Studies On Fast Algorithms For Some Direct And Inverse Scattering Problems

Studies on Fast Algorithms for Some Direct and Inverse Scattering ProblemsScattering and inverse scattering problems arise naturally in areas such as sonar, radar, geophysical imaging and nondestructive testing. The theory of scattering and in-verse scattering problems for time-harmonic acoustic and electromagnetic waves plays an important role in modern mathematical physics. The direct scattering problem, given the information of the incident wave and the nature of the scatterer, is to find the scattered wave and in particular its behavior at large distances from the scatterer, i.e., its "far-field" behavior. The inverse scattering problem is to find the boundary or physical parameters of the scatterer with given scattered field or its far-field. The scat-tering problems can be solved numerically by integral equation method, finite element method and infinite element method, etc. Roughly speaking, numerical methods for solving inverse obstacle scattering problems can be classified into three groups:opti-mization methods, iterative methods and sampling methods. In this dissertation, some fast and robust algorithms for scattering and inverse scattering problems are proposed. We specify the results as follows: I. Some domain decomposition methods employing the PML technique for the Helmholtz equationLet D∈R2 be a bounded domain with Lipschitz boundary (?)D. We consider the model scattering problem:find u∈Hloc1(R2\D), such that u satisfies Here g∈H-1/2((?)D) is determined by the incident wave, and v is the unit normal to (?)D, directed into the interior of D. We assume the wave number k>0 is a constant. The Sommerfeld radiation condition (3) ensures uniqueness for the obstacle scattering problem (1)-(3).Now we consider the perfectly matched layer (PML) technique for solving problem (1)-(3).Let D be contained in the interior of the circle The interested bounded domain is where 0< R<ρand σ(r) is the fictitious medium property and is usually taken as power functions: for some constantσ0>0 and integer m E N+.Letα(r)=1+iσ(r). Denote byγ(r) the complex radius defined byThe PML equation is defined by where K is a matrix which satisfies, in polar coordinatesLetГρ= (?)Bρ, then the PML solution u corresponding to (1)-(3) inΩρ=Bρ\D is defined as the solution of the following PML problemWe note that both the solution of (5) and its derivative have no jump across the interfaceГR.i. Domain decomposition and well-posedness results for subproblemsFor p=0,1, the operators are defined by where Ap: p=0,1, are injective and is the adjoint of Ap. Now we introduce the DD problem:find satisfyingThe corresponding variational problem for (6) is:Givenλ0∈H-1/2(ГR), find u0∈H1(ΩR), such that whereLemma 0.1 For anyλ0∈H-1/2(ГR), the variational problem (9) has an unique solution.Let The corresponding variational problem for (7) is:Given find u1∈H(0)1(Ωp), such that where Lemma 0.2 Suppose that u1=0 is the unique solution of the local problem (7) forλ1= 0, then for any the variational problem (10) has an unique solution.With the above two lemmas, we have the main result of this subsection:Theorem 0.1 If local problems (6) and (7) are uniquely solvable for any given then problem (6)-(8) is equivalent to problem (5).ii. The equivalent interface problemLet u denote the solution of problem (5) and set u0=u|ΩR, u1=u|Ωp. We define the Lagrange multipliers onГR by Following Theorem 0.1, u0, u1,λ0 andλ1 serve as the solution for problem (6)-(8).Let Operator is defined as follows:u= E(λ) if and only if where E0 and E1 give the solutions to boundary value problems (6) and (7), respec-tively. Following Lemma 0.1 and Lemma 0.2, operator E is well-defined.S:L2(ГR)→L2(ГR) is defined byLet I denote the identity operator, then is defined byWe can also define satisfies boundary value problem (6) with then (6)-(8) can be reformulated as a transmission problem: Theorem 0.2 is solution of(6)-(8),thenλ=(λ0,λ1)T is solution of(Ⅱ)Reciprocally,ifλ=(λ0.λ1)T is solution of(11).then u=v+E(λ)is solution of(6)-(8). Furthermore.the elimination ofλ1 from(11)gives interfaee problem forλ0: iii.Domain decomposition in matrix and vector formsThe Galerkin approximations for(9)and(10)give rise to linear systems: and where Correspondingly.the discretization of(8)leads toThe Schur complement methodLet The PML problem(5)can be written asThe bloek factorization of the maltrix of(16)leads to the Schur complement system whereOnce uГis found,the internal components uI(0) can be found by solvingThe dual Schur complement methodBy Gaussian elimination of the matrix of(13)and(14),we get Substituting the solutions uГ(0) and uГ(1) of(19)into(15).we derive the dual Schur complement systems forλГ=(λГ(0),λГ(1))TwhereOnceλГis known,we can find uI(0) and uI(1) by solving(13)and(14),respectively. Furthermore,by eliminatingλГ(1) from(20),we can derive the following system forλГ(0): where OnceλГ(0) is known, uI(0)can be obtained by solving (13).The choices of transmission operators(Ⅰ) A commonly used choice of continuous operators A0 and A0 is as followsThe discretization leads to the discrete operators A(0)=A(1)=iKMГ,where MГdenotes the mass matrix onГR. Now. F0 and do can be written as(Ⅱ) Matrix A(0) and A(1) can also be chosen as With these choices, (21) can be rewritten as where So, the simple iterative algorithm for (24) converges in at most one iteration.This dissertation also includes applications of the domain decomposition methods to open cavity scattering problems, as well as some numerical results which show the effectiveness of the proposed methods.