eq. (12)] the equality2 κ Z B ( π ) ( n − n ) E E d x = Z R S (cid:2) ν × (cid:0) W − − W +1 (cid:1) × ν (cid:3) [ N − N ] (cid:2) ν × (cid:0) W − − W +2 (cid:1)(cid:3) d s holds true with the operator( N j a )( x ) = 2 ν × Z R S w j ( x, y ) a ( y ) d y for a sufficiently smooth and j = 1 , N j ). Again by [15, eq. (11)] we know there exists a constant c such that (cid:13)(cid:13) ν × (cid:0) W − j − W + j (cid:1)(cid:13)(cid:13) L ( R S ) ≤ c k E j k L ( B (3 R/ . Estimating the L -operator norm of N − N by the L -norm of its kernel gives theassertion. (cid:3) This result can now be used to derive bounds on Fourier coefficients of n − n .Comparing Lemma 3.3 below with [15, Lemma 6], note that our choice of t ≥ t yields a better control of the difference of the Fourier coefficients in ̺ since in ourcase the first summand is independent of ̺ and in the second summand we havea factor of ̺ instead of ̺ . Moreover, the additional factor k n − n k H m will beuseful. That our estimate depends on ̺ is the main difference between this resultand the comparable result [18, Lemma 3.3] in the acoustic case. This will alsobe the major difference in the proof of the variational source condition for the VSC AND STABILITY OF INVERSE EM SCATTERING electromagnetic compared to the acoustic case leading to the smaller exponent inthe index function.
Lemma 3.3.
Let
R > π , m > / and n and n be two refractive indices suchthat n j ∈ D ∩ H m with k n j k H m ≤ C m for some C m ≥ with corresponding nearfield data w j for j = 1 , . Let t ≥ t with t as in (11) and ≤ ̺ ≤ √ κ + t .Then there exists a constant M depending only on R, κ, b and C m such that | ( c n − c n ) ( γ ) | ≤ M (cid:16) k w − w k ( L ( R S × R S )) × e Rt + k n − n k H m ̺t (cid:17) holds true for all γ ∈ Z with | γ | ≤ ̺ .Proof. Let γ ∈ Z with | γ | ≤ ̺ be given. Choose a and a in R such that { γ/ | γ | , a , a } form an orthonormal system of R . Define for t ≥ t the followingvectors in C :(13) ζ := − γ + i ta + s κ + t − | γ | a , η := 1 | γ | γ − i | γ | t a ,ζ := − γ − i ta − s κ + t − | γ | a , η := 1 | γ | γ + i | γ | t a . These vectors satisfy the relations ζ · ζ = ζ · ζ = κ , ζ · η = ζ · η = 0 , ζ + ζ = − γ. Hence by Theorem 3.1 for j = 1 , E j , H j to (8a) in B (2 R )with n replaced by n j , which have the form (12) in B (3 R/
2) with ζ, η replaced by ζ j , η j . The product of these solutions has the form:(14) E · E = e − i γ · x [ η + f ζ + V ] · [ η + f ζ + V ]= e − i γ · x " | γ | t − | γ | ( f + f ) + η · V + η · V + f f (cid:18) | γ | − κ (cid:19) + f ζ · V + f ζ · V + V · V , By the definition of the Fourier transform together with (14) this implies that(15) (2 π ) / | ( c n − c n ) ( γ ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12) Z B ( π ) ( n − n ) E E