【作者】 王玉洁；

【导师】 马富明；

【作者基本信息】 吉林大学 ， 计算数学， 2015， 博士

【摘要】 声波和电磁波的散射与反散射理论在数学物理领域发挥了重要作用,并且在实际生活中有着很多应用,如医学成像,雷达探测,无损探伤等.本文主要讨论了障碍散射问题和单缝散射问题的数值求解及贝叶斯方法在内腔体反散射问题和单缝反散射问题中的应用.我们所考虑的散射问题模型均为Helmholtz方程.由于障碍散射问题和单缝散射问题都是无界域上的散射问题,进行数值求解时首先要考虑无界域截断问题.在本文中,我们利用Dirichlet-to-Neumann算子(简称DtN算子)将无界域问题转化为有界域问题.与常用的吸收边界条件相比；这种转化是精确的,没有近似.因此,在第一章的绪论中,我们介绍了Hehmholtz方程和 Dirichlet-to-Neumann算子及文中所要探讨的几种散射问题的研究现状,并列出了本文的主要工作.第二章中我们提出了一种处理DtN边界条件的间断有限元方法,以障碍散射问题为模型介绍了算法的具体格式,证明了显波数k的DG模和L2模误差估计,并给出了相应的数值实验结果.第三章中我们利用DtN算子将单缝散射(single slit scattering)问题转化为单缝处的一个算子方程,证明了算子方程解的存在性、唯一性及稳定性.应用Galerkin法对算子方程进行数值求解,证明了数值解的存在性、唯一性和收敛性.最后进行了数值模拟,其结果与实际的物理现象一致,说明了算法的有效性.第四章中介绍了两个反散射问题的贝叶斯方法.以内腔体反散射问题为例,给出了贝叶斯方法的具体步骤及方法适定性分析,数值实验结果说明了方法的可行性.同时,我们还利用贝叶斯方法反演了单缝散射问题中点源的位置,给出了相应的数值算例.本文的最后一章为总结,对全文内容进行了总结并列出了后续研究的几个方向.1.处理DtN边界条件的间断有限元方法考虑障碍散射问题,设区域Ω (?)R2为Lipschitz区域,我们用表示区域Ω的外部.设Ω为包含原点的星形域且边界(?)Ω充分光滑,例如,(?)Ω是C2类的.我们考虑如下的具有Dirichlet边界条件和Sommerfeld辐射条件的Helmholtz方程外问题：给定入射场ui,求散射场us使得对于任意的函数f∈H1/2((?)BR),定义DtN算子如下：借助DtN算子我们将问题(1)转化为如下有界域上的问题其中上的单位外法向量,为了方便,我们将us简写成u.这里,BR1和BR2为包含区域Ω的两个大圆,将问题(1)转化为有界域上的问题(3)后,计算区域为ΩR2.为了处理DtN边界条件我们需要在边界ΓR2附近剖分出一个环形区域AR1R2,在其内部使用环形网格剖分和谱方法基底.在剩余的部分,即ΩR1内采用标准的三角网格剖分和多项式基底.记ΩR2上的整体剖分T=T1+T2,其中T1表示ΩR1中的三角剖分,T2表示AR1R2中的环形剖分.对于任意的单元K∈T1,我们定义其直径若单元我们定义其直径为圆环的宽度相应地,对于剖分T1中的三角单元K的边e,定义he为其长度he：=diam e.在我们的方法中采用拟一致网格剖分,即存在常数C1>0,C2> 0,使得设Fh表示剖分T1的“骨架”,即跃度与平均值的定义与其它标准DG方法相同.定义基于剖分T的分片Sobolev空间：其中那么问题(3)的变分形式如下：求u∈X使得其中ah(·,·)是如下定义的DG双线性形式：且G(v)的定义如下：这里a,b和λ是待定的正参数.从上面的定义可以看出,如果u是问题(3)的解,那么u满足下式定义DG模如下：对于单元K∈T1,无论它是直边三角形还是曲边三角形,在计算中我们都可以将它映射为参考单元K,其顶点为(0,0),(0,1),(1,0).用P1(K)表示单元K上的全体线性多项式的集合.我们定义DG近似空间其中则Vh(?)X是有限维空间.下面我们给出基于变分形式(4)的DG格式：求uh∈Vh使得可以证明DG格式(6)的解是存在且唯一的.定理1.(存在唯一性定理)设则DG格式(6)存在唯一解.为了给出误差估计,我们首先证明了空间Vh的逼近性质及双线性形式定理2.设函数对于(?)φh∈Vh2,如下的迹逆不等式成立其中Gti是与h无关的正的常数,这里“ti”表示“trace inverse",定理3.