Transmission eigenvalues for the electromagnetic scattering problem in pseudo-chiral media and a practical reconstruction method
|
|
- Kenneth Watkins
- 5 years ago
- Views:
Transcription
1 Transmission eigenvalues for the electromagnetic scattering problem in pseudo-chiral media and a practical reconstruction method Tiexiang Li School of Mathematics, Southeast University, Nanjing 289, People s Republic of China. txli@seu.edu.cn Tsung-Ming Huang Department of Mathematics, National Taiwan Normal University, Taipei 6, Taiwan. min@ntnu.edu.tw Wen-Wei Lin Department of Applied Mathematics, National Chiao Tung University, Hsinchu 3, Taiwan. wwlin@math.nctu.edu.tw Jenn-Nan Wang Institute of Applied Mathematical Sciences, NCTS, National Taiwan University, Taipei, 6, Taiwan. jnwang@math.ntu.edu.tw Abstract. In this paper, we consider the two-dimensional Maxwell s equations with the TM mode in pseudo-chiral media. The system can be reduced to the acoustic equation with a negative index of refraction. We first study the transmission eigenvalue problem (TEP) for this equation. By the continuous finite element method, we discretize the reduced equation and transform the study of TEP to a quadratic eigenvalue problem by deflating all nonphysical zeros. We then estimate half of the eigenvalues are negative with order of O() and the other half of eigenvalues are positive with order of O( 2 ). In the second part of the paper, we present a practical numerical method to reconstruct the support of the inhomogeneity by the near-field measurements, i.e., Cauchy data. Based on the linear sampling method, we propose the truncated singular value decomposition to solve the ill-posed near-field integral equation, at one wave number which is not a transmission eigenvalue. By carefully chosen an indicator function, this method produce different jumps for the sampling points inside and outside the support. Numerical results show that our method is able to reconstruct the support reliably. Keywords: Two-dimensional transmission eigenvalue problem, pseudo-chiral model,
2 Transmission eigenvalues and practical reconstruction method 2 transverse magnetic mode, linear sampling method, singular value decomposition
3 Transmission eigenvalues and practical reconstruction method 3. Introduction The transmission eigenvalue problem (TEP) has attracted a lot of attention recently in the study of direct/inverse scattering problems in inhomogeneous media [3, 4, 5, 6, 9,, 2, 2, 25]. The existence of transmission eigenvalues is intimately connected to the bijectivity of the far-field operator, which is crucial in some reconstruction methods such as the linear sampling method (LSM) [, 8] and the factorization method [2]. Conversely, the transmission eigenvalues (also eigenvalues) carry the information of the scatterer and can be estimated by the far-field data [2] or the near-field data (Cauchy data) [27]. This observation leads to the development of some reconstruction methods using the transmission eigenvalues or eigenvalues, see for example, [, 29]. In [29], an eigenvalue method using multiple frequency near-field data (EM 2 F) was proposed to detect Dirichlet or transmission eigenvalues and to reconstruct the support of the scatterer. In this paper, we would like to propose a reliable numerical method to investigate the distribution of transmission eigenvalues for the 2d acoustic equation with an inhomogeneous index of refraction. The model is derived from the Maxwell s equations with the TM mode in pseudo-chiral media. It turns out the index of refraction of the reduced acoustic equation decreases to negative infinity when we increase the charity parameter. Precisely, we consider the TEP u + λε(x)u =, in D, (a) v + λv =, in D, (b) u v =, on D, (c) u ν v =, ν on D (d) for the scattering of acoustic wave on a bounded and simply connected inhomogeneous domain D R 2, where ν is the outer normal to the smooth boundary D, u, v L 2 (D) with u v H(D) 2 = {w H 2 w =, w = }, and ε(x) is the index of refraction. Any ν λ C such that () has nontrivial solutions u and v is called a transmission eigenvalue and u, v are called the corresponding transmission eigenfunctions for D. Equation (a) can be considered as a reduced Maxwell s equations with transverse magnetic (TM) mode. For the standard Maxwell s equations, ε(x) is the electric permittivity corresponding to the product of the dielectric constant and the free-space dielectric constant. In practice, ε can not be taken as large as we wish. In [23] where the Maxwell s equations for Tellegen media is studied, ε(x) is the sum of the electric permittivity and the square of Tellegen parameter. By choosing large Tellegen parameter, we can enlarge the parameter ε(x) as we want. On the contrary, here ε(x) is the sum of the electric permittivity minus the square of pseudo-chiral media parameters. Therefore, by selecting the pseudo-chiral parameters sufficiently large, ε(x) becomes negative (see Section 2 below). In recent years, several papers have been devoted to developing efficient algorithms
4 Transmission eigenvalues and practical reconstruction method 4 for computing transmission eigenvalues of 2d/3d TEP [3,, 4, 6, 7, 2, 2, 22, 23, 24, 26, 27]. Three finite element methods (FEMs) and a coupled boundary element method were developed to solve the 2d/3d TEP in [, 4, 7] (see the book [3] for more details). Two iterative methods and the corresponding convergence analysis were given in [28]. In [7], a mixed finite element method for 2d TEP was proposed, which leads to a non-hermitian quadratic eigenvalue problem (QEP) that was solved by an adaptive Arnoldi method. Furthermore, a multilevel correction method was used to reduce the solution of TEP into some linear boundary value problems which could be solved by the multigrid method [8]. For results related to our current work, in [23, 24], the TEP for general inhomogeneous media are discretized into QEP with symmetric coefficient matrices. For such QEP, a secant-type iteration for computing several smallest positive transmission eigenvalues accurately was proposed in [24], and a quadratic Jacobi-Davidson method with nonequivalent deflation technique for computing a large number positive transmission eigenvalues was developed in [23]. Remind that the index of refraction ε(x) increases to infinity as we increase the Tellegen parameter. Numerical simulations demonstrate that the positive transmission eigenvalues are densely distributed in an interval near the origin. Similar to the method in [23], the TEP () is discretized into a QEP and a quadratic Jacobi-Davidson method with nonequivalence deflation is applied to compute a large number of positive eigenvalues. It turns out in the case here there exists an eigenvalue-free interval near the origin. The existence of the eigenvalue-free interval motivates us to study the reconstruction of the support of ε(x) from the near-field data in the spirit of LSM. Corresponding to the TEP(), the scattering problem is described by u + k 2 ε(x)u =, in R 2 \ {x }, (2a) u = u i + u s, (2b) ( ) u s lim r r r ikus =, (2c) where D := supp(ε(x) ) and u i is the incident field due to a point source at x, i.e., u i (x, x ) := Φ(x, x ), x C, (3) where Φ(x, x ) = i 4 H() (k x x ), the fundamental solution of the Helmholtz equation in R 2. Our reconstruction method is based on the set up in [29]. We assume that the target D is inside some domain Ω which itself is inside a curve C (see Figure ). Suppose that u s is measured on Γ = Ω for all point sources x on C. We define the near-field operator N : L 2 (Γ) L 2 (C) for v L 2 (Γ) by (N v)(x) = u s (y, x)v(y)ds(y), x C. (4) Γ The LSM with near-field data relies on solving the ill-posed integral equations (N v)(x) = Φ(x, z) for all x C, (5)
5 Transmission eigenvalues and practical reconstruction method 5 Figure. The target D is inside some domain Ω (Γ := Ω) which itself is surrounded by a curve C. The scattered field u s is due to the scattering of the incident field u i having a point source at x C. where z T is a sampling point and T is a sampling domain inside Ω containing the target D (see Figure ). In general, the above ill-posed equation do not have a solution. The following theorem serves as the backbone of the LSM. Theorem.. [8, 7, 29] Assume that λ = k 2 is not a transmission eigenvalue of (). Let N be the near-field operator defined by (4). If z D, then there exists a convergent sequence v n in L 2 (D), such that lim N v n = Φ(, z). n If z Ω \ D, then for every sequence v n satisfying we have lim N v n = Φ(, z), n lim v n L n 2 (D) =. Our reconstruction method is a direct application of Theorem.. By this theorem, we can see that if k 2 is not a transmission eigenvalue then we can determine whether z D or z Ω \ D from the behaviors of the solutions to (4). Since there exists an eigenvalue-free interval near the origin in TEP (), we choose a k near the origin so that k 2 is not a transmission eigenvalue, we then discretize the integral equation (5). The strategy here is to choose m points x,, x m C forming a m-vector b = [Φ(x, z),, Φ(x m, z)] C m and n unknowns [v(y ),, v(y n )] := v satisfying the linear system Av = b. (6)