d x (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) Z B ( π ) ( n − n ) | γ | t − | γ | ( f + f ) + η · V + η · V ! d x (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) Z B ( π ) ( n − n ) (cid:18) f f (cid:16) | γ | − κ (cid:17) + f ζ · V + f ζ · V + V · V (cid:19) d x (cid:12)(cid:12)(cid:12)(cid:12) To estimate further we need the moduli of the vectors in (13) which are given by | ζ | = | ζ | = p t + κ , | η | = | η | = q | γ | / | t | . Since t ≥ t ≥ max { , κ } we can estimate that ̺ t ≤ t + κ ) t ≤ | η j | ≤ j = 1 , SC AND STABILITY OF INVERSE EM SCATTERING 9
Applying Lemma 3.2 to the first integral of the right hand side of (15) oneobtains(17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B ( π ) ( n − n ) E E d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ M k w − w k ( L ( R S × R S )) × e Rt (cid:16) k k L ( B (3 R/ + 3 M (cid:17) , since t ≥ max { , κ } implies that k E j k L ( B (3 R/ ≤ Rt/ (cid:18) k k L ( B (3 R/ + M | ζ | + 1 t (cid:19) ≤ Rt/ (cid:16) k k L ( B (3 R/ + 3 M (cid:17) , since √ t + κ t ≤ j = 1 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B ( π ) ( n − n ) | γ | t − | γ | ( f + f ) + η · V + η · V ! d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k n − n k L ( C ( π )) (cid:18) √ ̺t k k L ( B ( π )) + ̺ (cid:16) k f k L ( B ( R )) + k f k L ( B ( R )) (cid:17) + 3 (cid:16) k V k L ( B ( R )) + k V k L ( B ( R )) (cid:17) (cid:19) ≤ k n − n k H m ̺t (cid:18) √ k k L ( B ( π )) + 24 M (cid:19) by the estimate on f j and V j in Theorem 3.1 and by (16). Similarly for the thirdintegral(19) (cid:12)(cid:12)(cid:12)(cid:12) Z B ( π ) ( n − n ) (cid:18) f f (cid:16) | γ | − κ (cid:17) + f ζ · V + f ζ · V + V · V (cid:19) d x (cid:12)(cid:12)(cid:12)(cid:12) ≤ k n − n k L ∞ ( C ( π )) (cid:18)(cid:16) | ̺ | κ (cid:17) k f k L ( B ( R )) k f k L ( B ( R )) + p t + κ (cid:16) k f k L ( B ( R )) k V k L ( B ( R )) + k V k L ( B ( R )) k f k L ( B ( R )) (cid:17) + k V k L ( B ( R )) k V k L ( B ( R )) (cid:19) ≤ L m k n − n k H m (cid:18) M ̺t + 36 M t + 9 M t (cid:19) ≤ L m M k n − n k H m ̺t , where we used in addition that t ≥ κ implies1 t (cid:16) ̺ κ (cid:17) ≤ . (20)Combining (15) to (19) one sees, that there exists a constant M such that | ( c n − c n ) ( γ ) | ≤ M (cid:16) k w − w k ( L ( R S × R S )) × e Rt + k n − n k H m ̺t (cid:17) , where M depends only on R, κ, b and C m . (cid:3) Proof of Theorem 1.1
Before we proof the main theorem of this paper we will rewrite the variationalsource condition (4) in an equivalent way(21) ℜ (cid:10) n † , n † − n (cid:11) X ≤ − β (cid:13)(cid:13) n − n † (cid:13)(cid:13) X + ψ (cid:16)(cid:13)(cid:13) F ( n ) − F ( n † ) (cid:13)(cid:13) Y (cid:17) , since this form is more convenient for the proof. Proof of Theorem 1.1.
