The Numeircal Computation Of Direct And Inverse Scattering Problems For Helmholtz Equation With Periodic Structure

It is well known that the Cauchy problem of the partial differential equations are severely ill-posed in the sense of Hadamard. The Cauchy problem suffers from the insta-bility of the solution in the sense that small perturbations in the input data may result in an enormous deviation in the solution. Therefore, there is considerable interest in estab-lishing accurate, stable, reliable and fast numerical algorithm for the Cauchy problem.This paper mainly studies the ill-posedness of the initial problem of the Laplacian equation and the Cauchy problem of the Helmholtz equation, proposes the mathematical models and investigates effective and stable numerical methods. The paper is divided into five chapters.Chapter1introduces the physical background of the Helmholtz equation and gives a short review about the study on the Helmholtz equation.Chapter2studies the ill-posed problem and the numerical methods for the ill-posed problem. We introduce the Cauchy problem of the partial differential equations and present a short review of the current research situation of the Cauchy problems. We show some results of the numerical method for solving the inverse problems.Chapter3investigates the numerical solution to the Cauchy problem with periodic structure which is also the elliptic equation initial problem. The Cauchy problem is trans-formed into the equivalent operator equation. The spectrum of the corresponding operator is analyzed. we divide the eigenvalue space into two subspaces and proposes a numerical method with regularization process for one of the subspace is severely ill-posed.Seek U(y)=(v(y),w(y))T, U∈C1([0,1]; Hp0×Hp0such thatThen analyze the posedness of this problem and use regularization. Theorem1The spectrum of operator Ais σ(A)={(2nπ)；n=0,±1,±2,…},The corresponding eigenfunctions are Let H+=span{Vn(1),Vn(2)}n=1+∞,H+=span{Vn(1),Vn(2)}n-1=+∞,H0=span{V0},V0=H+,H and Ho is invariable subspace of operator A。.(1)may decompose into: d/dyU+(y)=A+U+(y),(2) d/dyU0+(y)=A0U0(y),(3) d/dyU-(y)=A-U-(y),(4)Theorem2The operator-A+,A0and Agenerate the operator semigroups E+(y),Eo(y)andE(y) on the spaces H+,H0and H-respectively. And the solutions U+(y),U0(y)andU-(y)to equa-tions(2),(3)and(4)satisfy the equation systemThen we can obtain the expression of solution to the regularization problem:This method is new and the numerical experiments demonstrate that the method is easy and effective. Chapter4considers the reconstruction of the grating profile by the incident plane wave and the scattered waves measured on Γb,which is a straight line above the grating. First,we obtain the normal derivatives of the scattered waves on Γb by the Dirichlet-to-Neumann map.Then we obtain the total field below Γb by solving the Cauchy problem of the Helmholtz equation,which is changed into an initial value problem.The initial value problem is solved by the spectral method which is used to solve the initial value problem of the elliptic equation.Finally,we use the curve,which is composed by the zeros of the total field,to approximate the grating profile.Assume the total field satisfies where uα(x+l,y),uα(x,y),Ωlb={(x,y);b>y>f(s),x∈[0,l]},and (?)u/(?)n is outgoing normal derivative on boundaryrΓb.is the known Dirichlet-to-Neumann operator, u0is known,α=k sinθ,θis incident angle,and kis wavenumber.For(u,w)T∈Hp0×Hp0define operator A:Hp0×Hp0→Hp0×Hp0,The domain of definition is D(A)=Hp2×Hp0.where Lis differential operator.Then the problem(8)can be expressed into the following initial problem:seek U(t)= (u(t),ω(t))T,u∈C1([0,1]；Hp0×Hp0),such that Theorem3The spectrum of operaor Ais σ(A)={λn;n=0,±1,±2,…}, whereThe corresponding eigenfunctions areMake σi(A)={λn;n∈Zi),σ+(A)={λn;λn>0,n∈Zr},σ-(A)={λn;λn<0,n∈Zr},H+=span{Vn,λn∈σ+(A)}n∈Zr.H-=span{Vn,∈n∈σ-(A)∪σi(A)}n∈Zi∪Zr,则D(A)∈H+(?)H-,A(H+)∈H+,A(H-)∈H-,so that H+sndH-invariable subspace of operator A. A+=A|H+,A-=A|H-,(11)may decompose into: d/dtU+(t)=A+U+(t),(12) d/dtU-(t)=4-U-(t).(13)Theorem4The operator-A+and A-generate the operator semigroups E+(t) and E-(t) on thee spaces H+and H-.The solutions U+(t) andU-(t)to the equations (12)and(13)sat-isfy the operator equation system Theorem5Whent>0,the operatorsE+(t):H+→H+and E-(t):H-→H_are compact operators.So the regularization solution to the problem(7)isThe numerical results show that the method is effective.Chapter5gives the conclusions and prospects. We also give some discussions about how to solve the direct and inverse problems of Helmholtz equation with periodic structure.