6 Transmission eigenvalues and practical reconstruction method 6 The idea is to take m n, i.e., (6) is overdetermined. To determine whether z D or z Ω \ D, we look at the distance between the vector b and the space A = span(a), the subspace spanned by the columns of A, namely, dist(b, A) min v C n Av b 2. If dist(b, A) >, then, intuitively, any solution v satisfying (6) must contain some sufficiently large components. Mimicing the second case of Theorem., we thus assign z Ω \ D. On the other hand, if dist(b, A) =, i.e., b A, then the norm of v is clearly finite. We thus say that z D in view of the first case of Theorem.. An easy application of truncated SVD will quickly determine the value of dist(b, A). Our criterion is very simple and easily to be implemented. This paper is organized as follows. In Section 2, we demonstrate the derivation of 2d TEP () from the Maxwell s equations with TM mode in pseudo-chiral media. A corresponding discretized QEP and its spectral analysis are given in Section 3. In Section 4, a practical numerical method based on the LSM and truncated SVD for the reconstruction of the the target D is developed. Related numerical results are presented in Section 5. A concluding remark is given in Section 6. Finally, in Appendix A, we present a framework of solving the direct scattering problem using FEM. The purpose of solving the direct problem is to obtain synthetic data for numerical simulations. 2. Maxwell s equations with the TM mode in pseudo-chiral media Physically, the governing equations for the propagation of electromagnetic wave in biisotropic materials with complex media is modeled by the 3d source-free frequency domain Maxwell s equations E = ik (µh + ζe), H = ik (ε r E + ξh), (7a) (7b) where E and H are the electronic field and magnetic field respectively, k is the frequency, ε r and µ are electric permittivity and magnetic permeability respectively, ξ and ζ are 3- by-3 magnetoelectric parameter matrices in various forms for describing different types of complex media. We consider E and H in (7) in the transversal magnetic (TM) mode as E = [,, E 3 (x)], H = [H (x), H 2 (x), ] (8) with x = (x, x 2 ) R 2, ζ and ξ are pseudo-chiral media of the forms ζ ξ ζ = ζ 2, ξ = ξ 2 (9) ζ ζ 2 ξ ξ 2
7 Transmission eigenvalues and practical reconstruction method 7 with ξ = ζ = iγ, ξ 2 = ζ 2 = iγ 2, and γ, γ 2 being real numbers. We assume in this paper µ =. Then the equation (7a) can be simplified to x2 E 3 H ζ E 3 x E 3 = ik H 2 + ζ 2 E 3. () Substituting () into (7b) yields (ik) ( x 2 = ik ε r which implies that E x 2 2 ) E 3, ξ H + ξ 2 H 2 x (ζ 2 E 3 ) x 2 (ζ E 3 ) E 3 [ ( = k 2 ε r E 3 ξ (ik) ) ( E 3 ζ E 3 + ξ 2 (ik) )] E 3 + ζ 2 E 3 x 2 x [ + ik (ζ 2 E 3 ) ] (ζ E 3 ) x x 2 [ = k 2 (ε r + ξ ζ + ξ 2 ζ 2 ) E 3 + ik (ζ 2 E 3 ) ] (ζ E 3 ) + ξ E 3 ξ 2 E 3 x x 2 x 2 x =k 2 (ε r γ 2 γ 2 2)E 3 =k 2 εe 3 with ε = ε r (γ 2 + γ 2 2). Let E = [,, E,3 (x)] and H = [H, (x), H,2 (x), ] be, respectively, the electronic and magnetic plane waves with TM mode in vacuum which are governed by the free Maxwell s equations E = ikµ H, H = ikε E (a) (b) with ε = µ =. We now suppose the Maxwell s equations (7) and () are defined on a cylindrical set D R satisfying the boundary conditions E ν = E ν, (2a) (H + ζe) ν = H ν, (2b) where ν = [ν, ν 2, ] = [ν, ] is the outer unit normal to the smooth boundary D R. It is easily seen from (2a) that the Dirchlet boundary condition E 3 = E,3 on D (3a)
8 Transmission eigenvalues and practical reconstruction method 8 holds. Multiplying (2b) by ik and using (a) we get the Neumann boundary condition i.e., E ν = E ν, E 3 ν = E,3 on D. (3b) ν In view of the boundary conditions (3a), (3b), choosing u = E 3 and v = E,3, we arrive at the TEP () with λ = k 2 and ε(x) = ε r (γ 2 + γ2). 2 The index of refraction will become negative as long as either γ or γ 2 are sufficiently large. 3. Spectral analysis of discretized TEP In this section, we first briefly introduce the discretized TEP and give its spectral analysis. Let {φ i } n i= and {ψ i } m i= be standard nodal bases for spaces of continuous piecewise linear functions on D R 2, that have vanishing DoF on D and D, respectively, where DoF denotes the degree of freedom. Applying the standard piecewise linear FEM (we refer to [] for details) to () with n u = u i φ i + w i ψ i, (4a) v = i= n v i φ i + i= i= w i ψ i. i= (4b) Hereafter, we denote and I as a zero matrix/submatrix and the identity matrix, respectively, with appropriate dimensions if it is clear in the context. Now, setting u = [u,, u n ], v = [v,, v n ] and w = [w,, w m ] appeared in (4), we can derive a generalized eigenvalue problem (GEP) L(λ)z = (A λb)z =, (5) in which λ = k 2, where K E M ε F ε u A = K E, B = M F, z = v (6) E E Fε F G ε G w with K, E, M, M ε, F, F ε, G and G ε being given in Table. Here, A means A is symmetric positive definite. We now define M = M ε + M, G = G ε + G, F = F ε + F, (7a) K = K EG F, M = M F G F, M = M F G F, (7b) and S = [ K [ ] ] M E, T = [M F, M = F ] F. (7c) G
9 Transmission eigenvalues and practical reconstruction method 9 stiffness matrix for interior meshes stiffness matrix for interior/boundary meshes mass matrices for interior meshes mass matrices for interior/boundary meshes mass matrices for boundary meshes K = [( φ i, φ j )] R n n E = [( φ i, ψ j )] R n m M = [(φ i, φ j )] R n n M ε = [ (εφ i, φ j )] R n n F = [(φ i, ψ j )] R n m F ε = [ (εφ i, ψ j )] R n m G = [(ψ i, ψ j )] R m m G ε = [ (εψ i, ψ j )] R m m Table. Stiffness and mass matrices with ε(x) < for x D. Lemma 3.. Let M and M be defined in (7a), (7b). Then M, M. Proof. Because [ ] M F F, G we see that [ M M = F On the other hand, observe that [ ] [ I F G M I which implies M. [ M ε Fε F ε G ε ] [ ] [ F M F = G F + G F ] M ε Fε, F ε G ε ] [ ] F I G G F = I ]. ] [ M, G Theorem 3.2. For λ, the GEP (5) can be reduced to the following QEP Q(λ)p = (λ 2 A 2 λa A )p =, (8) where p = u v, A 2, A and A are all n n symmetric matrices given by A 2 = M M M M F G F (9a) = M T M T, A = K K M M M M K EG F F G E (9b) = K SM T T M S, A = K M K + EG E = SM S. (9c)
10 Transmission eigenvalues and practical reconstruction method Proof. By (6), subtracting the st equation by the 2nd equation in (5), and rewriting the 3rd equation in (5) we have [ ] [ ] [ ] [ ] K u E p λ p = λ. (2) v M F M ε + M F ε + F Fε + F G ε + G On the other hand, from the st equation of (5) and the st equation of (2), we have [ ] [ ] u [ ] [ ] u K E λ M F + λm p = Kp. (2) w w From (7a) and (7c), equations (2) and (2) can be rewritten as [ ] (S λt ) u p = λm, (22a) w [ ] u (S λt ) + λm p = Kp. (22b) w [ ] u Expressing in terms of p in (22a) and plugging it into (22b), we end up with a w QEP [ λ 2 M (S λt )M (S λt ) ] p = λkp. (23) Rewriting (23) as [ λ 2 (M T M T ) + λ( K + SM T + T M S ) SM S ] p = and using the fact that [ M M = F ] [ F M = G G F M G ] [ ] I F G. I We can show by careful calculation that the coefficient matrices in (23) satisfy those in (9). Corollary 3.3. [5] Let L(λ) and Q(λ) be defined in (5) and (8), respectively. Then σ(l(λ)) = σ(q(λ)) {,, }, }{{} m where σ( ) denotes the spectra of the associated matrix pencil. Theorem 3.4. Let Q(λ) in (8) with matrices A 2, A and A defined in (9). Then A 2 and A are positive definite. Furthermore there are n negative and n positive eigenvalues for Q(λ).
11 Transmission eigenvalues and practical reconstruction method Proof. From Lemma 3. and S being of full column rank, we get that the matrix A in (9c) is positive definite. On the other hand, from [ ] [ ] M F M F, ε F ε G G ε F ε it follows Let M F M F C + M ε F ε = M ε F ε. F G Fε G ε F Fε G I n I n L = I n I n, L 2 = I n F G, I m I m I n M M I n F G L 3 = I n, L 4 = I n. I m I m Then L 4 L 3 L 2 L CL L 2 L 3 L 4 M F I n I n = L 4 L 3 L 2 M M ε F I n L 2 L 3 L 4 F Fε G I m M M F = L 4 L 3 M I n M I n L 3 L 4 F F G F I m M M M M F I n = L 4 M M M M I n L 4 F G I m A 2 = M, G which implies that A 2. Let (λ, p) be an eigenpair of (8), then λ 2 (p H A 2 p) λ(p H A p) (p H A p) =. (24) Here and hereafter, the superscript H in (24) denotes the conjugate transpose. Because A is symmetric and A 2, A, we have a 2 p H A 2 p >, a p H A p R, a p H A p >, (25)
12 Transmission eigenvalues and practical reconstruction method 2 which implies that the roots of the quadratic equation (24) are λ + = a + a 2 + 4a 2 a >, λ = a a 2 + 4a 2 a <. (26) 2a 2 2a 2 Hence, there are n negative and n positive eigenvalues for (8) and the associated eigenvectors are real vectors. Theorem 3.5. Let [ M W = and Suppose that F ] /2 [ ] [ F K M G E, W = F d = W 2, d = W 2. ] /2 [ ] F M G F, (27) a = λ min (K) d d >, (28a) a = λ min (A ), ā = λ max (A ) = d 2, (28b) a 2 = λ min (A 2 ), ā 2 = λ max (A 2 ). (28c) Then the n negative and n positive eigenvalues of (8) are, respectively, in the intervals ( β, ) and (β, ), where β = 2ā a 2 + 4a 2 a + a, β = a + a 2 + 4a 2 a 2ā 2. (29) Proof. By the definitions of W and W, A in (9b) can be represented as A = K W W W W. (3) Given an orthogonal vector p, from (27) and (28a), equation (3) implies that p H A p = p H Kp p H W W p p H W W p λ min (K) d d = a >. (3) From (25)-(27) and (3), it follows that λ = 2a a 2 + 4a 2 a + a 2ā a 2 + 4a 2 a + a = β, and λ + = a + a 2 + 4a 2 a 2a 2 a + a 2 + 4a 2 a 2ā 2 = β.