Let n ∈ D ∩ H m be given. To be able to apply the previouslemmata we need that k n k H m ≤ C m for some C m >
0. Thus consider the case k n − n † k H m > C s . Then by applying Cauchy-Schwarz we obtain ℜ (cid:10) n † , n † − n (cid:11) H m ≤ (cid:13)(cid:13) n † (cid:13)(cid:13) H m (cid:13)(cid:13) n † − n (cid:13)(cid:13) H m ≤ (cid:13)(cid:13) n − n † (cid:13)(cid:13) H m which implies (21).Hence, it remains to treat the case k n k H m ≤ C s . Let us introduce the notation h γ i := 1 + | γ | for γ ∈ Z . We choose t ≥ t with C m = 5 C s in Lemma 3.3 and1 ≤ ̺ ≤ √ κ + t and set δ = k w n − w n † k ( L ( R S × R S )) × to obtain(22) ℜ (cid:10) n † , P ̺ ( n † − n ) (cid:11) H m = ℜ X γ ∈ Z ∩ B ( ̺ ) h γ i m c n † ( γ ) (cid:16)c n † − b n (cid:17) ( γ ) ≤ M (cid:16) δ e Rt + (cid:13)(cid:13) n † − n (cid:13)(cid:13) H m ̺t (cid:17) X γ ∈ Z ∩ B ( ̺ ) h γ i m (cid:12)(cid:12)(cid:12)c n † ( γ ) (cid:12)(cid:12)(cid:12) , where P ̺ ( f ) is the projection onto the Fourier coefficients of f with | γ | ≤ ̺ . ByLemma 4.3 of [18] there exists a constant M depending on m and s such that theremaining sum on the right hand side of (22) can be bounded by(23) X γ ∈ Z ∩ B ( ̺ ) (cid:12)(cid:12)(cid:12) h γ i m c n † ( γ ) (cid:12)(cid:12)(cid:12) ≤ s X γ ∈ Z h γ i s | c n † ( γ ) | s X γ ∈ Z ∩ B ( ̺ ) h γ i m − s ≤ M C s ̺ τ with τ = max { m + 3 / − s, } for s = 2 m + 3 / n † similar to [16, 18, 20]. Therefore, we first apply Cauchy-Schwarz and then Young’sinequality to obtain(24) ℜ (cid:10) n † , ( I − P ̺ )( n † − n ) (cid:11) H m = ℜ X γ ∈ Z \ B ( ̺ ) h γ i m c n † ( γ ) (cid:16)c n † − b n (cid:17) ( γ ) ≤ vuut X γ ∈ Z \ B ( ̺ ) h γ i m (cid:12)(cid:12)(cid:12)c n † ( γ ) (cid:12)(cid:12)(cid:12) vuut X γ ∈ Z \ B ( ̺ ) h γ i m (cid:12)(cid:12)(cid:12)(cid:16)c n † − b n (cid:17) ( γ ) (cid:12)(cid:12)(cid:12) ≤ X γ ∈ Z \ B ( ̺ ) h γ i m (cid:12)(cid:12)(cid:12)c n † ( γ ) (cid:12)(cid:12)(cid:12) + 18 (cid:13)(cid:13) n † − n (cid:13)(cid:13) H m . The smoothness assumption on n † now implies that(25) X γ ∈ Z \ B ( ̺ ) h γ i m (cid:12)(cid:12)(cid:12)c n † ( γ ) (cid:12)(cid:12)(cid:12) ≤ ̺ ) s − m X γ ∈ Z \ B ( ̺ ) h γ i s (cid:12)(cid:12)(cid:12)c n † ( γ ) (cid:12)(cid:12)(cid:12) ≤ ̺ m − s ) C s . SC AND STABILITY OF INVERSE EM SCATTERING 11
Combining (22) to (25) we arrive at(26) ℜ (cid:10) n † , n † − n (cid:11) H m ≤ (cid:13)(cid:13) n † − n (cid:13)(cid:13) H m + 2 ̺ m − s ) C s + M M C s ̺ τ (cid:16) δ e Rt + (cid:13)(cid:13) n † − n (cid:13)(cid:13) H m ̺t (cid:17) ≤ (cid:18)
18 + 18 ̺ τ + s − m ) εt (cid:19) (cid:13)(cid:13) n † − n (cid:13)(cid:13) H m + 2 C s (1 + εM M ) ̺ m − s ) + M M C s ̺ τ δ e Rt for all ε > ̺, t and ε in dependence of δ > ε = (9 R ) , Rt = ln(3 + δ − ) = ̺ τ + s − m . Then the constraint ̺ ≥ τ + s − m ≥ m +5 / >
