Stability Estimates In Inverse Scattering And Numerical Realization Using Point Source Method

Here we consider the scattering problem and inverse scattering problem.Next will introduce the theory and numerical experiment for scattering of time-harmonic acoustic waves by an impenetrable obstacle in some penetrable inhomogeneous medium of compact support..Let the time harmonic acoustic plane waveuⁱ(x) = e^ikx·dwhere i = (?), x∈R^m, k is wave number, d∈Ω:= {x∈R^m; |x| = 1, m = 2,3} the direction of propagation.Note u^s(x),x∈R^m is scattering field,total field u (x) := uⁱ(x) + u^s(x)∈H_loc¹(R^m\D).Then the direct scattered problem is find the u which satisfies the exterior Dirichlet boundary value problem:â–³u(x) + k²u(x) =0 x∈R^m\D (1)u(x) = 0 x∈(?)D (2)(?)r((?)u^s/(?)r-iku^s)) = 0 (3)the boundary condition (2) is corresponds to a sound-soft obstacle,condition (3) assure there is unique solution of equations (1)-(2).The scattering field u^s(x) has the asymptotic behavioru^s(x) =e^ik|x|/(?){u^∞(x,d)+O(1/|x|)},|x|→∞where x, d∈Ω.u^∞(x, d), (x, d)∈Ω×Ωis the far field pattern of u^s(x).we consider m=2 here,defineΦ(x, z)Φ(x,z) :=i/4H₀¹(k|x - z|),x≠z, m, = 2is the fundamental solution satisfies the Helmholtz equation in R^m\z.For density function (?)(x)∈C((?)D),define Single-layer operator S:(S(?))(x) := 2∫_(?)DΦ{x, y)(?)(y)ds(y), x∈(?)D,Double-layer operator K:(K(?))(x) := 2∫_(?)D(?)Φ(x,y)/(?)v(y) (?)(y)ds(y),x∈(?)D,v(y) is exterior normal derivative.For the numerical solution of the boundary integral equations for two-dimensional problems,we would use Nystrom method.Let scattering obstacle D∈(?),scattering fieldΦ^s(x, z),x∈R² of point sourceΦ(·, z).define operator Q(Qu^∞)(x,z):=1/γ_m∫_Ω∫_Ωf(x,d)g(z,ξ)u^∞(-d,ξ)ds(d)ds(ξ)引理2 For scattering of a point-sourceΦ(·z) by a domain D∈(?) in a strip 0 < d(z, D) <Ï„with a constantÏ„> 0 there is the lower bound|Φ^s(z,z)|≥c|lnd(z,D)|. (5)with constant c > 0.For all z∈B\D we have the upper bound|Φ^s(z,z)|≤c|lnd(z,D)|+E. (6) with constants C,E > O.In R³ the corresponding estimates are|Φ^s(z,z)|≥c/|d(z,D)| (7)and|Φ^s(z,z)|≤c/|d(z,D)| (8)All constants hold uniformly for domains D∈(?) .定理1 (Regularization properties). For the reconstruction of the scattered fieldΦ^s(x,z),(?)x, (?)z∈B\D_ρ,from disturbed far field data u_δ^∞with error bound‖u^∞-u_δ^∞‖L²(Ω×Ω)≤δusing the operator Q with geometrical parameterρ> 0 and regularization parametersη,ε> 0 we have the error estimate|Φ^s(x,z)-(Qu_δ^∞)(x,z)|≤aε+b‖g(z,·)‖L²(Ω)^η+1/γm‖f(x,·)‖L²(Ω)‖g(z,·)‖L²(Ω)^δ(9)for x, z∈B\D_ρwith some constants a,b depending onβ,ρand on R_e.定理2 (Explicit stability estimate). Let D₁,D₂∈(?) be domains with scatteringdata u₁^∞(x, d) and u₂^∞(x, d) for all x,d∈Ω.If‖u₁^∞-u₂^∞‖L²(Ω×Ω)≤δ(10)we haved(H(D₁),H(D₂))≤C/|lnδ|^c (11)with constants C > 0 and 0 < c < 1.Estimate (11) holds in two or three dimensions. Prom the behaviour ofΦ^s as estimated in lemma 2,the set of zeros A of (Qu^∞)(z,z)-c,is the boundary of D. Assume D (?)B_R(0), z∈B_R(0), from theorem 1,we can use the operator Q to compute the approximate value ofΦ^s(z, z) .Theorem 2 give the explicit stability estimate for inverse scattering problem.In chapter four,we give numerical example using point source method and detailed describe the main idea.