13 Transmission eigenvalues and practical reconstruction method 3 4. Numerical method for reconstructing the unknown domain In this section, we will propose a practical numerical method for the reconstruction of the support of the target D based on the LSM and the truncated SVD technique. As described in the Introduction, let z be a sampling point in the sampling domain T (see Figure ) and k 2 is not a transmission eigenvalue for TEP (). According to Theorem., from (4) and (5), we can determine whether z is inside D or not by solving v L 2 (Γ) which satisfies the near field integral equation u s (x, y)v(y)ds(y) = Φ(x, z) for all x C, (32) Γ where Φ(x, z) is given in (3). To solve (32), we first discretize (32) as follows. For each point source x i C, i =,, m, suppose we have measured the scattered fields on the discrete points y,, y n on Γ. With these scattered fields, the discretized equation of (32) can be formulated as the overdetermined linear system Av = b, (33) where b = [Φ(x, z),, Φ(x m, z)] C m, A C m n depends on x i and y j for i =,, m and j =,, n, and the measured scattered fileds. v C n is the approximation to the unknown values [v(y ),, v(y n )]. In general, the linear system (33) is very sensitive due to the ill-posedness of (32). For the case of m < n, the problem (33) can be solved by the Tikhonov regularization method which gives min v C n{ Av b 2 + δ v 2 } (34) with δ > be a regularization parameter. The proper choice of regularization parameters plays an important role in (34) in order to achieve accurate and stable numerical results. Actually, over the last four decades, many different methods for selecting regularization parameters have been proposed [3, 9, 3]. Even so, the selection of the optimal regularization parameter remains a difficult question in solving the discretized ill-posed linear system (33). In this paper, we will propose a novel numerical method to reconstruct the domain D by solving the linear system (33) for the case of m n. Let A = span(a) be the subspace spanned by the columns of A. Recall from Section that dist(b, A) = min v C n Av b 2. From the perspective of matrix analysis, we reinterpret Theorem. in the following way. Assume that k 2 is not an eigenvalue of (8), for instance, we choose < k 2 < β. If dist(b, A) >, then v satisfying (33) is in the sense that some components of v are infinity. In other words, we have v =. We then assign z Ω \ D. On the other hand, if dist(b, A) =, then we can find a regular vector v with v < satisfying
14 Transmission eigenvalues and practical reconstruction method 4 (33). In this case, we say that z D. Our reconstruction method is based on this interpretation. We will use a truncated SVD to determine the size of v. Let A = U [ be the SVD of A, where U C m m and V C n n are unitary, Σ = diag(σ,, σ n ) is diagonal matrix of singular values with σ σ n. Usually, the discretized linear system (33) is sensitive that is inherited from the ill-posedness of (32), and the coefficient matrix A in (33) is not of full column rank. Suppose rank(a) = r < n, which means that σ r > σ r+ = = σ n =. Denote Σ r = diag(σ,, σ r ) and partition U and V with respect to Σ r as U = [U, U 2 ] and V = [V, V 2 ] with U C m r and V C n r, respectively. Obviously, the following statements hold true. and Σ ] A = U Σ r V H, A = span(u ) the subspace spanned by columns of U, (35) dist(b, A) 2 = min v C n Av b 2 2 = min v C r Σ rv U H b U H 2 b 2 2 = U H 2 b 2 2, where U Σ r V H is called the truncated SVD of A. Let v be a vector satisfying (33). Then we have [ ] [ ] [ ] [ ] [ ] ] [ˆb Av = b Σ r V H v V2 H v = U H b U H 2 b V H Σ r ˆv ˆv 2 = ˆb 2. (36) We interpret that v = is the solution of the equation v = c if c. Thus, from (36) follows that Σ rˆv = ˆb ˆv = Σ r ˆb ˆv <, and This implies that ˆv + ˆv 2 = ˆb 2 if ˆb 2, then ˆv =. if b span{u } rank(a) = rank([a, b]), then v < ; if b / span{u } rank(a) rank([a, b]), then v =. (37a) (37b) On the other hand, let  = [A, b], then [ ] [ ] [ ]  H V Σ r Σ r V H ˆb V Σ 2  = ˆb H ˆb H = rv H V Σ r ˆb. (38) 2 ˆb2 ˆb H Σ r V H b 2 2 It is easy to see that ÂH  in (38) is similar to Σ 2 r Σ r ˆb ˆb H Σ r b 2 2 ] [ Σ2 (39)
15 Transmission eigenvalues and practical reconstruction method 5 [ ] I by similarity transformations V I and. Here denotes the direct sum I of matrices. Let ˆσ ˆσ r+ be eigenvalues of Σ. It follows immediately that the singular values of  are ˆσ ˆσ r+ ˆσ r+2 = = ˆσ n+ =. In view of the special structure of matrix Σ in (39), we apply the interlacing theorem for singular values of Σ and Σ which leads to ˆσ σ ˆσ 2 σ 2 ˆσ r σ r ˆσ r+ σ r+ =. (4) For convenience, we set / =. Let σ n+ =, then we have ˆσ j /σ j < for j =,, r, and ˆσ j /σ j = for j = r + 2,, n +. (4) Combing (4) and (4), the statements (37) can be represented as b span{u } ˆσ r+ =, thus ˆσ r+ /σ r+ = / = ; b / span{u } ˆσ r+, thus ˆσ r+ /σ r+ = ˆσ r+ / =. (42a) (42b) In conclusion, in our method, to decide whether z D or not relies heavily on effectively determining the ranks of A and Â. However, since the integral equation (33) is ill-posed, the matrices A and  are normally ill-conditioned and there is often no gap in the spectrums of singular values. From numerical point of view, it is difficult to compute the truncated SVD for A. Consequently, we usually set r = n. We now propose a practical criterion to numerically realize (42). As above, let the singular values of A be σ σ n and those of  be ˆσ ˆσ n+. We first compute and define j(z) arg max j n ˆσ j σ j I z = ˆσ j(z). Here I z is used as an indicator to determine whether z is in D or not. Precisely, we pick a small threshold parameter < M. Then we choose the following dichotomy: if I z < M, then we set z D, if I z M, then we set z / D. The algorithm of computing the indicator I z is summarized in Algorithm below. 5. Numerical experiments In what follows, we will demonstrate the efficiency of our numerical method for four different domains shown in Figure 2: (a) a disk centered at (, ) with radius.5; (b) an ellipse region centered at the origin with axes.6 and.4; (c) a peanut-like region
16 Transmission eigenvalues and practical reconstruction method 6 Algorithm Practical numerical reconstruction method based on the singular values Input: Discrete point sources x i C, detection points y j Γ for i =,, m and j =,, n, respectively, with m > n. The wave number k R and sampling points z T. Output: The indicator I z. : For each point sources x i, collect all the measured scattered field u s ij for i =,, m, j =,, n. 2: Using the measured scattered field u s ij to discretize the integral of (32), then construct the coefficient matrix A C m n of (33). 3: Compute the SVD of A = U Σ n V H as in (35) with Σ n = diag(σ,, σ n ). 4: For the sampling point z, compute the vector on the right hand side of (33), b = [Φ(x[, z),, Φ(x m ], z)] T, where Φ(, ) is given in (3). 5: Let Σ Σ 2 n Σ nˆb with ˆb H Σ n b ˆb 2 = U H b. 2 6: Calculate the singular values of Σ as ˆσ ˆσ n+. 7: Determine the index j(z) = arg max j nˆσ j /σ j. 8: Set I z = ˆσ j(z). (a) Disk (b) Ellipse (c) Peanut (d) Heart Figure 2. Four model domains that represent the region D. enclosed by the equation ((x.7) 2 + x 2 2)((x +.7) 2 + x 2 2) =.72 4 ; (d) a heart-like region enclosed by the equation x 2 + x x 2 =.5 x 2 + x 2 2. All computations for numerical test examples are carried out in MATLAB 27a. For the hardware configuration, we use an HP server equipped with the RedHat Linux operating system, two Intel Quad-Core Xeon E GHz CPUs and 96 GB of main memory. 5.. Distribution of the transmission eigenvalues We carry out the FEM method described in Section 3 to the TEP () on four different domains as in Figure 2 with the regular mesh size h.4 for triangles of each domain D. We set ε r = 6, γ =, γ 2 = 4, and thus ε(x) =. The associated dimensions n and m of matrices given in Table are shown in Table 2. For each domain, we derive a corresponding QEP as given in (8). Because of the two negative signs in (8), we should modify the quadratic Jacobi-