Ⅱ. Optimization methods for acoustic inverse obstacle scatter-ing problems with multiple incident wavesi. The illuminated regions and the reconstruction methodConsider the scattering of a time-harmonic acoustic wave with incident direction d and positive wave number k by a given impenetrable obstacle D∈R2 imbedded in a homogeneous isotropic medium. Then the direct obstacle scattering problem is to find the total field u= ui+us such that u satisfies the following Helmholtz equationIt can be shown that the scattered field us has the asymptotic behaviour uniformly in all directions x=x/|x|, where u∞is defined on the unit circleΩand known as the far field pattern.Now the inverse scattering problem is to determine (?)D, or part of (?)D, from a knowledge of the far field pattern u∞for one or several incident plane waves. The focus of this paper is on the issues of determining (?)D with N(N≥2) incident waves u1i, u2i,…,uNi and the corresponding far field patterns u∞,i,u∞.2,…,u∞,N.We make the general assumption that the incident plane waves with different directions either uniformly distributed over the unit circleΩ(in the full aperture case) or uniformly distributed over the measuring angleδ(in the limited aperture case).Analogous to the method of Kirsch and Kress [103,104], our method is divided into two steps.Step 1. Having a mild a priori information about the unknown scatterer D, we can choose an auxiliary closed C2 curveГcontained in D, such that K2 is not a Dirichlet eigenvalue for the negative Laplacian in the interior ofГ. Then we try to represent each scattered field as an acoustic single-layer potential with an unknown density denotes the funda-mental solution to the Helmholtz equation and H0(1) is the Hankel function of the first kind of order zero. Given the (measured) far field patterns u∞,j,j=1,2,…,N, we have to solve the integral equations of the first kind for the densitiesφj. The far field integral operator S∞:L2(Г)→L2(Ω) is defined by whereFor stable numerical solutions ofφj, the associated densitiesφα,j are evaluated via the standard Tikhonov regularization with a regularization parameterα>0 and the adjoint operator S∞*:L2(Ω)→L2(Г) of S∞.Step 2. Once we getφα,j as solutions of (30). the corresponding approximation uapprox.α,j for the scattered fields are obtained. Then we can seek the boundary of the scattcrer D as the location of a zero level curves of uji+uapprox,α,j s in a minimum norm sense, i.e., we can choose D as the minimizer of over some suitable class U of admissible curves A. The illuminated regions AjN in (31) are defined as follows:Definition 0.5 (Illuminated region) Let N denotes the number of incident plane waves, A is a curve with the parametric representation where r∈C2[0,2π].(Ⅰ) For N= 1 and incident angleθ,θ∈[0,2π], the illuminated region A1 is defined by (Ⅱ) For N≥2 and incident anglesθj,θj∈[0,2π], j=1,…,N, the illuminated region AjN corresponding to the j-th incident wave is defined by Particularly, we also denote AjN byΩjN for r= 1, i.e whereν= 2π/N denotes the aperture angle for N. Assume that (?)D is star-like and can be parametrized in the form where r∈C2(Ω) is the radial function which satisfies the a priori assumption with given constants a and b. Thus the set U of admissible curves may be chosen to be a compact subset (with respect to the C1,βnorm,0<β<1) of the set of all star-like curves described by (35) and (36).The illuminated regions on a circle for N=2,4,6 and 8 are illustrated in Fig 2. ii. Convergence resultsFor simplicity, we shall denote AjN by Aj for fixed N. To analyse the convergence properties, we define the cost functional whereφ=(φ1,φ2,…,φN), A=∪n=1N Aj.Hereα>0 denotes the regularization parameter,γ>0 denotes a coupling parameter and S:L2(Г)→L2(A) denotes the single-layer operator.Definition 0.6 Given the incident fields u1i,u2i,…,uN2, the (measured) far field patterns u∞,1,u∞,2,…,u∞,N and a regularization parameterα>0, a curve A0 from the compact set U is called optimal if there exists such thatφ0 and A0 minimize the cost functional (37) simultaneously over allφ0<∈(L2(Г))N and A∈U that is. we have whereThe following properties for the optimization method have been proved:Theorem 0.3 For eachα>0 there exists an optimal curve A∈U.Theorem 0.4 Let u∞j,j=1,2,…,N, be the exact far field patterns of a domain D such that (?)D belongs to U. Then we have convergence of the cost functional Theorem 0.5 Let{αn} be a null sequence and let{An} be a corresponding sequence of optimal curves for the regularization parameterαn. Then there exists a convergent subsequence of{An}. Assume that u∞,j,j=1,2,…,N,are the exact far field patterns of a domain D such that (?)D is contained in U. Then every limit point A* of{An} represents a curve on which the total fields vanish.ii. The limited aperture problemLetΩ' be a closed subset ofΩ, then the inverse scattering problem in this section is to recover (?)D from the knowledge of u∞,j(x), x∈Ω', j=1,2,…, N. Analogously to (29), we can define the corresponding far field operator S'∞:L2(Г)→L2(Ω').In the limited aperture case, we propose the following cost functional Here the weightsγj>0 can be chosen according to incident directions dj and are possibly distinct. The reason for weighting the total fields with distinctγj is that. for each given incident plane wave with fixed wave number k, its contribution to the reconstruction essentially relies onΩ' and the corresponding illuminated regionΩjN.Thus, we recommend that the value ofγj is directly proportional to the arclength ofΩ'(?)ΩjN.Numerical experiments show that our method accelerates the computations without losing the accuracy of the reconstructions.