Downloaded 11/20/13 to Redistribution subject to SIAM license or copyright; see

Transcription

1 SIAM J. SCI. COMPUT. Vol. 33, No. 1, pp c 2011 Society for Industrial and Applied Mathematics A SIMPLE NUMERICAL METHOD FOR COMPLEX GEOMETRICAL OPTICS SOLUTIONS TO THE CONDUCTIVITY EQUATION KUI DU Dedicated to Professor Zhi-hao Cao on the occasion of his 75th birthday Abstract. This paper concerns numerical methods for computing complex geometrical optics (CGO) solutions to the conductivity equation σ u(,k)=0inr 2 for piecewise smooth conductivities σ, where k is a complex parameter. The key is to solve an R-linear singular integral equation defined in the unit disk. Recently, Astala et al. [Appl. Comput. Harmon. Anal., 29 (2010), pp. 2 17] proposed a complicated method for numerical computation of CGO solutions by solving a periodic version of the R-linear integral equation in a rectangle containing the unit disk. In this paper, based on the fast algorithms in [P. Daripa and D. Mashat, Numer. Algorithms, 18 (1998), pp ] for singular integral transforms, we propose a simpler numerical method which solves the R-linear integral equation in the unit disk directly. For the resulting R-linear operator equation, a minimal residual iterative method is proposed. Numerical examples illustrate the accuracy and efficiency of the new method. Key words. complex geometrical optics solution, R-linear integral equation, singular integral, fast algorithm, global R-linear GMRES AMS subject classifications. 45E10, 65N22, 65R20 DOI / Introduction. This paper is concerned with the numerical computation of complex geometrical optics (CGO) solutions to the conductivity equation [5, 3] (1.1) σ u(,k)=0inr 2 together with the asymptotical condition u(ζ,k)=e ikζ (1+O ( )) 1, ζ +. ζ Here k is a complex parameter, i is the imaginary unit, and σ(ζ) is a piecewise continuous conductivity satisfying 0 <c σ(ζ) < and σ(ζ) = 1 outside a compact set Ω. For simplicity we let Ω be the unit disk; this is not a significant loss of generality, as a large class of more general settings can be reduced to this case. The CGO solutions to (1.1) are used to solve the inverse conductivity problem (Calderón s problem) [6]. We describe the details of this problem as follows. Define the Dirichlet-to-Neumann (DtN) map corresponding to σ by Λ σ : H 1/2 ( Ω) H 1/2 ( Ω), Λ σ φ, g = σ u v, Submitted to the journal s Methods and Algorithms for Scientific Computing section July 15, 2010; accepted for publication (in revised form) November 22, 2010; published electronically February 22, This work was supported by Academy of Finland grant and the Väisälä Foundation of the Finnish Academy of Science and Letters. Institute of Mathematics, Aalto University, P.O. Box 11100, FI Aalto, Finland (kuidumath@yahoo.com). 328 Ω

2 A SIMPLE NUMERICAL METHOD FOR CGO SOLUTIONS 329 where v H 1 (Ω) with trace g H 1/2 ( Ω) and u H 1 (Ω) is the unique solution of the Dirichlet problem σ u = 0 in Ω, u Ω = φ. The physical interpretation of Λ σ is the knowledge of the resulting current distributions on Ω corresponding to all possible voltage distributions on Ω. The inverse conductivity problem is to show that the map σ Λ σ is injective and find a reconstruction algorithm for the conductivity σ from the knowledge of the DtN map Λ σ. The inverse conductivity problem has many applications, such as the recent medical imaging technique known as electrical impedance tomography (EIT) [7, 16]. Many uniqueness results have been obtained for the inverse conductivity problem; see, for example, [27, 25, 20, 4, 5, 32] and the references therein. In particular, for the two-dimensional case, Astala and Päivärinta [5] proved that the DtN map Λ σ uniquely determines the conductivity σ L (Ω), 0 <c σ. The so-called D-bar methods [29, 30, 26, 25, 24, 18, 8, 21, 22, 23] were proposed to reconstruct the conductivity σ. The scattering transform t(k) [27] plays an important role in the D-bar methods, which can be computed through the CGO solver; see (5.2) and (5.5). Therefore, the CGO solver can be used to check the intermediate results when developing reconstruction algorithms. The CGO solutions to (1.1) can be constructed via the R-linear Beltrami-type equation (1.2) f = μ f together with the asymptotical condition (1.3) f(ζ,k)=e ikζ (1 + ω(ζ,k)), ω(ζ,k)=o ( ) 1, ζ +, ζ where f denotes the complex conjugate of f, = 1 ( ) +i, = 1 ( ) i, ζ = ζ 1 +iζ 2, 2 ζ 1 ζ 2 2 ζ 1 ζ 2 and μ(ζ) = 1 σ(ζ) 1+σ(ζ). Recently, Astala et al. [3] proposed a complicated numerical method to compute ω(ζ,k) in (1.3) via solving an R-linear integral equation based on periodization, truncation of a Neumann series, discretization, fast Fourier transform (FFT), and the GMRES method [28]. In this paper we propose a simpler numerical method for solving the same R-linear integral equation via a simple substitution, which avoids periodization and truncation of the Neumann series; see Remark 2.2. More precisely, we discretize an equivalent R-linear integral equation in the unit disk directly. We also propose a minimal residual method to solve the R-linear operator equation arising from the new discretization scheme, which is an extension of the R-linear GMRES method [14] for R-linear systems to R-linear operator equations. We show that the new minimal residual method is more efficient than the existing method for a related C-linear operator equation; see Theorem 4.3 and Remark 4.5. The new CGO solver