17 Transmission eigenvalues and practical reconstruction method 7 Table 2. Dimensions n, m (K R n n, E R n m ) of matrices for the benchmark problems with the mesh size h.4. Domain Disk Ellipse Peanut Heart (n, m) (2463, 5) (7546, 976) (495, 87) (68548, 492) Heart Peanut Ellipse Disk Transmission eigenvalues Figure 3. The eigenvalues λ of the QEP (8) in the intervel [ 8, 25] for the four domains in Figure 2 with ε(x) =. The arrows point to the first positive eigenvalue of the QEP corresponding to each domain. Davidson method in [23] to solve these QEPs. Of course, the partial locking and partial deflation schemes of Algorithm 2 and Algorithm 3 in [23] can be employed as well. The numerical results are shown in Figure 3. Owing to the negative inhomogeneous medium ε(x) = ε r (γ 2 + γ 2 2) in TEP () derived from the pesudo-chiral model. From Figure 3, we immediately observe that there indeed exists an eigenvalue-free interval (, β ) = (,.3) (disk), (, 28.3) (ellipse), (, 3.3) (peanut) and (, 7.3) (heart), which verifies Theorem 3.5. Consequently, it is legitimate to say that k 2 is not a transmission eigenvalue if k 2 (, β ). In comparison, we note that the TEP (a) with the Tellegen model in [23] is reduced to E 3 = ω 2 (ε r + (γ 2 + γ 2 2))E 3 ω 2 ε(x)e 3 (43) with a large positive inhomogeneous medium ε(x), and its transmission eigenvalues are densely distributed on the interval (, O()) Reconstruction of the unknown domain from the near-field measurements In this subsection, we apply Algorithm to reconstruct the domain D from the near-field measurements for four distinct shapes as in Figure 2. As described above, our method is based on the LSM and the SVD technique. For LSM, as shown in Figure, we make the following preparations for numerical experiments. Consider each domain in Figure 2 to be the target D, and let the circles
18 Transmission eigenvalues and practical reconstruction method 8 with radii 3 and 6 be Γ and C, respectively. Choose the rectangle domain [, ] [, ] to be the sampling domain T, which contains all possible targets D. From Figure 3, we choose a k (, β ), in other words, k 2 is not a transmission eigenvalue. Different k (, β ) s are chosen for testing in our numerical simulations. Divid C and Γ into m and n segments uniformly, and denote the nodes as x,, x m and y,, y n, respectively. Place a uniform grid on T by drawing vertical and horizontal lines through the points with coordinates (x i, x 2j ), where x i = + ih for i =,, p and x 2j = + jh 2 for j =,, q, respectively. Let each mesh point (x i, x 2j ) to be the sampling point z ij. In our experiment, for these parameters, we choose m = 269, n = 693 and p = q = 2, respectively. To construct the discretized near-field integral equation, we need to collect all the scattered fields u s on the detection points y,, y n for each point source x i, i =,, m. For the purpose of numerical experiments, for a given point source x i, we can obtain the scattered fields in row [u s i,, u s in] by solving the direct scattering problem (2) using the FEM in Appendix A. Here, the FEM is applied to the domain enclosed by C, and the corresponding dimensions of E in Table are n = and m = 269, respectively. Adding 3% noise to the computed scattered fields u s to produce the detected waves, with which, we discretize (32) into an ill-posed overdetermined linear system (33) with A = [δ Γ u s ij(x i, y j )] m n C m n, where δ Γ = 6π/n is the arc length of the uniform segment of Γ. Applying Algorithm to each sampling point z ij T with all the above parameters to obtain the corresponding indicator I zij for i =,, p and j =,, q. From numerous numerical experiments by tuning k (, β ), we find that the reconstruction results will be heavily affected by the wave number k. In Figure 4, we show the 3d surface figures and the corresponding 2d contour figures by plotting I zij with respect to (x i, y j ) T, and the actual targets in Figure 2 are plotted by red lines. In fact, Figure 4 shows the best results for reconstructing targets listed in Figure 2 by choosing appropriate wave numbers k (, β ). In Figure 5, we show the 2d contour figures for other wave numbers k (, β ). According to our experimental experience, the areas of the reconstructed domains are less than the areas of the actual domains by approximately % 2%. Our numerical experiments also suggest that the reconstructions of convex domains are more stable with respect to the choice of the wave number k and noise Tellegen model positive index of refraction In comparison, we demonstrate some numerical results of reconstructing the unknown target D for the inverse scattering problem with positive index of refraction. In [24], we considered the Maxwell s equations with the TM mode in Tellegen media and derived the acoustic equation (a) with a positive ε(x) increasing with respect to the Tellegen parameters. When ε(x) is large enough, we have shown that the transmission eigenvalues are densely distributed near the origin. In this subsection, we show some numerical
19 Transmission eigenvalues and practical reconstruction method (a) Disk (b) Disk (c) Ellipse (d) Ellipse (e) Peanut (f) Peanut (g) Heart (h) Heart Figure 4. Reconstruction results of four targets in 3d surface figures and in 2d contour figures with ε(x) = and k (, β ). The domains enclosed by red curves are the exact targets D. The scattered fields used have 3% noise.
20 Transmission eigenvalues and practical reconstruction method (a) Disk (b) Ellipse (c) Peanut (d) Heart Figure 5. Reconstruction results of four targets in 2d contour figures with ε(x) = and the different k (, β ). The domains enclosed by red curves are the exact targets D. The scattered fields have 3% noise. reconstruction results based on our method when k 2 is an eigenvalue of the associated QEP (presumably a transmission eigenvalue). We test two different kinds of index of refraction on the disk domain, one is a normal index of refraction ε(x) = 6 [24] and the other is ε(x) = 5 (Tellegen model) [23]. As shown in [24], when ε(x) = 6, the lowest transmission eigenvalue is approximately.988. In Figure 6(a), 6(b), we see that the reconstruction results are very sensitive to the choice of the wave number k. For the case of ε(x) = 5, our numerical result in [23] indicates that the transmission eigenvalues are distribute densely in the interval (, 5). Figure 6(c) (choosing k = 5) shows that the disk is completely unrecognizable. The numerical results in this subsection verify the need of avoiding the transmission eigenvalues in the LSM. 6. Conclusion In this paper, we propose a practical numerical method based on the LSM and the truncated SVD to reconstruct the support of the inhomogeneity in the acoustic equation with negative index of refraction, resulting from in pseudo-chiral media. It turns out
21 Transmission eigenvalues and practical reconstruction method 2.8 Disk domain with (x)=6 and k= (a) Reconstruction result of a disk (k = 2).8 Disk domain with (x)=6 and k= (b) Reconstruction result of a disk (k = 2.) (c) Reconstruction result of a disk (large index of refraction) Figure 6. Numerical reconstruction results of a disk when k 2 is a transmission eigenvalue. The scattered fields have 3% noise. the index of refraction is in the form ε(x) = εr (γ2 + γ22 ). We are also interested in the distribution of transmission eigenvalues for the corresponding TEP. The associated discretized TEP can be reduced into a QEP with symmetric coefficient matrices whose all the nonphysical zero eigenvalues are deflated. We prove that the corresponding QEP has half of negative eigenvalues and half of positive eigenvalues in ( β, ) and (β, ), respectively, and there exists an eigenvalue-free interval (, β ). We then apply the LSM using the near-field data to reconstruct the domain D with ε(x) 6= and propose an
22 Transmission eigenvalues and practical reconstruction method 22 easily implemented truncated SVD technique to determine whether a sampling point is inside or outside of D according to the size of the indicator function. Numerical results show that the effectiveness of our method depends on the wave numbers of incident fields and the convexity of the target. 7. Acknowledgments Li is supported in parts by the NSFC Huang was partially supported by the Ministry of Science and Technology (MOST) 5-25-M-3-9-MY3, National Center of Theoretical Sciences (NCTS) in Taiwan. Lin was partially supported by MOST, NCTS and ST Yau Center in Taiwan. Wang was partially supported by MOST 5-25-M-2-4-MY3. Appendix A. An FEM framework for the direct scattering problem In this appendix, we will present a numerical framework of solving the direct scattering problem (2) by the FEM given in Section 3. As described in Figure, let u i = Φ(x, x ) be an incident field with x C. Suppose C is far from the origin, since u i + k 2 u i =, we have (u i + u s ) + k 2 ε(x)(u i + u s ) = u s + k 2 ε(x)u s + u i + k 2 u i + k 2 (ε(x) )u i =( + k 2 ε(x))u s + k 2 (ε(x) )u i =, x R 2 \ {x }, x C, (44a) and u s r = ikus C. (44b) Denote C the closed domain enclosed by C. Applying the standard FEM framework in Section 3 on C, we define in addition the stiffness matrix for boundary meshes as S = [( ψ i, ψ j )] R m m, (45) and the mass matrices for boundary condition as R = [ ψ i ψ j ds] = [ 6 ( + δ ij) e ij ] R m m, (46) C where e ij is the length of boundary edge e ij and δ ij being the Kronecker delta function. Let n u s = w i φ i + v i ψ i, and i= u i = Φ(x, x ) = i= n f i φ i + i= g i ψ i, i=
