Studies On Numerical Methods For Some Direct And Inverse Scattering Problems

The scattering and inverse scattering problems for acoustic and electromagnetic waves play an important role in the field of mathematical physics. There are also many practical applications about scattering and inverse scattering problems, such as medical imaging, radar detection, nondestructive testing and so on. In this thesis, the numerical methods for obstacle scattering problem and single slit scattering problem are proposed and the applications of Bayesian method in two inverse scattering problems are inves-tigated. All scattering problems considered in this thesis are modelled by Helmholtz e-quation. The obstacle scattering problem and the single slit scattering problem are both in unbounded domain. Therefore, we should truncate the unbounded domain before applying the numerical method. In this thesis, we use Dirichlet-to-Neumann operator (DtN operator) to reformulate the scattering problem into bounded domain. Comparing with the absorbing boundary condition, this reformulation is exact, without approxima-tion. In Chapter 1, we introduce Helmholtz equation, Dirichlet-to-Neumann operator and the research status of the scattering problems considered in this thesis. We also outline the main work of this thesis. In Chapter 2, a discontinuous Galerkin method for DtN boundary condition is proposed. We apply this numerical method to solve the obstacle scattering problem, and give the DG-norm and L2-norm error estimate explic-itly with wave number k. Some numerical experiments are presented for our method as well. In Chapter 3, by use of the DtN operator on the slit, we reformulate the scattering problem for a single slit to an operator equation. The well-posedness of the solution is proved. Galerkin method is applied to solve this operator equation numerically and the existence, uniqueness and convergence of the numerical solution are proved. Finally, some computational examples are presented. The numerical results are in accord with the physical phenomenon. It is shown that the Galerkin method is effective. In Chapter 4, we introduce the application of Bayesian method in two inverse scattering problems. For inverse scattering problem of interior cavity, we give the steps of Bayesian method and prove the well-posedness of this method. The numerical examples show that our method is viable. We also employ the Bayesian method to seek the location of point source in inverse single slit problem. Some numerical experiments are presented to il-lustrate the effectiveness of our method. In Chapter 5, we give a conclusion of the thesis and point out some directions of our future work.1. A discontinuous Galerkin method for DtN boundary conditionConsider obstacle scattering problem, For the bounded Lipschitz domain Ω(?)R2, we denote its exterior Ωc by Ωc=R2\Ω. Suppose Ω, is starlike and (?)Ω is sufficiently smooth, e.g., of class C2. We consider the exterior Dirichlet problem with Sommerfeld radiation condition:given incident field ui,find the scattered field us, such thatDefine DtN operator as follows: By use of DtN operator we reformulate problem (1) on a bounded domain. where g:=-ui, n is the outward normal derivative of ΓR2, and we use u instead of us. Here, BR1 and BR2 are two circles containing domain Ω,For bounded problem (3), the computational domain is ΩR2. In order to cope with DtN boundary condition we need an annular domain AR1R2 near ΓR2, and use annular mesh in it. In QRl we use standard triangular mesh. Let partition T=τ1+τ2, where τ1 is a family of triangulations of the domain ΩR1 and τ2 is a family of annulations of the domain AR1R2. For any element K ∈τ1, we define hK:= diam K. When Let h:=K∈max Hk.Similarly, for each edge e of KeTi, define he:=diam e. In this paper, we use shape regular and quasi-uniform meshes, so he(?)h, that is there exist Let Th be the skeleton of the partition τ1, The definitions of averages and jumps are the same as standard DG method.Based on the partition τ we can introduce broken Sobolev spaces:X=X1+ H2(Ariri+1)}. Then the weak formulation of problem (3) is as follows:find u∈X such mat where ah(.,.) is the DG-bilinear form defined byand G(v) is defined byHere, a, b and λ are positive parameters to be chosen. From the definition, if u is the analytic solution of problem (3), thenFor K ∈τ71, no matter it is a straight triangle or a curved triangle, in computation we can map it to a reference triangle K with vertices (0,0), (0,1), (1,0). Let P1(K) denote the set of all linear polynomials on K. We define our DG approximation spaceThen Vh C X is a finite dimensional space.We are now ready to define our DG method based on the weak formulation (4): find uh ∈Vh such thatThen we give the uniqueness and existence of problem (6).Theorem 1. (uniqueness and existence) Ifb> 0,α> 0, andλ> 0, the discrete variational problem (6) possesses a unique solution.In order to give the error analysis, we first prove the approximation properties of Vh and two properties of bilinear form bh(u, v)= ah(u, v)+k2(u, v).Theorem 2. Ifφ∈H2(K), K∈T,and h·N= O{1), thenFor (?)