6, there existsa t ′ such that 2 t ≥ (9 Rt ) / (1+ τ + s − m ) for all t ≥ t ′ . Hence by strengthening theconstraint on t to t ≥ t ′ := max { t , t ′ } the upper bound on ̺ is satisfied since then2 √ κ + t > t ≥ (9 Rt ) / (1+ τ + s − m ) = ̺ (see (20) for the first inequality). However t ≥ t ′ is only satisfied for δ ≤ δ max , where δ max := (exp(9 Rt ′ ) − − / (or δ max = ∞ if exp(9 Rt ′ ) ≤ δ > δ max has to be handled subsequently. For δ ≤ δ max plugging our choice (27) into (26) yields ℜ (cid:10) n † , n † − n (cid:11) H m ≤ (cid:13)(cid:13) n † − n (cid:13)(cid:13) H m + 2 C s (1 + (9 R ) M M ) (cid:0) ln(3 + δ − ) (cid:1) − ν + M M C s (cid:0) ln(3 + δ − ) (cid:1) λ δ (3 + δ − ) / with ν := min (cid:26) s − mm + 5 / , s − ms − m + 1 (cid:27) , λ = max (cid:26) , m + 3 / − sm + 5 / (cid:27) . Since the term in the second line tends faster to 0 then the last one in the first for δ ց A such that for δ ≤ δ max ℜ (cid:10) n † , n † − n (cid:11) H m ≤ (cid:13)(cid:13) n † − n (cid:13)(cid:13) H m + ˜ A (cid:0) ln(3 + δ − ) (cid:1) − ν showing (21) for δ ≤ δ max .If on the other hand δ > δ max applying again Cauchy-Schwarz and Young’sinequality yields(28) ℜ (cid:10) n † , n † − n (cid:11) H m ≤ (cid:13)(cid:13) n † − n (cid:13)(cid:13) H m + C s . Therefore, Theorem 1.1 holds true with A = max { ˜ A, C s (ln(3 + δ − )) ν } . (cid:3) From near to far field data
In this section we show that the difference of the near field measurements for tworefractive indices can be bounded by a function of their far field measurements. Theidea of the proof of Theorem 1.3 is then to insert this bound into the variationalsource condition for the near field data.Since both kinds of data are measured on spheres we will express them in termsof a series representation using spherical harmonics. Let Y kl for l ∈ N and k ∈ Z , | k | ≤ l denote the spherical harmonics and set for convenience M := { ( l , k , l , k ) ∈ N × Z × N × Z : | k | ≤ l , | k | ≤ l } . Then we will define the (matrix valued) Fourier coefficients of the far field by α ( l , m , l , m ) = Z S Z S e ∞ n (ˆ x, d ) Y k l (ˆ x ) Y k l ( d ) dˆ x d d, ( l , m , l , m ) ∈ M. Denoting by h (1) l the spherical Hankel function of first kind of order l now gives usa series representation for w s n in terms of the Fourier coefficients of the far field. Ashort introduction on the functions Y kl and h (1) l can be found in [7]. Lemma 5.1.
Let
R > π and n ∈ D ∩ H m for m > / be a refractive index with e ∞ n and w n the corresponding far and near field. Denote by α ( l , m , l , m ) theFourier coefficients of e ∞ n . Then the scattered part of the near field data has therepresentation w s n ( x, y ) = − κ π X ( l ,k ,l ,k ) ∈ M i l − l α ( l , k , l , k ) h (1) l ( κR ) h (1) l ( κR ) Y k l (cid:18) x | x | (cid:19) Y k l (cid:18) y | y | (cid:19) , | x | > | y | = R. Furthermore the series converges absolutely and uniformly on compact sets.Proof.
See [15, Lemma 10]. (cid:3)
A similar result for the acoustic case has been derived by Stefanov in [26]. In [16]this results was used together with stability estimates for the inverse of compactlinear operators under spectral source conditions to estimate the difference of nearfield data by the difference of far field data. Since the series representation of thenear field for the acoustic and the electromagnetic case coincide up to the fact thatthe electromagnetic Fourier coefficients are matrix valued, one obtains the followinglemma along the lines of [16]:
Lemma 5.2.