设ah(·,·)定义式(5)中待定的常数a，b，λ分别满足则双线性形式bh(·,·)满足下式,|bh(v,v)|≥Ccope,‖v‖DG2,(?)v∈Vh,其中Ccoer是与波数k和剖分h无关的正的常数.定理4.存在与波数k无关的常数C>0，使得借助空间Vh的逼近性质及双线性形式bh(·,·)的性质,我们证明了DG模和L2模的误差估计如下：定理5.(DG-模误差估计)设区域Ω为光滑星形域,边界(?)Ω是解析的,定理3的条件成立,当k2h充分小时,我们可以得到如下的误差估计其中C>0是与网格剖分h和波数k无关的常数.定理6.(L2模误差估计)设定理5的条件满足,当k2h充分小吐我们可以得到如下的误差估计其中C>0是与网格剖分h和波数k无关的常数.最后,我们应用所提出的间断有限元方法对障碍散射问题进行了数值求解.实验结果说明我们的方法对于大波数情形也是有效的,并且其数值误差收敛阶与前面的理论证明相吻合.2.单缝散射问题的Galerkin法设ΓUΓc表示R2中带有单缝的良导体平面,其所在直线设为x轴,其中表示平面上的单缝,}＼r表示良导体平面.令单缝的左端点为原点,单缝长度为L.ui为入射场,ur为平面产生的反射场,即y=0时,ui(x,y)+ur(x,y)=0.若入射场是入射角为θ(关于y轴)的平面波,那么其中当y≥0时,设参考场全场这里us表示散射场.当y<0时,全场由于Γc为良导体平面,那么由其物理性质可知在Γc上ut=0.于是全场ut满足：其中波数由于入射场ui和反射场ur都满足Helmhotz方程,且在故散射场us满足：对(10)关于x做Fourier变换可以解得：当y>0时,当y<0时,其中为零扩张算子,定义如下：定义Γ上的Dirichlet-to-Neumann算子T如下：利用Γ上的DtN悄算子T和全场ut及其法向导数(?)ut/(?)y在Γ上的连续性,我们可以将问题(10)-(12)化成如下r上的算子方程：其中于(16)中解得uΓs后,可由(13),(14)式得到散射场us.为了简单,我们记uΓs为u.于是方程(16)可以改写为：对于实数s,定义Sobolev空间：相应的模为当不发生混淆时,用分别代替借助上面的空间定义,可以证明方程(17)的解存在、唯一并且稳定.其中C>0为与g无关的常数.设为H*2(Γ)的有限维子空间,且形成空间VN的一组基底.则解方程(17)的Galerkin法为：求使得可以证明数值解un。存在、唯一并且收敛到真解u.定理8.方程(18)的解存在且唯一定理9.设u是方程(17)的解,un是方程(18)的解,且在中终归稠密,即则当N充分大时,下面的估计式成立,其中C>0是与u，N无关的常数.从而在我们所考虑的单缝散射问题中,故所以,我们选取不失一般性,我们假设则算子方程(17)相应的Galerkin方程可化为：解上面的方程可以得到uΓs的数值解通过uΓs的数值解即可得散射场us(x,y)的数值解.最后,我们证明了{VN}N∞=在空间H*1/2(r)中终归稠密.定理10.有限维子空间列{VN}N=1∞在H*1/2(Β)中终归稠密.我们应用Galerkin法对单缝散射问题进行了数值模拟,其结果与实际的物理现象相吻合,说明了算法的有效性.3.两个反散射问题的贝叶斯方法考虑下面的问题：已知变量y∈RJ,求q∈Rn满足如下方程这里我们称y为观测值,q为未知量.这是一个典型的反问题模型,通常情况下观测值y是带有误差的,即我们实际考虑的模型应该是其中δ∈RJ为观测误差.应用贝叶斯方法从统计学的角度来思考以上问题.首先将变量q,y和δ均视为随机变量,那么反问题的解就是给定y,求变量q的概率分布,即求条件分布q|y.这样虽然不知道误差δ的具体值,我们却可以将它的统计性质写入到了模型中.设随机变量q的概率密度函数为π0(q),随机变量δ与q相互独立,且δ的概率密度函数为π(δ).当q已知时,y由(19)式决定,故y|q的概率密度函数为)).于是随机变量y和q的联合概率密度为由贝叶斯公式,我们就可以得到条件分布q|y的概率密度函数.π0(q)通常称为先验密度函数,称为似然函数,πy(q)称为后验密度函数.设脚是具有密度函数πy的后验测度,μ0是具有密度函数π0的先验测度,那么.我们称dμy/dμ0为Radon-Nikodym导数,它关于似然函数成正比,即利用Radon-Nikodym导数的概念可以将上面的分析推广到无穷维的情形,即q∈X,y∈Y,X和Y均为无穷维空间.