23 Transmission eigenvalues and practical reconstruction method 23 it follows from (44a) that =(( + k 2 ε(x))u s, φ j ) + (k 2 (ε(x) )u i, φ j ) =( u s, φ j ) + k 2 (ε(x)u s, φ j ) + k 2 (ε(x)φ(x, x ), φ j ) k 2 (Φ(x, x ), φ j ). (47) Utilizing the integration by parts and setting w = [w,, w n ], v = [v,, v m ], f = [f,, f n ] and g = [g,, g m ], (47) can be written as ( u s, φ j ) + k 2 (ε(x)u s, φ j ) = ( u s, φ j ) + k 2 (ε(x)u s, φ j ) ( n ) ( n = w i ( φ i, φ j ) + v i ( ψ i, φ j ) + k 2 w i (εφ i, φ j ) + i= i= ( n = w i K ij + i= i= ) ( n v i E ji k 2 w i M εij + i= i= ) v i F εji i= ) v i (εψ i, φ j ) = Kw Ev k 2 M ε w k 2 F ε v, (48) and k 2 (ε(x)φ(x, x ), φ j ) k 2 (Φ(x, x ), φ j ) ( n ) ( n ) =k 2 f i (εφ i, φ j ) + g i (εψ i, φ j ) k 2 f i (φ i, φ j ) + g i (ψ i, φ j ) i= i= i= i= n m n m = k 2 f i M εij k 2 g i F εji k 2 f i M ij k 2 g i F ji i= i= i= = k 2 (M ε + M )f k 2 (F ε + F )g, (49) where the stiffness matrices K, E and mass matrices M, M ε, F, F ε are defined as in Table with domain D being replaced by C. Continuing to apply the FEM and integration by parts to (44a), combining with i= i=
24 Transmission eigenvalues and practical reconstruction method 24 (44b), we obtain that ( u s, ψ j ) + k 2 (ε(x)u s, ψ j ) = ( u s, ψ j ) + n u s ψ j ds + k 2 (ε(x)u s, ψ j ) C ( n ) = w i ( φ i, ψ j ) + v i ( ψ i, ψ j ) + iku s ψ j ds i= + k 2 ( n i= w i (εφ i, ψ j ) + n = w i E ij i= i= k 2 ( n i= w i F εij + i= ) v i (εψ i, ψ j ) i= ( n v i S ij + ik w i ) v i G εij i= i= C ) φ i ψ j ds + v i ψ i ψ j ds C i= C = (E T + k 2 Fε T )w (S + k 2 G ε )v + ikrv (5) and k 2 (ε(x)φ(x, x ), ψ j ) k 2 (Φ(x, x ), ψ j ) ( n ) ( n =k 2 f i (εφ i, ψ j ) + g i (εψ i, ψ j ) k 2 f i (φ i, ψ j ) + i= i= i= n m n m = k 2 f i F εij k 2 g i G εij k 2 f i F ij k 2 g i G ij i= i= i= i= ) g i (ψ i, ψ j ) = k 2 (F T + F T ε )f k 2 (G + G ε )g, (5) i= where S and R are given in (45) and (46), and similarly, the mass matrices G and G ε are defined as in Table on domain C. Finally, we can arrange (48), (49), (5) and (5) into a linear system [ ] [ ] [ ] [ ] K + k 2 M ε E + k 2 F ε w E T + k 2 Fε T S + k 2 = k 2 M ε + M F + F ε f G ε ikr v F T + Fε T. (52) G + G ε g The approximate scattered field u s is then obtained by solving (52). References [] F. Cakoni and D. Colton. Qualitative methods in inverse scattering theory: An introduction. Springer Science & Business Media, 25. [2] F. Cakoni, D. Colton, and H. Haddar. On the determination of Dirichlet or transmission eigenvalues from far field data. C. R. Math. Acad. Sci. Pairs, 348(7-8): , 2. [3] F. Cakoni, D. Colton, P. Monk, and J. Sun. The inverse electromagnetic scattering problem for anisotropic media. Inv. Prob., 26(7):744, 2.
25 Transmission eigenvalues and practical reconstruction method 25 [4] F. Cakoni, D. Gintides, and H. Haddar. The existence of an infinite discrete set of transmission eigenvalues. SIAM J. Math. Anal., 42(): , 2. [5] F. Cakoni and H. Haddar. On the existence of transmission eigenvalues in an inhomogeneous medium. Appl. Anal., 88(4): , 29. [6] F. Cakoni and H. Haddar. Transmission eigenvalues in inverse scattering theory. In G. Uhlmann, editor, Inverse Problems and Applications: Inside Out II, volume 6 of Math. Sci. Res. Inst. Publ., pages Cambridge University Press, Cambridge, 22. [7] D. Colton and H. Haddar. An application of the reciprocity gap functional to inverse scattering theory. Inv. Prob., 2:383 98, 25. [8] D. Colton and A. Kirsch. A simple method for solving inverse scattering problems in the resonance region. Inv. Prob., 2(4):383, 996. [9] D. Colton and R. Kress. Inverse Acoustic and Electromagnetic Scattering Theory, volume 93 of Applied Mathematical Sciences. Springer, New York, 3rd edition, 23. [] D. Colton and P. Monk. A novel method for solving the inverse scattering problem for timeharmonic acoustic waves in the resonance region. SIAM J. Appl. Math., 45:39 53, 985. [] D. Colton, P. Monk, and J. Sun. Analytical and computational methods for transmission eigenvalues. Inv. Prob., 26:45, 2. [2] D. Colton, L. Päivärinta, and J. Sylvester. The interior transmission problem. Inv. Prob. Imaging, ():3 28, 27. [3] P.C. Hansen. Discrete Inverse Problems: Insight and Algorithms. SIAM, Philadelphia, PA, 2. [4] G. C. Hsiao, F. Liu, J. Sun, and L. Xu. A coupled BEM and FEM for the interior transmission problem in acoustics. J. Comput. Appl. Math., 235: , 2. [5] T.-M. Huang, W.-Q. Huang, and W.-W. Lin. A robust numerical algorithm for computing Maxwell s transmission eigenvalue problems. SIAM J. Sci. Comput., 37:A243 A2423, 25. [6] T.-M. Hwang, W.-W. Lin, W.-C. Wang, and W. Wang. Numerical simulation of three dimensional pyramid quantum dot. J. Comput. Phys., 96:28 232, 24. [7] X. Ji, J. Sun, and T. Turner. Algorithm 922: A mixed finite element method for Helmholtz transmission eigenvalues. ACM Trans. Math. Software, 38:29, 22. [8] X. Ji, J. Sun, and H. Xie. A multigrid method for Helmholtz transmission eigenvalue problems. J. Sci. Comput., 6(2): , 24. [9] M.E. Kilmer and D.P. O leary. Choosing regularization parameters in iterative methods for illposed problems. SIAM J. Matrix Anal. Appl., 22(4):24 22, 2. [2] A. Kirsch. On the existence of transmission eigenvalues. Inv. Prob. Imaging, 3(2):55 72, 29. [2] A. Kirsch and N. Grinberg. The factorization method for inverse problems. Number 36. Oxford University Press, 28. [22] A. Kleefeld. Numerical methods for acoustic and electromagnetic scattering: Transmission boundary-value problems, interior transmission eigenvalues, and the factorization method. Habilitation Thesis, May 25. [23] T. Li, T. M. Huang, W. W. Lin, and J. N. Wang. An efficient numerical algorithm for computing densely distributed positive interior transmission eigenvalues. Inv. Prob., 33:359, 27. [24] T. Li, W.-Q. Huang, W.-W. Lin, and J. Liu. On spectral analysis and a novel algorithm for transmission eigenvalue problems. J. Sci. Comput., 64():83 8, 25. [25] L. Päivärinta and J. Sylvester. Transmission eigenvalues. SIAM J. Math. Anal., 4(2): , 28. [26] A. Serdyukov, I. Semchenko, S. Tretyakov, and A. Sihvola. Electromagnetics of Bi-anisotropic Materials: Theory and Applications. Gordon and Breach Science, 2. [27] J. Sun. Estimation of transmission eigenvalues and the index of refraction from cauchy data. Inv. Prob., 27():59, 2. [28] J. Sun. Iterative methods for transmission eigenvalues. SIAM J. Numer. Anal., 49(5):86 874, 2. [29] J. Sun. An eigenvalue method using multiple frequency data for inverse scattering problems. Inv.
26 Transmission eigenvalues and practical reconstruction method 26 Prob., 28(2):252, 22. [3] J. Sun and A. Zhou. Finite Element Methods for Eigenvalue Problems. Boca Raton: CRC Press, 27. [3] A.N. Tikhonov and V.Y. Arsenin. Solutions of Ill-Posed Problems. Winston Sons, Washington, D.C., 977.
An eigenvalue method using multiple frequency data for inverse scattering problems
An eigenvalue method using multiple frequency data for inverse scattering problems Jiguang Sun Abstract Dirichlet and transmission eigenvalues have important applications in qualitative methods in inverse
More informationEstimation of transmission eigenvalues and the index of refraction from Cauchy data
Estimation of transmission eigenvalues and the index of refraction from Cauchy data Jiguang Sun Abstract Recently the transmission eigenvalue problem has come to play an important role and received a lot
More informationA coupled BEM and FEM for the interior transmission problem
A coupled BEM and FEM for the interior transmission problem George C. Hsiao, Liwei Xu, Fengshan Liu, Jiguang Sun Abstract The interior transmission problem (ITP) is a boundary value problem arising in
More informationA probing method for the transmission eigenvalue problem
A probing method for the transmission eigenvalue problem Fang Zeng Jiguang Sun Liwe Xu Abstract In this paper, we consider an integral eigenvalue problem, which is a reformulation of the transmission eigenvalue
More informationInverse Scattering Theory: Transmission Eigenvalues and Non-destructive Testing
Inverse Scattering Theory: Transmission Eigenvalues and Non-destructive Testing Isaac Harris Texas A & M University College Station, Texas 77843-3368 iharris@math.tamu.edu Joint work with: F. Cakoni, H.