Let
R > π , m > / , C m > and < θ < . Then there existconstants ω, ̺, δ max > such that for any two refractive indices n , n ∈ D ∩ H s with k n j k H s ≤ C s for some C s > , we have k w − w k L (2 R S × R S )) × ≤ ̺ exp − − ln k u ∞ − u ∞ k ( L ( S × S )) × ω̺ ! θ if k u ∞ − u ∞ k ( L ( S × S )) × ≤ δ max where w j and u ∞ j denote near and far fieldscattering data for n j , j = 1 , . Proof of Theorem 1.3
Having a variational source condition for near field data and a way to boundnear field data by far field data, the proof of Theorem 1.3 now proceeds similar tothe acoustic case.
Proof of Theorem 1.3.
The case k n † − n k H m > C s can be treated as in the proofof Theorem 1.1.For k n k H m ≤ C s we can apply Lemma 5.2. Setting δ := k F f ( n † ) − F f ( n ) k ( L ( S × S )) × , ϕ ( t ) := ̺ exp( − ( − ln( √ t ) + ln( ω̺ )) θ ) , it follows from Theorem 1.1 and the monotonicity of ψ n that the variational sourcecondition (4) holds true with ψ ( t ) = ψ n ( ϕ ( t )) if δ ≤ δ max . Bounding ψ n ( t ) ≤ A (ln t − ) − ν for t < ψ n ( ϕ ( t )) ≤ A (cid:18) − (cid:16) − ln( √ t ) + ln( ̺ω ) (cid:17) θ − ln ̺ (cid:19) − ν for √ t ≤ min (cid:26) δ max , (cid:27) . SC AND STABILITY OF INVERSE EM SCATTERING 13
Hence, it is easy to see that there are constants
B > e δ max ∈ (0 , min { δ max , } ]such that ψ n ( ϕ ( t )) ≤ B (ln(3 + t − )) − νθ for √ t ≤ e δ max . This shows (4) for δ ≤ e δ max .The case δ > e δ max is again treated as in the proof of Theorem 1.1, see (28). (cid:3) Appendix A. Complex geometrical optics solutions forelectromagnetic inverse scattering
In the following we sketch the construction of CGO solution for electromagneticscattering. We will mostly follow the arguments in [15] (a more detailed version bythe same author can be found in [14]), which is based on an idea in [6]. However,we will make the lower bound on |ℑ ( ζ ) | more explicit which will enable us to derivea better value of the exponent ν for fixed values of m and s , so the limit for s → ∞ is the same.In this section we will always assume that π < R ′ < R ′′ , n ∈ D ∩ H m for m > / κ > ζ, η ∈ C such that ζ · ζ = κ and ζ · η = 0.Our goal is to construct solutions to the Maxwell equations with an electric fieldof the form E ( x ) = e i ζ · x ( η + r ( x, ζ, η )) where the term r decompses into a boundedand a decaying part as | ζ | → ∞ . In order to do so one needs to constructs an(unphysical) fundamental solution Ψ ζ to the Helmholtz equation in a ball aroundthe origin. A construction using periodic spaces was developed in [13] and can alsobe found in the textbook [7]. One obtains that the corresponding volume integraloperator G ζ with kernel e − i ζ ( x − y ) Ψ ζ ( x − y ) fulfills the estimate(29) k G ζ f k L ( B ( R ′ )) ≤ R ′′ / ( π ℑ ( ζ )) k f k L ( B ( R ′ )) and hence is a contraction for |ℑ ( ζ ) | large enough.Similar to the construction of CGO solutions in the acoustic case, one can nowuse the Lippmann-Schwinger equation where one replaces the usual fundamentalsolution by the unphysical Ψ ζ and uses η e i ζ · x as an incident field to construct asolution to (8a): Lemma A.1.