φh∈Vh2, the trace inverse inequality also holds where Cu is a positive constant independent of h and "ti" stands for "trace inverse". where Ccoer is a positive constant independent of k and h.Theorem 4. There exists a positive constant C independent of wave number k such thatUsing the above theorems, we prove the DG-norm and L2-norm error estimates as follows: Theorem 5. (DG-norm error estimate) Assume Ω is a starlike domain, the boundary (?)Ω is analytic, and the conditions of Theorem 3 are satisfied. When k2 h is small enough, we have the following error estimate where C> 0 is independent of the mesh and wave number k.Theorem 6. (L2-norm error estimate) Assume the conditions of Theorem 5 are satis-fied. When k2h is small enough, we have the following error estimate where C> 0 is independent of the mesh and wave number k.Finally, we apply our discontinuous Galerkin method to solve the obstacle scatter-ing problem numerically. The results show mat our method is effective for the condition of large wave number as well and the convergence rate complies with Ihe theoretical result we have proved.2. Galerkin method for scattering problem of a slitWe assume Γ∪Γc is a perfectly conducting flat surface with a single slit in R2 and the line of Γ∶Γc is x axis. Denote the slit by and the flat surface by Assume the left endpoint of the slit is origin and the width of the slit is L. ul is the incident wave and ur is the reflected wave, If the incident wave is plane wave with angle θ (with respect to y axis), then , where α= k sin δ, β= k cos 9. When y≥0, we set the reference field uref=ui+ur and the total field ut=uref+us, where us represents the scattered field. When y< 0, ut= us. Because the surface Fc is perfectly conducting material. It follows from the physics that the the total field ut= 0 on Γc. Then ul satisfies Δut+k2ut=0, inR2\Γc, (7) ut=0, on Γc, (8) lim (?)r ((?)r/(?)un-ikus)=0. (9) where k> 0 is the wave number and r=(?)x2+y2.Because the incident wave u2 and the reflected wave ur both satisfy the Helmhotz equation as well, and u2+ur=0 on Δc. Then we can get thatTaking the Fourier transform of (10) with respect to x, we haveEo is zero extension operator defined by the followingDefine Dirichlet-to-Neumann operator T onThen we use Dirichlet-to-Neumann map on the slit Γ and the continuity of ut and (?)ut/(?)y on Γ to reformulate the problem (10)-(12) to the following operator equation on Γ: whereAfter solving uΓs from (16), we can get us from (13) or (14).For simplicity, we use u instead of uΓs. Then the equation (16) can be rewritten as:For any real number s, define Sobolev space: with normWhen there is no confusion, we use ‖u‖1/2 and ‖u‖-1/2 instead of ‖u‖1/2,*,Γ, r and ‖u‖-1/2,*,Γ for short.With the spaces and norms defined above, we can prove the well-posedness of the solution to equation (17). where C> 0 is α constant independent of g.s a finite dimensional subspace of H*1/2 (Γ) and φ1,φ2,…, φN form a basis of VN. The Galerkin method for (17) is:finding un ∈ Vn, such thatThen the numerical solution un is uniquely existed and convergent to u. Theorem 8. The equation (18) has a unique solution in VN. Theorem 9. Assume u is the solution to (17), un is the solution to (18) an are dense in H*1/2 (Γ), that is When N is large enough, the following result holds, where C is a constant independent of u and N. ThenIn our problem, ut(0,0)= ut(L,0)= 0 and uref(0,0)= uref (L,0)= 0. Then us(0,0)= us(L,0)= 0. Thus we choose cpn=sin(nπx/L). Without loss of generality, we assume L=π,φn= sin(nx). The Galerkin equation for operator equation (17) is: From the above equation we obtain the numerical solution of uΓs, that is un=∑n=1N cn sin (nx). From the numerical solution of uΓs we can get the approximation of us(x, y).Finally we show the spaces {Vat}N=1∞ are dense in H*1/2 (Γ). Theorem 10. The finite dimensional spaces {VN}∞=1 are dense in H*1/2 (Γ).We employ Galerkin method to give the numerical simulation of single slit scat-tering problem and the result is in accord with the physical phenomenon. It is shown that our numerical method is effective.3. Bayesian method for two inverse scattering problemsConsider the following problem:given y ∈RJ, seek q ∈Rn such thatHere, y is observational data, q is unknown. This is a typical model of inverse problem. Usually the observational data y is perturbed by noise and we should really consider the equationwhere δ∈RJ represents the observational noise. We look at the above problem from Bayesian perspective. First, we regard q, y and δ as random variables and define the "solution" of the inverse problem to be the probability distribution of q given y, de-noted q|y. If we do not know the exact value of the noise entering the given data 8, we can model the noise via its statistical properties. Assume the probability density of q is π0(q),δ is independent of q, and the density of δ is π(δ). When q is given, y is decided by (19). Thus the density of y|q is π(y-g(q)). The joint probability density for y and q is π(y-g(q))π0(q). From Bayes’formula, we can get the density of q|y, π0(q) is called the prior density. π(y-g(q)) is the likelihood, and πy(q) is the posterior density.Let μy be posterior measure with density πy, and μ0 be prior measure with density π0. Thendμy/dμc is Radon-Nikodym derivative, proportional to the likelihood,By