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Studies On The Numerical Methods For Inverse Scattering Problems With Phaseless Data

Posted on:2020-01-27Degree:DoctorType:Dissertation
Country:ChinaCandidate:P GaoFull Text:PDF
GTID:1360330575478814Subject:Computational Mathematics
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We are considered with analysis and computation on several inverse scattering problems for cracks and obstacles with phaseless data,all scattering problems consid-ered are modelled by the Helmholtz equation.In practical applications,it is expensive and difficult to acquire the phased data of the scattered field or far field,therefore the phaseless inverse scattering problems attract more attention from both the mathematics and physics.In first chapter,we introduce the background and status of our research,we also give a brief introduction to the structure of this thesis.In second chapter,we introduce some preliminary knowledge,which includes some related concepts of acoustic scatter-ing,the regularization methods ofill-posed problems and the Nystrom methods.In third chapter,we consider the inverse scattering problem for a crack with phaseless data,we check the translation invariance,which implies that we can can not recover the location of the crack for one incident plane wave with the modulus of the far field pattern as data.For the inverse scattering problem,we present the nonlinear integral equations method which is based on a system of nonlinear and ill-posed integral equations,and our scheme is easy and simple to implement.Also the numerical examples are present-ed to illustrate the feasibility of the proposed method.In fourth chapter,we study the reconstruction for a crack from phaseless data with a reference ball.As we known,we can not determine the location of the crack with phaseless data.To deal with this dif-ficulty,we add a sound-soft reference ball in the scattering system,so we can recover the shape and location of the crack via the nonlinear integral equations method,also the numerical examples are presented to illustrate the feasibility of the method.In fifth chapter,we consider inverse scattering for obstacles with a reference ball from phase-less data,the obstacles satisfy the Neumann boundary condition.Due to the translation invariance,the location of the obstacle can not be determined,so we add a sound hard reference ball.Similar to the case of cracks,we solve the inverse problem to recover the shape and location of the obstacle via the nonlinear integral equations method.The last chapter in this thesis is the conclusion.The main work of this thesis is as follows:1.Inverse scattering for a crack from phaseless dataAssume that ?c ={z(s):s?[-1,1]} is a crack,z1:z(1)and z-1:= z(-1)are the end points of ?c,we consider the following model problem:The inverse scattering problem:determine the shape of the sound-soft crack ?c with the given|u?| which is the modulus of the far field pattern for one incident plane wave ui.Since the solution to the inverse scattering problem is not unique,we present the following result.Theorem 1.(Translation invariance)Assume that u?(x)is the far field pattern of scatteringfrom a sound-soft crack Tc.Then,for the cracks ?c?:= {x +?h:x??c}with h ? R2,the far field pattern u?? have the form that is,the inverse scattering problemfor the sound-soft crack with the modulus of the far field pattern has the translation invariance.Due to the translation invariance,we can not recover the location of the sound-soft crack for one incident plane wave with the modulus of the far field pattern as data.For the inverse scattering problem,we use the nonlinear integral equations method We introduce the single-layer operator Sc:and the far field operator Sc,?We can observe that the unknown curve ?c and density ? satisfy the following equationsWe transform the integral operator S,into the parameterized operator C,given by where Analogously,we introduce the parameterized far field operator C?where In addition,we parameterize the incident field ui and the far field pattern u? by the form of ?c=ui oz,?c,?=u? o z?.Then the parametric form of equations is given by The Frechet derivative of C?(z,?)with respect to z:has the following representation The derivative of C?C? with respect to z:is given by The linearization of(10)leads to whereThe relative error estimator for our iterative procedure isThe suggested iterative procedure is the following:(?)Emanate an incident plane wave with a fixed wavenumber k>0,and a fixed incident direction d??,then collect the phaseless far field data|u?|.(?)Make an initial guess ?c0 for the curve ?c,set k=0.(?)For the curve ?ck,find the density? from(9).(?)Solve(12)to obtain the update ?ck+1 =?ck+q for the curve approximation and evaluate Ek.(?)If Ek???,then set k =k+ 1,go to(?),Otherwise ?ck+1 is served as the final reconstruction of ?c,Our iterative procedure is little different from Newton method.It is easy to see that our scheme can be easily realized and reduce the computational cost.The numerical examples are presented to illustrate the feasibility of the proposed method.2.Inverse scattering for a crack from phaseless data with a refer-ence ballDue to the translation invariance,we can only recover the shape but can not de-termine the location of the sound-soft crack for one incident plane wave with phaseless data.To deal with this difficulty,we add a sound-soft reference ball in the scattering system.Assume that ?1={z(s):s[-1,1]} is a sound-soft crack,D(?)R2 is a sound-soft reference ball,and D ? ?1 =(?),?2:=(?)D.The inverse scattering problem:given an incident plane wave ui for a fixed wavenum-ber and a single incident direction d,together with the phaseless far-field data |uD??1?(x)|,x?? determine the location and the shape of ?1.We introduce the single-layer operator:and the far field operator We can observe that the unknown curve ?1 and density ?j satisfy the following equa-tions The crack ?1 is parameterized by and the boundary ?2 is parameterized by We transform the integral operator Sjl,Sj? into the parameterized operator Cjl,Cj?,and parameterize the right hand by the form of Thus we can obtain the parameterized integral equations(14)-(16)in the form The linearization of(19)leads to where We present some numerical examples to illustrate the feasibility of the proposed method.3.Inverse scattering for obstacles from phaseless data with a refer-ence ballAssume that D(?)R2,we consider following scattering problem:The inverse problem we are concerned with is to determine the location and the shape of D with the given phaseless far-field data |uD?(x)|,x??,for incident plane wave ui with a fixed wavenumber and a single incident direction d.Due to the relation that is,the inverse scattering problem for the sound-hard obstacle with the modulus of the far field pattern has the translation invariance,so we can not recover the location of the obstacle for one incident plane wave with phaseless far field data.Our goal is to overcome this difficulty by adding a reference ball.We assume that the sound-hard reference ball B(?)R2,D?B =(?).The inverse scattering problem:given an incident plane wave ui for a fixed wavenum-ber and a single incident direction d,together with the phaseless far-field data|uD?B(x)|,x?determine the shape and the location of D.We introduce the normal derivative operator and the far field operator:We can observe that the unknown obstacle D and density?j satisfy the following equa-tions The boundary ?1 is parameterized by and the boundary ?2 is parameterized by We transform the integral operator Tjl,Tj? into the parameterized operator Ajl,Aj?,and parameterize the right hand by the form of Thus we can obtain the parameterized integral equations in the form The linearization of(29)leads to where Some numerical examples are presented to illustrate the feasibility of the proposed method.
Keywords/Search Tags:inverse scattering problem, Helmholtz equation, phaseless data, crack, obstacle, the nonlinear integral equations method
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