Suppose |ℑ ( ζ ) | ≥ κ ( R ′′ /π ) k − n k L ∞ ( R d ) + 1 . Set E i = η e i ζ · x and H i = (i κ ) − ∇ × E i . Let E ∈ C ( B ( R )) be a solution to E ( x ) = E i ( x ) − κ Z B ( π ) Ψ ζ ( x − y ) (1 − n ( y )) E ( y ) d y + ∇ Z B ( π ) Ψ ζ ( x − y ) 1 n ( y ) ∇ n ( y ) · E ( y ) d y, x ∈ B ( R ′ ) . (30) Then E ∈ C ( B ( R ′ ))) , and E and H := ( iκ ) − ∇ × E satisfy the perturbed Maxwellequation (8a) in B ( R ′ ) .Proof. See [15, Lemma 13]. (cid:3)
To show uniqueness and the form E = e i ζ · x ( η + r ) of solutions to (30), oneneeds a further characterization of these solution in the form of a Helmholtz typeequation. In [6] a matrix-valued function Q : R → C × with the same support as n was introduced such that if ( E, H ) fulfills (8a), then the field ( ˜ E, ˜ H ) = ( n / E, H )fulfills the Helmholtz equation(∆ + κ ) (cid:18) ˜ E ˜ H (cid:19) = Q (cid:18) ˜ E ˜ H (cid:19) where (cid:18) Q (cid:18) AB (cid:19)(cid:19) ( x ) := Q ( x ) (cid:18) A ( x ) B ( x ) (cid:19) for x ∈ R . The matrix Q is defined such that Q (cid:18) AB (cid:19) = (cid:18) κ (1 − n ) A − i κn − / ∇ n × B − ( A ·∇ ) (cid:0) n ∇ n (cid:1) + (cid:0) n − / ∆ n / (cid:1) Aκ (1 − n ) B + i κn − / ∇ n × A (cid:19) or more explicitly Q = κ (1 − n )1 + i κ √ n (cid:18) −∇ n ×∇ n × (cid:19) + (cid:18) − D (cid:0) ∇ nn (cid:1) + ( n − / ∆ n / )1 (cid:19) where 1 k and 0 k denote the k × k unit and zero matrix respectively, D ( V ) theJacobian of a vector field V , and( ∇ n × ) = − ∂n∂z ∂n∂y∂n∂z − ∂n∂x − ∂n∂y ∂n∂x . Lemma A.2.
Let n ∈ D ∩ H m with m > / and k n k H m ≤ C m , and let L m begiven by (9) . Then kQ ( x ) k ≤ κ ) b − ( L m C m ) for all x ∈ B ( π ) . Proof.
Estimating summand by summand, bounding the k·k norm of non-diagonalmatrices by the Frobenius norm, and using n − / ∆ n / = n − ∆ n − n − |∇ n | we obtain kQ ( x ) k ≤ κ (1 + L m C m ) + 2 κ √ b − / ( L m C m )+ 3 (cid:0) b − ( L m C m ) + b − ( L m C m ) (cid:1) + 14 b − ( L m C m ) + 12 b − ( L m C m ) ≤ (4 √ κ ) b − ( L m C m ) ≤ κ ) b − ( L m C m ) where we have used that L m C m ≥ /b ≥ n ( x ) = 1 for | x | ≥ π . (cid:3) Lemma A.3.
Let the assumptions of Lemma A.1 be fulfilled and let E be thesolution to (30) . Define E ′ ( x ) = e − i ζ · x n / ( x )( E − E i )( x ) and H ′ ( x ) = e − i ζ · x ( H − H i )( x ) for x ∈ B ( R ′ ) . Then (31) (cid:18) E ′ H ′ (cid:19) + G ζ Q (cid:18) E ′ H ′ (cid:19) = (cid:18) F ( · , ζ, η ) F ( · , ζ, η ) (cid:19) where (cid:18) F ( · , ζ, η ) F ( · , ζ, η ) (cid:19) := − G ζ (cid:18) ( − i n − / ζ · ∇ n − ∆ n / ) η (cid:19) − G ζ Q (cid:18) n / ηκ − ζ × η (cid:19) . Proof.
See [15, Lemma 14]. (cid:3)
There is only a sketch of the main theorem of existence of CGO solutions in[15] and also the dependence of the constants t and M appearing below on theother constants is not specified. This has been done more explicitly in [14]. Forconvenience we include the proof here. Proposition A.4.