3 330 KUI DU results in remarkable computational savings; see Example 2 in section 5.2. Furthermore, the scattering transform τ (k) considered by Astala and Päivärinta [5] is readily obtained by our approach; see the formulas (5.2) (5.5) in section 5.2. The rest of this paper is organized as follows. In section 2 we give the R-linear integral equation used in this paper. In section 3 we present the new discretization scheme for the R-linear integral equation. The minimal residual method for the R-linear operator equation arising from our new discretization scheme is given in section 4. Numerical examples illustrating the accuracy and efficiency of the new method are reported in section 5. We present brief concluding remarks in section The R-linear integral equations. Define the Cauchy transform and the Beurling transform by Pf(ξ) = 1 f(ζ) π C ζ ξ dζ 1dζ 2 and Sf(ξ) = 1 f(ζ) π C (ζ ξ) 2 dζ 1dζ 2, respectively. We refer the reader to [2] for basic properties of P and S. Let α(ζ,k)= ikν(ζ,k), ν(ζ,k)=e i(kζ+kζ) μ(ζ), μ(ζ) = 1 σ(ζ) 1+σ(ζ). Then α L (Ω), supp(α) Ω, ν L (Ω), supp(ν) Ω, and ν γ<1. Take 2 <p<1+1/γ, and define the operator K : L p (C) L p (C) by (2.1) Kg = P(I νs) 1 (αg), where I is the identity operator, S denotes the R-linear operator Sf = Sf, andg denotes the complex conjugate of g. Letχ Ω denote the indicator function of Ω, i.e., { 0, ζ / Ω, χ Ω (ζ) = 1, ζ Ω. Then K : L p (C) W 1,p (C), I K is invertible in L p (C), and the equation (2.2) (I K)ω = Kχ Ω has a unique solution satisfying ω(ζ,k) = O(1/ζ) as ζ + ; see [5] for a proof. Note that this ω(ζ,k) satisfies (1.2) (1.3). Astala et al. [3] proposed a quite complicated numerical method for solving (2.2); see [3, sections 2 4] for details. We present an equivalent integral equation expression of (2.2) in the following. Let (2.3) u = (I νs) 1 (αω + αχ Ω ). Then we have (2.4) αω = u νsu α. Multiplying both sides of (2.2) by α yields (2.5) αω αkω = αkχ Ω.

4 A SIMPLE NUMERICAL METHOD FOR CGO SOLUTIONS 331 Substituting (2.1), (2.3), and (2.4) into (2.5) yields (2.6) u (αp + νs)u = α. It follows from supp(ν) Ω and supp(α) Ω that supp(u) Ω. equivalent to (2.7) u (αt 1 + νt 2 )u = α, Then (2.6) is where the integral operators T m, m =1, 2, are defined as follows: T m f(ξ) = 1 f(ζ) π Ω (ζ ξ) m dζ 1dζ 2, ζ = ζ 1 +iζ 2. The uniqueness of the solution u of (2.7) follows from the same arguments as those of Theorem 3.2 and Proposition 4.1 of [5]. Moreover, one can prove that the solution u of (2.7) has the same regularity as that of α and ν [1]; i.e., if α, ν C m,δ,then u C m,δ. It follows from (2.2) and (2.3) that w = K(w + χ Ω )=P(I νs) 1 (αw + αχ Ω )=Pu. Then we have the following proposition. Proposition 2.1. The solution of (2.2) is given by ω = Pu, where u is the unique solution of the R-linear integral equation (2.7). Remark 2.2. The numerical method proposed in [3] requires the numerical computation of (I νs) 1 (αg), which is approximated by truncating the Neumann series of (I νs) 1. By Proposition 2.1, the numerical computation of (I νs) 1 (αg) is avoided. Remark 2.3. A discrete version of Proposition 2.1 was proposed in [17, section 3]. In that paper the R-linear integral equation (2.6) was discretized in the rectangle [ 1, 1) 2 by collocating on a uniform grid of points. Since the inverse conductivity problem is defined in the unit disk and the known data are measured on its boundary, it is natural to develop a reconstruction algorithm which recovers the conductivity distribution in the unit disk directly. The CGO solver presented in this paper is helpful for this purpose. 3. A new discretization scheme. Based on the fast algorithms for T m f, m = 1, 2, proposed in [9, 10, 11, 15], we present a new discretization scheme for the R-linear integral equation of the form (3.1) u +(ν 1 T 1 + ν 2 T 2 )u = ν 3, defined in the unit disk Ω = {ζ C : ζ < 1}, whereν i L (Ω), supp(ν i ) Ω, i =1, 2, 3, and ν 2 γ< Fast algorithms for T m f, m =1, 2. Represent f and T m f, m =1, 2, as Fourier series f(ρe iθ )= f n (ρ)e inθ, T m f(ρe iθ )= n= n= T n,m (ρ)e inθ.