More informationThe Imaging of Anisotropic Media in Inverse Electromagnetic Scattering
The Imaging of Anisotropic Media in Inverse Electromagnetic Scattering Fioralba Cakoni Department of Mathematical Sciences University of Delaware Newark, DE 19716, USA email: cakoni@math.udel.edu Research
More informationRecursive integral method for transmission eigenvalues
Recursive integral method for transmission eigenvalues Ruihao Huang Allan A. Struthers Jiguang Sun Ruming Zhang Abstract Recently, a new eigenvalue problem, called the transmission eigenvalue problem,
More informationFactorization method in inverse
Title: Name: Affil./Addr.: Factorization method in inverse scattering Armin Lechleiter University of Bremen Zentrum für Technomathematik Bibliothekstr. 1 28359 Bremen Germany Phone: +49 (421) 218-63891
More informationITERATIVE METHODS FOR TRANSMISSION EIGENVALUES
ITERATIVE METHODS FOR TRANSMISSION EIGENVALUES JIGUANG SUN Abstract. Transmission eigenvalues have important applications in inverse scattering theory. They can be used to obtain useful information of
More informationDiscreteness of Transmission Eigenvalues via Upper Triangular Compact Operators
Discreteness of Transmission Eigenvalues via Upper Triangular Compact Operators John Sylvester Department of Mathematics University of Washington Seattle, Washington 98195 U.S.A. June 3, 2011 This research
More informationThe Interior Transmission Eigenvalue Problem for Maxwell s Equations
The Interior Transmission Eigenvalue Problem for Maxwell s Equations Andreas Kirsch MSRI 2010 epartment of Mathematics KIT University of the State of Baden-Württemberg and National Large-scale Research
More informationON THE EXISTENCE OF TRANSMISSION EIGENVALUES. Andreas Kirsch1
Manuscript submitted to AIMS Journals Volume 3, Number 2, May 29 Website: http://aimsciences.org pp. 1 XX ON THE EXISTENCE OF TRANSMISSION EIGENVALUES Andreas Kirsch1 University of Karlsruhe epartment
More informationNEW RESULTS ON TRANSMISSION EIGENVALUES. Fioralba Cakoni. Drossos Gintides
Inverse Problems and Imaging Volume 0, No. 0, 0, 0 Web site: http://www.aimsciences.org NEW RESULTS ON TRANSMISSION EIGENVALUES Fioralba Cakoni epartment of Mathematical Sciences University of elaware
More informationComputation of Maxwell s transmission eigenvalues and its applications in inverse medium problems
Computation of Maxwell s transmission eigenvalues and its applications in inverse medium problems Jiguang Sun 1 and Liwei Xu 2,3 1 Department of Mathematical Sciences, Michigan Technological University,
More informationInverse Scattering Theory and Transmission Eigenvalues
Inverse Scattering Theory and Transmission Eigenvalues David Colton Department of Mathematical Sciences University of Delaware Newark, DE 19716, USA email: colton@udel.edu Research supported a grant from
More informationTransmission Eigenvalues in Inverse Scattering Theory
Transmission Eigenvalues in Inverse Scattering Theory Fioralba Cakoni Department of Mathematical Sciences University of Delaware Newark, DE 19716, USA email: cakoni@math.udel.edu Jointly with D. Colton,
More informationTransmission Eigenvalues in Inverse Scattering Theory
Transmission Eigenvalues in Inverse Scattering Theory David Colton Department of Mathematical Sciences University of Delaware Newark, DE 19716, USA email: colton@math.udel.edu Research supported by a grant
More informationPolynomial Jacobi Davidson Method for Large/Sparse Eigenvalue Problems
Polynomial Jacobi Davidson Method for Large/Sparse Eigenvalue Problems Tsung-Ming Huang Department of Mathematics National Taiwan Normal University, Taiwan April 28, 2011 T.M. Huang (Taiwan Normal Univ.)
More informationA Direct Method for reconstructing inclusions from Electrostatic Data
A Direct Method for reconstructing inclusions from Electrostatic Data Isaac Harris Texas A&M University, Department of Mathematics College Station, Texas 77843-3368 iharris@math.tamu.edu Joint work with:
More informationA Domain Decomposition Based Jacobi-Davidson Algorithm for Quantum Dot Simulation
A Domain Decomposition Based Jacobi-Davidson Algorithm for Quantum Dot Simulation Tao Zhao 1, Feng-Nan Hwang 2 and Xiao-Chuan Cai 3 Abstract In this paper, we develop an overlapping domain decomposition
More informationScattering of electromagnetic waves by thin high contrast dielectrics II: asymptotics of the electric field and a method for inversion.
Scattering of electromagnetic waves by thin high contrast dielectrics II: asymptotics of the electric field and a method for inversion. David M. Ambrose Jay Gopalakrishnan Shari Moskow Scott Rome June
More informationTransmission eigenvalues and non-destructive testing of anisotropic magnetic materials with voids
Transmission eigenvalues and non-destructive testing of anisotropic magnetic materials with voids Isaac Harris 1, Fioralba Cakoni 1 and Jiguang Sun 2 1 epartment of Mathematical Sciences, University of
More informationarxiv: v1 [math.na] 1 Sep 2018
On the perturbation of an L -orthogonal projection Xuefeng Xu arxiv:18090000v1 [mathna] 1 Sep 018 September 5 018 Abstract The L -orthogonal projection is an important mathematical tool in scientific computing
More informationFINITE-DIMENSIONAL LINEAR ALGEBRA
DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H ROSEN FINITE-DIMENSIONAL LINEAR ALGEBRA Mark S Gockenbach Michigan Technological University Houghton, USA CRC Press Taylor & Francis Croup
More informationDomain Decomposition Preconditioners for Spectral Nédélec Elements in Two and Three Dimensions
Domain Decomposition Preconditioners for Spectral Nédélec Elements in Two and Three Dimensions Bernhard Hientzsch Courant Institute of Mathematical Sciences, New York University, 51 Mercer Street, New
More informationThe inverse transmission eigenvalue problem
The inverse transmission eigenvalue problem Drossos Gintides Department of Mathematics National Technical University of Athens Greece email: dgindi@math.ntua.gr. Transmission Eigenvalue Problem Find (w,
More informationHomogenization of the Transmission Eigenvalue Problem for a Periodic Media
Homogenization of the Transmission Eigenvalue Problem for a Periodic Media Isaac Harris Texas A&M University, Department of Mathematics College Station, Texas 77843-3368 iharris@math.tamu.edu Joint work
More informationNotes on Transmission Eigenvalues
Notes on Transmission Eigenvalues Cédric Bellis December 28, 2011 Contents 1 Scattering by inhomogeneous medium 1 2 Inverse scattering via the linear sampling method 2 2.1 Relationship with the solution
More informationComplex geometrical optics solutions for Lipschitz conductivities
Rev. Mat. Iberoamericana 19 (2003), 57 72 Complex geometrical optics solutions for Lipschitz conductivities Lassi Päivärinta, Alexander Panchenko and Gunther Uhlmann Abstract We prove the existence of
More informationThe Factorization Method for Inverse Scattering Problems Part I
The Factorization Method for Inverse Scattering Problems Part I Andreas Kirsch Madrid 2011 Department of Mathematics KIT University of the State of Baden-Württemberg and National Large-scale Research Center
More informationDELFT UNIVERSITY OF TECHNOLOGY
DELFT UNIVERSITY OF TECHNOLOGY REPORT -09 Computational and Sensitivity Aspects of Eigenvalue-Based Methods for the Large-Scale Trust-Region Subproblem Marielba Rojas, Bjørn H. Fotland, and Trond Steihaug
More informationTHE PERTURBATION BOUND FOR THE SPECTRAL RADIUS OF A NON-NEGATIVE TENSOR
THE PERTURBATION BOUND FOR THE SPECTRAL RADIUS OF A NON-NEGATIVE TENSOR WEN LI AND MICHAEL K. NG Abstract. In this paper, we study the perturbation bound for the spectral radius of an m th - order n-dimensional
More informationThe inverse transmission eigenvalue problem
The inverse transmission eigenvalue problem Drossos Gintides Department of Mathematics National Technical University of Athens Greece email: dgindi@math.ntua.gr. Transmission Eigenvalue Problem Find (w,
More informationCompleteness of the generalized transmission eigenstates
Completeness of the generalized transmission eigenstates Eemeli Blåsten Department of Mathematics and Statistics, University of Helsinki Research carried out at the Mittag-Leffler Institute Joint work
More informationA DIVIDE-AND-CONQUER METHOD FOR THE TAKAGI FACTORIZATION
SIAM J MATRIX ANAL APPL Vol 0, No 0, pp 000 000 c XXXX Society for Industrial and Applied Mathematics A DIVIDE-AND-CONQUER METHOD FOR THE TAKAGI FACTORIZATION WEI XU AND SANZHENG QIAO Abstract This paper
More informationBEHAVIOR OF THE REGULARIZED SAMPLING INVERSE SCATTERING METHOD AT INTERNAL RESONANCE FREQUENCIES
Progress In Electromagnetics Research, PIER 38, 29 45, 2002 BEHAVIOR OF THE REGULARIZED SAMPLING INVERSE SCATTERING METHOD AT INTERNAL RESONANCE FREQUENCIES N. Shelton and K. F. Warnick Department of Electrical
More informationSPECTRAL APPROXIMATION TO A TRANSMISSION EIGENVALUE PROBLEM AND ITS APPLICATIONS TO AN INVERSE PROBLEM
SPECTRAL APPROXIMATIO TO A TRASMISSIO EIGEVALUE PROBLEM AD ITS APPLICATIOS TO A IVERSE PROBLEM JIG A JIE SHE 2,3 Abstract. We first develop an efficient spectral-galerkin method and an rigorous error analysis
More informationThe Singular Value Decomposition (SVD) and Principal Component Analysis (PCA)
Chapter 5 The Singular Value Decomposition (SVD) and Principal Component Analysis (PCA) 5.1 Basics of SVD 5.1.1 Review of Key Concepts We review some key definitions and results about matrices that will
More informationLecture 6. Numerical methods. Approximation of functions
Lecture 6 Numerical methods Approximation of functions Lecture 6 OUTLINE 1. Approximation and interpolation 2. Least-square method basis functions design matrix residual weighted least squares normal equation
More informationThe Singular Value Decomposition
The Singular Value Decomposition Philippe B. Laval KSU Fall 2015 Philippe B. Laval (KSU) SVD Fall 2015 1 / 13 Review of Key Concepts We review some key definitions and results about matrices that will
More informationComparative Analysis of Mesh Generators and MIC(0) Preconditioning of FEM Elasticity Systems
Comparative Analysis of Mesh Generators and MIC(0) Preconditioning of FEM Elasticity Systems Nikola Kosturski and Svetozar Margenov Institute for Parallel Processing, Bulgarian Academy of Sciences Abstract.
