The Linear Sampling Method Without Sampling And Its Numerical Experiments

With the development of science and technology, inverse scattering theoryhas got successful application on medical science and military a?airs.CT whichwe know very well most is a example of the application of inverse problem. Andwith the complication of the problem, multi obstacle becomes an importantsubject gradually. It is not exaggerated saying that the inverse scatteringproblem has the vast development prospect ,and many problems demandingprompt solution require us to probe into.This paper first introduces the fundamental model of direct and inversescattering problem and Helmholtz equation, gives the basic implement ofstudying inverse scattering problem ,such as uniqueness of solution of the in-verse scattering problem, far field operator, the Herglotz wave function. Sub-sequently we concentrate on having introduced the linearing sampling methodwithout sampling .In artical[29], R.Aramini and others has presented a newimplementation of the linear sampling method ,i.e.,the linear sampling methodwithout sampling . The advantage of the new method is that the set of far-field equations for all sampling points is substituted by a single-functionalequation which is formulated in a Hilbert space , from the point of view ofcomputational , pointwise regularization course has been avoided. Moreover,the new method supplied a analytical form of indicator function ,and from thepoint of view of applications, this method provides some hints about the spa-tial resolution gained by the method. In this paper,we begin with the Fouriertransform of indicator function, analysis the spatial resolution,give the methodresolution aboutλ/4,and extends the solution to factorization method. About the theory of Helmholtz equation ,associate conclusions are follows:Theorem2.1.Let D be a bound domain of class C2 and letνdisplacethe unit normal vector to the boundary ?D directed to the exterior of D. Letu∈C~2(D)∩C(D|~) be a function that possesses a normal derivative on theboundary in the sense that the limitexists uniformly in ?D. Then we have Green's formulaIn particular, if u is a solution of the Helmholtz equationthenTheory2.2. Suppose the bounded set D is the open complement of anunbounded domain of C~2 and letνdenote the unit normal vector to theboundary (?)D directed to the exterior of D. Let u∈C~2(R~3 \ D|~)∩C(R~3 \ D)be a radiating solution to Helmholtz equationwhich possesses a normal derivative on the boundary in the sense that thelimit exists uniformly in (?)D. Then Green's formula is gainedIn this paper,Let us consider the far-field equation:The main results are as follows.Theory3.4. (General theorem)Let us suppose that k~2 is not an eigen-value for the negative Laplacian in scatterer D. Then, if F is the far-fieldoperator defined as (3-3-7), we have that(a)if z∈D, then for every > 0, there exists a solution gz(·)∈L2(Ω) satisfiedthe inequalitywhere vgz(·) is the Herglotz wave function with kernel gz(·);(b)if z∈D, then for every > 0 andδ> 0, there exists a solution gz(·)∈L2(Ω) satisfied the inequality Theory3.5. The linear operator Fh is continuous and its kernel N(Fh)ishere we have denoted with N(F_h) as the kernel of the linear operator F_h.Moreover, if g(·)∈[L~2(T_A~B )]~N is satisfied that g(z)∈N(F_h)~⊥f.a.a.z∈T_A~B ,then g(·)∈N(F_h)~⊥. Here f.a.a.denotes for almost all.Theory3.6. The generalized and regularized solutions of problem (3-3-35) are given asAs the spatial resolution,we have Shannon-Nyquist representation (4-1-16), i.e.,from Shannon-Nyquist representation,the estimate of the spatialresolution is aboutλ/4.