Suppose that n ∈ D ∩ H m with k n k H m ≤ C m and (32) |ℑ ( ζ ) | ≥ max ( R ′′ π max x ∈ B ( π ) kQ ( x ) k , κ R ′′ π k − n k L ∞ ( R d ) + 1 ) . SC AND STABILITY OF INVERSE EM SCATTERING 15
Then both the equations (30) and (31) have unique solutions. Moreover, (33) (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) E ′ H ′ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L ( B ( R ′ )) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) F ( · , ζ, η ) F ( · , ζ, η ) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L ( B ( R ′ )) . Proof. If ζ satisfies (32), it follows from (29) that (cid:13)(cid:13)(cid:13)(cid:13) G ζ Q (cid:18) E ′ H ′ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L ( B ( R ′ )) ≤ R ′′ sup x kQ ( x ) k π |ℑ ( ζ ) | (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) E ′ H ′ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L ( B ( R ′ )) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) E ′ H ′ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L ( B ( R ′ )) for all ( E ′ , H ′ ) ∈ L ( B ( R ′ )) . Hence k G ζ Qk ≤ /
2, and by the Neumann serieseq. (31) has a unique solution for all right hand sides. Moreover, k ( I + G ζ Q ) − k ≤ E to the homogeneous equation (30) thefunction (e − i ζ · x n / E, e − i ζ · x (i κ ) − ∇ × E ) yields a solution to the homogeneousequation (31), the solution to (30) is unique. Since the integral operator in (30) iscompact, existence of solutions to (30) now follows from Riesz-Fredholm theory. (cid:3) From Proposition A.4 it is not yet clear how large we have to choose ℑ ( ζ ) interms of our set of parameters C m , b, κ and R . We will show that the explicitcondition (11) in Theorem 3.1 implies (32). Note that our choice of R ′ and R ′′ isarbitrary, as long as R ′′ > R ′ > R , but their values influence the choice of otherparameters. Proof of Theorem 3.1.
We first note from Lemma A.2 that for R ′ = R and R ′′ =2 R the condition ℑ ( ζ ) ≥ t with t defined in (11) implies (32) since L m C m ≥k n k L ∞ ≥ | ζ | = |ℜ ( ζ ) | + |ℑ ( ζ ) | = 2 |ℑ ( ζ ) | + κ , we can bound | ζ | in terms of |ℑ ( ζ ) | ,and we obtain using the definition of ( F ( · , ζ, η ) , F ( · , ζ, η )), and (29) that thereexists a constant ˜ M depending on C m , b, κ and R ′′ such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) F ( · , ζ, η ) F ( · , ζ, η ) (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L ( B ( R ′ )) ≤ ˜ M | η | . Therefore, k ( E ′ , H ′ ) k L ( B ( R ′ )) ≤ M | η | by (33). Writing down the equation for E ′ explicitly one obtains E ′ = − G ζ (cid:16) − i n − / ( ζ · ∇ nη + ∇ n × ( ζ × η )) (cid:17) − G ζ (cid:16) κ (1 − n ) ˜ E ′ − i κn − / ∇ n × H ′ − ( ˜ E ′ · ∇ )( 1 n ∇ n )+ ( n − / ∆ n / ) ˜ E ′ − ∆ n / η (cid:17) , where ˜ E ′ = E ′ + n / η . Using the vector identity a × ( b × c ) = b ( a · c ) − c ( a · b ) inthe first line one obtains E ′ = G ζ (cid:16) i n − / ∇ n · η (cid:17) ζ + V ′ ( · , ζ, η )with V ′ denoting the second summand of the previous equation. Since E ′ =e − i ζ · x n / ( E − E i ) this shows that E has the claimed form with f ( · , ζ, η ) := n − / G ζ (cid:16) i n − / ∇ n · η (cid:17) , V ( · , ζ, η ) := n − / V ′ ( · , ζ, η ) , and the desired estimate follows from the bound (29). (cid:3) Acknowledgments
Financial support through CRC 755, project C09 is gratefully acknowledged.
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