More informationA Multigrid Method for Two Dimensional Maxwell Interface Problems
A Multigrid Method for Two Dimensional Maxwell Interface Problems Susanne C. Brenner Department of Mathematics and Center for Computation & Technology Louisiana State University USA JSA 2013 Outline A
More informationREGULARIZATION PARAMETER SELECTION IN DISCRETE ILL POSED PROBLEMS THE USE OF THE U CURVE
Int. J. Appl. Math. Comput. Sci., 007, Vol. 17, No., 157 164 DOI: 10.478/v10006-007-0014-3 REGULARIZATION PARAMETER SELECTION IN DISCRETE ILL POSED PROBLEMS THE USE OF THE U CURVE DOROTA KRAWCZYK-STAŃDO,
More informationA new method for the solution of scattering problems
A new method for the solution of scattering problems Thorsten Hohage, Frank Schmidt and Lin Zschiedrich Konrad-Zuse-Zentrum Berlin, hohage@zibde * after February 22: University of Göttingen Abstract We
More informationA MODIFIED TSVD METHOD FOR DISCRETE ILL-POSED PROBLEMS
A MODIFIED TSVD METHOD FOR DISCRETE ILL-POSED PROBLEMS SILVIA NOSCHESE AND LOTHAR REICHEL Abstract. Truncated singular value decomposition (TSVD) is a popular method for solving linear discrete ill-posed
More informationDS-GA 1002 Lecture notes 0 Fall Linear Algebra. These notes provide a review of basic concepts in linear algebra.
DS-GA 1002 Lecture notes 0 Fall 2016 Linear Algebra These notes provide a review of basic concepts in linear algebra. 1 Vector spaces You are no doubt familiar with vectors in R 2 or R 3, i.e. [ ] 1.1
More informationIncreasing stability in an inverse problem for the acoustic equation
Increasing stability in an inverse problem for the acoustic equation Sei Nagayasu Gunther Uhlmann Jenn-Nan Wang Abstract In this work we study the inverse boundary value problem of determining the refractive
More informationRegularization methods for large-scale, ill-posed, linear, discrete, inverse problems
Regularization methods for large-scale, ill-posed, linear, discrete, inverse problems Silvia Gazzola Dipartimento di Matematica - Università di Padova January 10, 2012 Seminario ex-studenti 2 Silvia Gazzola
More informationCOMPLEX SPHERICAL WAVES AND INVERSE PROBLEMS IN UNBOUNDED DOMAINS
COMPLEX SPHERICAL WAVES AND INVERSE PROBLEMS IN UNBOUNDED DOMAINS MIKKO SALO AND JENN-NAN WANG Abstract. This work is motivated by the inverse conductivity problem of identifying an embedded object in
More informationInverse Obstacle Scattering
, Göttingen AIP 2011, Pre-Conference Workshop Texas A&M University, May 2011 Scattering theory Scattering theory is concerned with the effects that obstacles and inhomogenities have on the propagation
More informationPreliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012
Instructions Preliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012 The exam consists of four problems, each having multiple parts. You should attempt to solve all four problems. 1.
More informationWeighted Regularization of Maxwell Equations Computations in Curvilinear Polygons
Weighted Regularization of Maxwell Equations Computations in Curvilinear Polygons Martin Costabel, Monique Dauge, Daniel Martin and Gregory Vial IRMAR, Université de Rennes, Campus de Beaulieu, Rennes,
More informationUniqueness in determining refractive indices by formally determined far-field data
Applicable Analysis, 2015 Vol. 94, No. 6, 1259 1269, http://dx.doi.org/10.1080/00036811.2014.924215 Uniqueness in determining refractive indices by formally determined far-field data Guanghui Hu a, Jingzhi
More informationSTEKLOFF EIGENVALUES AND INVERSE SCATTERING THEORY
STEKLOFF EIGENVALUES AND INVERSE SCATTERING THEORY David Colton, Shixu Meng, Peter Monk University of Delaware Fioralba Cakoni Rutgers University Research supported by AFOSR Grant FA 9550-13-1-0199 Scattering
More informationFinite Element Modelling of Finite Single and Double Quantum Wells
Finite Element Modelling of Finite Single and Double Quantum Wells A Major Qualifying Project Report Submitted to the Faculty of the WORCESTER POLYTECHNIC INSTITUTE In partial fulfilment of the requirements
More informationDELFT UNIVERSITY OF TECHNOLOGY
DELFT UNIVERSITY OF TECHNOLOGY REPORT 10-12 Large-Scale Eigenvalue Problems in Trust-Region Calculations Marielba Rojas, Bjørn H. Fotland, and Trond Steihaug ISSN 1389-6520 Reports of the Department of
More informationElectromagnetic Modeling and Simulation
Electromagnetic Modeling and Simulation Erin Bela and Erik Hortsch Department of Mathematics Research Experiences for Undergraduates April 7, 2011 Bela and Hortsch (OSU) EM REU 2010 1 / 45 Maxwell s Equations
More informationBasic Elements of Linear Algebra
A Basic Review of Linear Algebra Nick West nickwest@stanfordedu September 16, 2010 Part I Basic Elements of Linear Algebra Although the subject of linear algebra is much broader than just vectors and matrices,
More informationNonlinear Integral Equations for the Inverse Problem in Corrosion Detection from Partial Cauchy Data. Fioralba Cakoni
Nonlinear Integral Equations for the Inverse Problem in Corrosion Detection from Partial Cauchy Data Fioralba Cakoni Department of Mathematical Sciences, University of Delaware email: cakoni@math.udel.edu
More informationThe Solvability Conditions for the Inverse Eigenvalue Problem of Hermitian and Generalized Skew-Hamiltonian Matrices and Its Approximation
The Solvability Conditions for the Inverse Eigenvalue Problem of Hermitian and Generalized Skew-Hamiltonian Matrices and Its Approximation Zheng-jian Bai Abstract In this paper, we first consider the inverse
More informationInverse wave scattering problems: fast algorithms, resonance and applications
Inverse wave scattering problems: fast algorithms, resonance and applications Wagner B. Muniz Department of Mathematics Federal University of Santa Catarina (UFSC) w.b.muniz@ufsc.br III Colóquio de Matemática
More informationDomain decomposition on different levels of the Jacobi-Davidson method
hapter 5 Domain decomposition on different levels of the Jacobi-Davidson method Abstract Most computational work of Jacobi-Davidson [46], an iterative method suitable for computing solutions of large dimensional
More informationSolving Differential Equations on 2-D Geometries with Matlab
Solving Differential Equations on 2-D Geometries with Matlab Joshua Wall Drexel University Philadelphia, PA 19104 (Dated: April 28, 2014) I. INTRODUCTION Here we introduce the reader to solving partial
More informationNumerical Methods I Non-Square and Sparse Linear Systems
Numerical Methods I Non-Square and Sparse Linear Systems Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 MATH-GA 2011.003 / CSCI-GA 2945.003, Fall 2014 September 25th, 2014 A. Donev (Courant
More informationON THE MATHEMATICAL BASIS OF THE LINEAR SAMPLING METHOD
Georgian Mathematical Journal Volume 10 (2003), Number 3, 411 425 ON THE MATHEMATICAL BASIS OF THE LINEAR SAMPLING METHOD FIORALBA CAKONI AND DAVID COLTON Dedicated to the memory of Professor Victor Kupradze
More informationAN INVERSE EIGENVALUE PROBLEM AND AN ASSOCIATED APPROXIMATION PROBLEM FOR GENERALIZED K-CENTROHERMITIAN MATRICES
AN INVERSE EIGENVALUE PROBLEM AND AN ASSOCIATED APPROXIMATION PROBLEM FOR GENERALIZED K-CENTROHERMITIAN MATRICES ZHONGYUN LIU AND HEIKE FAßBENDER Abstract: A partially described inverse eigenvalue problem
More information13-2 Text: 28-30; AB: 1.3.3, 3.2.3, 3.4.2, 3.5, 3.6.2; GvL Eigen2
The QR algorithm The most common method for solving small (dense) eigenvalue problems. The basic algorithm: QR without shifts 1. Until Convergence Do: 2. Compute the QR factorization A = QR 3. Set A :=
More informationThe linear sampling method for three-dimensional inverse scattering problems
ANZIAM J. 42 (E) ppc434 C46, 2 C434 The linear sampling method for three-dimensional inverse scattering problems David Colton Klaus Giebermann Peter Monk (Received 7 August 2) Abstract The inverse scattering
More informationHomework 1. Yuan Yao. September 18, 2011
Homework 1 Yuan Yao September 18, 2011 1. Singular Value Decomposition: The goal of this exercise is to refresh your memory about the singular value decomposition and matrix norms. A good reference to
More informationA Robust Preconditioner for the Hessian System in Elliptic Optimal Control Problems
A Robust Preconditioner for the Hessian System in Elliptic Optimal Control Problems Etereldes Gonçalves 1, Tarek P. Mathew 1, Markus Sarkis 1,2, and Christian E. Schaerer 1 1 Instituto de Matemática Pura
More informationσ 11 σ 22 σ pp 0 with p = min(n, m) The σ ii s are the singular values. Notation change σ ii A 1 σ 2