use of Radon-Nikodym derivative, the above analysis can be extended to the infinite dimensional case, i.e. q ∈ X,y ,X and Y are infinite dimensional spaces.From the analysis above, solving an inverse problem may be broken into three subtasks:1. Based on the prior information of the unknown q, find a prior probability density π0(q) to describe this prior information.2. Based on the property of observational error δ, find the liklihood function π(y-g{q)). It is indeed the density of y|q.3. Based on step 1 and 2, decide the posterior density from Bayes’ formula and develop proper sampling method to explore it.3.1. Bayesian method for inverse scattering problem of interior cavityConsider the following inverse interior scattering problem. Let D(?)R2 be a simply connected domain with C2,α boundary (?)D. We assume the point sources and the observational points all locate on curve C inside the domain D. Then the above scattering problem reduces to finding the scattered field us which satisfies where k> 0 is the wave number and u2 is the incident field given by:where i=(?)-1, d is a fixed point on C, and Φ(x, d) is the fundamental solution of two dimensional Helmhotlz equation. The operator B in boundary condition (22) repre-sents three different boundary conditions, i.e., Dirichlet, Neumann and Robin boundary conditions.The inverse problem is recovering the shape D for given measurements us and boundary condition (22). For simplification, we assume D is a starlike domain. Thus we can represent the boundary (?)D bywhere 0< r< R0. We set q= In r just for the convenience of the proof. Then our model can be written aswhere g:= (us(x1),…, us(xn)) (n is the number of measurements) is a finite di-mensional observational operator corresponding to equation (21), and vector y is the observational data with noise δ. Here q belongs to some function space X, and the choice of X will be introduced in the following part. Assume q satisfies the normal distribution (also called Gaussian distribution) N(mo,Co), and observational error δ obeys normal distribution N(0,Γ), where m0 is mean value, Co is covariance operator and T is covariance matrix. From the analysis above we can get the Radon-Nikodym derivative:As in [99] we give the prior density of q. Let A= -d2/dθ2 with definition domain:Because in the representation of starlike domain r(θ) is 2π periodic, here we use the square bracket in the definition of the domain [0,2π] to signify the periodicity. Assume q"(θ) satisfies the normal distribution N(0, A-1) onL2[0,27r]. According to Karhumen-Loeve expansion, we havewhere an and bn are an i.i.d. (independent and identically distributed) sequences with α1～N(0,1) and b1～N(0,1). In order to get q(θ), we integrate q"(θ). The function q(θ) is not unique when its second derivative q"(θ) is given. Here, we just use periodic form of q(θ) as the same in reference [99]. More specifically, we definewhere qo～N(μ, α2) is a Gaussian random variable. Then we can integrate the expan-sion of q"(θ) term by term to obtainThen q (θ) also satisfies normal distribution because the integral operator is linear and continuous.From the lemma 6.25 in [95], we have that q"(θ) is almost surly in C0,α[0,27r] for α< 1/2. So the integration above is reasonable, and q (θ) is almost surely C2,α. Therefore, let X= C2,α[0,2π].We can prove that the observational operator g satisfies the following Assumption 1:Assumption 1From Theorem 4.1,4.2 and Lemma 6.31 in [95] we can get the following theorem. Theorem 11. (well-posedness) Let the observational operator g satisfy Assumption 1 and q～μ0 with μ0(X)=1. Then(i) The posterior measure μy is absolutely continuous with respect to μ0 and has Radon-Nikodym derivative given by(ii) The posterior measure μy is a well-defined probability measure on L2[0,2π].(iii) The posterior measure μy is Lipschitz in the data y, with respect to the Hellinger distance, i.e., there exists a constant C(t)> 0 such that provided μ1 and μ2 are both absolutely continuous with μ0-3.2. Bayesian method for inverse scattering problem of single slitConsider the single slit scattering problem for point source incidence. The scat-tered fields on a straight line in the lower half-plane are given, and our task is to find the location of the point source. The width of slit Γ is π. Assume the scattered fields of some points on y= a are given, i.e., y=g(q)+δ is known, where δ represents the observational error, and g:= (us(x1), …,us(xn)) is decided by equations (10)-(12). We want to find the coordinate q= (x, y) of the point source. Suppose we know the location of the source is in [b, c] x [e,f] in prior. Then the prior distribution is uniform distribution U([b, c] x [e,f]). We assume error δ obeys normal distribution N(0,σ21). From Bayes’ formula, when (x, y) ∈ [b, c] ×[e,f], the posterior density is proportional to When (x, y) is not in [b, c]×[e,f], the posterior density is 0. Using MCMC method we can numerically describe the above posterior density of q, and the mean value can be viewed as the location of point source.We reconstruct the shape of interior cavity and the location of point source numer-ically by Bayesian method. The numerical experiments show that Bayesian method is effective.