由上面的分析可以看出,应用贝叶斯方法解反问题主要可以分为以下三步：1.依据关于未知量q的所有先验信息来确定先验密度函数π0(q).2.根据误差函数的性质来确定似然函数π(y-g(q)),实际上似然函数反映了当未知量q已知时观测值y所满足的分布.3.根据前两步的准备和贝叶斯公式,可以确定后验密度函数的表达形式,然后寻找合适的抽样方法将其描述出来即可.3.1.内腔体反散射问题的贝叶斯方法考虑如下的内腔体散射问题,设D(?)R2是单连通区域且具有C2光滑的边界.设点源和观测点都位于区域D内部的曲线C上,则内腔体散射问题可以描述如下：其中k>0是波数,ui为入射场,定义如下：这里,是C上的一个固定点,Φ(x,d)是二维Helmhotlz方程的基本解.边界条件(22)中的B代表三类不同的边界条件,即Dirichlet, Neumann 和 Robin边界条件.我们所要考虑的反问题就是已知测量曲线C上的散射场us及边界条件(22),重构腔体形状D.下面为了简化问题,我们设区域D是关于原点的星形域,即其中0<r< R0.为了后面证明的方便设q=lnr.故问题的模型可以写成如下形式其中为观测点的个数)是对应于散射场方程(21)的有限维观测算子,向量y是具有误差δ的观测值.这里q属于函数空间X,对于X的选择将在后面介绍.假设q满足正态分布(也称高斯分布)其中m0为期望,G0为协方差算子.误差δ满足正态分布N(0,Γ),其中Γ为有界协方差矩阵.由前面的分析可知Radon-Nikodym导数参照文献[99]我们给出q所满足的正态分布.设算子且定义域如下：由于星形域的表示中,r(θ)是2π周期的,所以这里我们在函数空间的符号中用方括号表示函数是2π周期的.假设q"(θ)满足L2[0,2π]上的正态分布N(0,A-1).则由Karhunen-Loeve展开,我们可以得到其中an和bn是独立同分布的随机变量,其满足的分布为Ⅳ(0,1).然后我们通过对q"(θ)积分来得到q(θ).而积分得到的结果并不是唯一的,我们选择其中的一种周期形式来定义q(θ),如文献[99]所示,定义其中q0～N(μ,σ2)是一个正态随机变量.对q"(θ)的展开式逐项积分可得由积分算子的线性和连续性可知这样得到的q(θ)依然是正态分布.根据文献[95]中的引理6.25,可知上面定义的q"(θ)对于α<1/2几乎一定落在中.因此上面的逐项积分是可以进行的,且g(θ)几乎一定是C2,a的,因此设可以验证观测算子g满足如下的假设1：假设1(i)对于任意ε>0,存在常数使得对于q∈X下式成立,(ii)对于任意t>0,存在常数K=K(t)>0,使得对于满足t的q1,q2∈X,下式成立,则根据文献[95]中的定理4.1,定理4.2和引理6.31可以得到如下的适定性定理.定理11.(适定性定理)设观测算子g满足假设1且则有(i)后验测度μg关于先验测度μ0绝对连续,且Radon-Nikodym导数(ii)后验测度μg是L2[0,2π]上的一个概率测度.(iii)后验测度μy关于观测值y是Lipschitz连续的,即存在常数使得其中这里，Hellinger距离定义如下:若μ1和μ2都关于μ0绝对连续,则3.2.单缝反散射问题的贝叶斯方法下面考虑点源入射时的单缝散射反问题：假设已知下半平面一条线上的散射场值,目标是反演点源的位置.单缝r的长度仍然设为π,已知y=a上若干点的散射场测量值其中δ为测量误差,由单缝散射问题中的方程(10)-(12)决定,目标是反演点源的位置q=(x,y).假设我们事先已知点源的大致位置在[b,c]×[e,f]的区域中,这样初始分布可以采用均匀分布依然假设误差δ满足正态分布Ⅳ(0,σ2I),则根据贝叶斯公式,当时,后验分布的密度函数正比于当(x,y)不在区域[b,c]×[e,f]中时,后验分布密度函数为0.利用MCMC方法抽样描述上述关于q的后验分布,取平均值后即可做为点源位置的估计.我们对内腔体散射中重构腔体形状的问题及单缝散射中反演点源位置的问题,应用贝叶斯方法分别进行了数值实验,实验说明了贝叶斯方法的有效性.更多还原

【Abstract】 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.更多还原