HE SINGULAR VALUE DECOMPOSIION he SVD existence - properties. Pseudo-inverses and the SVD Use of SVD for least-squares problems Applications of the SVD he Singular Value Decomposition (SVD) heorem For
More informationNonlinear Eigenvalue Problems: An Introduction
Nonlinear Eigenvalue Problems: An Introduction Cedric Effenberger Seminar for Applied Mathematics ETH Zurich Pro*Doc Workshop Disentis, August 18 21, 2010 Cedric Effenberger (SAM, ETHZ) NLEVPs: An Introduction
More informationKarhunen-Loève Approximation of Random Fields Using Hierarchical Matrix Techniques
Institut für Numerische Mathematik und Optimierung Karhunen-Loève Approximation of Random Fields Using Hierarchical Matrix Techniques Oliver Ernst Computational Methods with Applications Harrachov, CR,
More informationA posteriori error estimates for the adaptivity technique for the Tikhonov functional and global convergence for a coefficient inverse problem
A posteriori error estimates for the adaptivity technique for the Tikhonov functional and global convergence for a coefficient inverse problem Larisa Beilina Michael V. Klibanov December 18, 29 Abstract
More informationOPTIMAL SCALING FOR P -NORMS AND COMPONENTWISE DISTANCE TO SINGULARITY
published in IMA Journal of Numerical Analysis (IMAJNA), Vol. 23, 1-9, 23. OPTIMAL SCALING FOR P -NORMS AND COMPONENTWISE DISTANCE TO SINGULARITY SIEGFRIED M. RUMP Abstract. In this note we give lower
More informationLecture notes: Applied linear algebra Part 1. Version 2
Lecture notes: Applied linear algebra Part 1. Version 2 Michael Karow Berlin University of Technology karow@math.tu-berlin.de October 2, 2008 1 Notation, basic notions and facts 1.1 Subspaces, range and
More informationIntroduction. Chapter One
Chapter One Introduction The aim of this book is to describe and explain the beautiful mathematical relationships between matrices, moments, orthogonal polynomials, quadrature rules and the Lanczos and
More informationNotes on singular value decomposition for Math 54. Recall that if A is a symmetric n n matrix, then A has real eigenvalues A = P DP 1 A = P DP T.
Notes on singular value decomposition for Math 54 Recall that if A is a symmetric n n matrix, then A has real eigenvalues λ 1,, λ n (possibly repeated), and R n has an orthonormal basis v 1,, v n, where
More informationA Divide-and-Conquer Method for the Takagi Factorization
A Divide-and-Conquer Method for the Takagi Factorization Wei Xu 1 and Sanzheng Qiao 1, Department of Computing and Software, McMaster University Hamilton, Ont, L8S 4K1, Canada. 1 xuw5@mcmaster.ca qiao@mcmaster.ca
More informationMath 350 Fall 2011 Notes about inner product spaces. In this notes we state and prove some important properties of inner product spaces.
Math 350 Fall 2011 Notes about inner product spaces In this notes we state and prove some important properties of inner product spaces. First, recall the dot product on R n : if x, y R n, say x = (x 1,...,
More informationFAME-matlab Package: Fast Algorithm for Maxwell Equations Tsung-Ming Huang
FAME-matlab Package: Fast Algorithm for Maxwell Equations Tsung-Ming Huang Modelling, Simulation and Analysis of Nonlinear Optics, NUK, September, 4-8, 2015 1 2 FAME group Wen-Wei Lin Department of Applied
More informationTransmission eigenvalues with artificial background for explicit material index identification
Transmission eigenvalues with artificial background for explicit material index identification Lorenzo Audibert 1,, Lucas Chesnel, Houssem Haddar 1 Department STEP, EDF R&D, 6 quai Watier, 78401, Chatou
More informationFall TMA4145 Linear Methods. Exercise set Given the matrix 1 2
Norwegian University of Science and Technology Department of Mathematical Sciences TMA445 Linear Methods Fall 07 Exercise set Please justify your answers! The most important part is how you arrive at an
More informationPRODUCT OF OPERATORS AND NUMERICAL RANGE
PRODUCT OF OPERATORS AND NUMERICAL RANGE MAO-TING CHIEN 1, HWA-LONG GAU 2, CHI-KWONG LI 3, MING-CHENG TSAI 4, KUO-ZHONG WANG 5 Abstract. We show that a bounded linear operator A B(H) is a multiple of a
More informationMath Introduction to Numerical Analysis - Class Notes. Fernando Guevara Vasquez. Version Date: January 17, 2012.
Math 5620 - Introduction to Numerical Analysis - Class Notes Fernando Guevara Vasquez Version 1990. Date: January 17, 2012. 3 Contents 1. Disclaimer 4 Chapter 1. Iterative methods for solving linear systems
More informationProceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11-15th July 2005
Proceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11-15th July 2005 SOME INVERSE SCATTERING PROBLEMS FOR TWO-DIMENSIONAL SCHRÖDINGER
More informationTHE SINGULAR VALUE DECOMPOSITION MARKUS GRASMAIR
THE SINGULAR VALUE DECOMPOSITION MARKUS GRASMAIR 1. Definition Existence Theorem 1. Assume that A R m n. Then there exist orthogonal matrices U R m m V R n n, values σ 1 σ 2... σ p 0 with p = min{m, n},
More informationBlock Bidiagonal Decomposition and Least Squares Problems
Block Bidiagonal Decomposition and Least Squares Problems Åke Björck Department of Mathematics Linköping University Perspectives in Numerical Analysis, Helsinki, May 27 29, 2008 Outline Bidiagonal Decomposition
More informationPART IV Spectral Methods
PART IV Spectral Methods Additional References: R. Peyret, Spectral methods for incompressible viscous flow, Springer (2002), B. Mercier, An introduction to the numerical analysis of spectral methods,
More informationMain matrix factorizations
Main matrix factorizations A P L U P permutation matrix, L lower triangular, U upper triangular Key use: Solve square linear system Ax b. A Q R Q unitary, R upper triangular Key use: Solve square or overdetrmined
More informationReview of some mathematical tools
MATHEMATICAL FOUNDATIONS OF SIGNAL PROCESSING Fall 2016 Benjamín Béjar Haro, Mihailo Kolundžija, Reza Parhizkar, Adam Scholefield Teaching assistants: Golnoosh Elhami, Hanjie Pan Review of some mathematical
More informationGolub-Kahan iterative bidiagonalization and determining the noise level in the data
Golub-Kahan iterative bidiagonalization and determining the noise level in the data Iveta Hnětynková,, Martin Plešinger,, Zdeněk Strakoš, * Charles University, Prague ** Academy of Sciences of the Czech
More informationAMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 23: GMRES and Other Krylov Subspace Methods; Preconditioning
AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 23: GMRES and Other Krylov Subspace Methods; Preconditioning Xiangmin Jiao SUNY Stony Brook Xiangmin Jiao Numerical Analysis I 1 / 18 Outline
More informationAnnounce Statistical Motivation Properties Spectral Theorem Other ODE Theory Spectral Embedding. Eigenproblems I
Eigenproblems I CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Doug James (and Justin Solomon) CS 205A: Mathematical Methods Eigenproblems I 1 / 33 Announcements Homework 1: Due tonight
More informationCS168: The Modern Algorithmic Toolbox Lecture #8: How PCA Works
CS68: The Modern Algorithmic Toolbox Lecture #8: How PCA Works Tim Roughgarden & Gregory Valiant April 20, 206 Introduction Last lecture introduced the idea of principal components analysis (PCA). The
More informationAMS526: Numerical Analysis I (Numerical Linear Algebra)
AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 1: Course Overview & Matrix-Vector Multiplication Xiangmin Jiao SUNY Stony Brook Xiangmin Jiao Numerical Analysis I 1 / 20 Outline 1 Course
More informationComputational math: Assignment 1
Computational math: Assignment 1 Thanks Ting Gao for her Latex file 11 Let B be a 4 4 matrix to which we apply the following operations: 1double column 1, halve row 3, 3add row 3 to row 1, 4interchange
More informationJust solve this: macroscopic Maxwell s equations
ust solve this: macroscopic Maxwell s equations Faraday: E = B t constitutive equations (here, linear media): Ampere: H = D t + (nonzero frequency) Gauss: D = ρ B = 0 D = ε E B = µ H electric permittivity
More informationMATH 220: INNER PRODUCT SPACES, SYMMETRIC OPERATORS, ORTHOGONALITY
MATH 22: INNER PRODUCT SPACES, SYMMETRIC OPERATORS, ORTHOGONALITY When discussing separation of variables, we noted that at the last step we need to express the inhomogeneous initial or boundary data as
More informationOn uniqueness in the inverse conductivity problem with local data
On uniqueness in the inverse conductivity problem with local data Victor Isakov June 21, 2006 1 Introduction The inverse condictivity problem with many boundary measurements consists of recovery of conductivity
More information