The Study Of Symmetry Of Far-field Operation And Reconstruction For Boundary Of Obstacle With Incomplete Data

This paper is a series of computational research in the scattering of time-harmonic acoustic or electromagnetic waves by a bounded impenetrable obstacle.We also introduced the method of factorization , further more , give the proof of the far-field matrix's symmetrical.Considering first the case of acoustic waves, assume the incident field field is given by the time-harmonic acoustic plane wave:where k =ω/c₀ is the wave number,ωthe fre(?)uency, c₀ the speed of sound and d the direction of propagation. Then the direct scattered problem is find the v which satisfies the exterior Dirichlet boundary value problem:defineis the fundamental solution satisfies the Helmho .tz equation in R^N \ {y}, we consider N = 2 here.Given an integrable functionφ, the integrals are called, respectively, acoustic single-layer and acoustic double-layer potentials with densityφ.We can get theorem as follows[7]:定理1.6. For a given function g∈L²(Ω) the solution to the scattering problem for the incident waveis given byand has the far field patternConcerning above theorem we have the (F^*F)^1/4 method of exterior Dirichlet problem[16].定理1.7. Assume K² is not a Dirichlet eigenvalue of—△in D, Then the range R(G) of G : H^1/2 ((?)D)→L²(Ω) is given bywhere {σ_j,ψ_j,ψ_j} is a singul(?)r system of F ,Note that we consider G here as an operator on H^1/2((?)D).Moreover,the operator (F^*F)^-1/4G is a norm-isomorphism from H^1/2((?)D) onto L²(Ω) . 定理1.8. For any z∈R³ ,define the fun(?)tion r_z∈L²(Ω) byThen r₂∈R(G) if and only if z∈D .In our paper, we have studied the Symmetrical and the influence of the Symmetrical to reconstructing scattered obstacle concerned with the (F^*F)^1/4 method of far-field matrix.We solve the direct problem firstly, use method of Nystrom in order to get the far-field .Then we can get the far-field whose incident direction isθ_j , the observe direction isθ_l , so we have the far-field matrix A = (a_j,l),A(j,l) =Ï…_∞(θ_j,θ_l).Concerning the reciprocity relation, we have the following result:定理1.9. A := (a_j,l)∈C^2Mx2M is far-field matrix, a_j,l =Ï…_∞(θ_j,θ_l). is a plane wave whose incident direction isθ_j ,obse(?)ve direction isθ_l .θ_i = iÏ€/M,i = 1,…,2M defineand A₁ ,B ,C ,D are all matrix of degree M Alien定理1.10. defineI is unitary matrix of degree M .then A = A(?) is symmetrical matrix.Concerned of the method of (F^*F)^1/4, we can see the symmetrical of A give us a lot of convenient in computing the scattered obstacle especially the case of incomplete data. a:In the case of the lack data correspond(?)ng the different incident angle and observe angle, we can Makes up it use the symmetrical of A. We can get the obstacle of little different from original result.b:In the case of the lack data correspond(?)ng the different incident angle and observe angle. This case is a little difficult , bus we also can get the satisfied result in use of the symmetrical of A than A.The picture of the result reconstructed by A except one row and one line.The picture of the result reconstructed by A except one forth row and one forth line. Factorization method, as one of the efficiency method in inverse problems which was used in lots of area in our live. Further more , this new method may give us a lot of inspiration in other area of inverse problems, such as the reconstruction of obstacle without complete data. Even this paper can not solve this question after all, we also hope it can bring some inspiration to someone who interested in.