5 332 KUI DU Then, according to [11], 1 2 lim ρ T n,m (0) = 1 m f m (ρ)dρ, n =0, ɛ 0 ɛ 0, n 0, and for 0 <ρ 1, where T n,m (ρ) =B n,m (ρ)+c n,m (ρ), B n,m (ρ) = { 0, m =1, f n+2 (ρ), m =2, and 2( 1) m+1 ( ) ρ n 1 ( ρ ) m+n 1 fm+n ρ m 1 (t)dt, n m, m 1 0 t C n,m (ρ) = 0, m <n<0, 2 ( ) 1 ( m + n 1 ρ ) m+n 1 fm+n ρ m 1 (t)dt, n 0. m 1 ρ t Fast algorithms for the numerical computations of the integral transforms T m f, m =1, 2, were proposed in [11] using FFT. We describe the details of the fast algorithms as follows. Let Ω h = { (ρ i,θ j ):ρ i = i 1 Fig. 1. The grid Ω h with M =6,N =8. M 1, θ j = j2π N }, i =1,...,M, j =1,...,N denote the discretization of the unit disk using M N lattice points with M equidistant points in the radial direction and N equidistant points in the angular direction; seefigure1. ForeffectiveuseofFFT,wechooseN to be powers of two. The following recursive formulas were given in Corollary 2.2 of [11] to simplify the full integration in C n,m (ρ): C n,m (ρ j )= ( ρj ρ i ) n C n,m(ρ i )+C i,j n,m, n m,

6 A SIMPLE NUMERICAL METHOD FOR CGO SOLUTIONS 333 C n,m (ρ i )= ( ρi ρ j ) n C n,m(ρ j ) C i,j n,m, n 0, where ρ j >ρ i and 2( 1) m+1 ( ) ρj n 1 Cn,m i,j = ρ m 1 m 1 j ρ ( ) i 2 ρj m + n 1 ρ m 1 m 1 i ρ i ( ρj ( ρi t ) m+n 1 fm+n (t)dt, n m, t ) m+n 1 fm+n (t)dt, n 0. Algorithm 1: Fast algorithms for T m f, m =1, 2. Input: m = 1 or 2, M,N, andf(ρ i e iθj ), i =1,...,M, j =1,...,N.SetK = N/2. Output: T m f(ρ i e iθj ), i =1,...,M, j =1,...,N. (1) Compute the Fourier coefficients f n (ρ i )fromf(ρ i e iθj ) using FFT, where i = 1,...,M, j =1,...,N,andn = K + m,...,k + m 1. (2) Compute the Fourier coefficients T n,m (ρ i ), i =1,...,M, n = K,...,K 1. If m =1,setB n,m (ρ i )=0,i =1,...,M, n = K,...,K 1. If m =2,set B n,m (ρ i )=f n+2 (ρ i ), i =1,...,M, n = K,...,K 1. do i =1,...,M 1 do n = K,..., m C i,i+1 n,m = 2( 1)m+1 ρ m 1 i+1 do n =0,...,K 1 Cn,m i,i+1 = 2 ρ m 1 i ( n 1 m 1 ( m + n 1 m 1 ) ρi+1 ρ i ) ρi+1 ρ i ( ρi+1 Set C n,m (ρ i )=0,i =1,...,M, n = K,...,K 1. do n = K,..., m C n,m (ρ 2 )=Cn,m 1,2 do i =3,...,M C n,m (ρ i )= do n =0,...,K 1 do i = M 1,...,1 ( ) n i 1 C n,m(ρ i 1)+Cn,m i 1,i i 2 ( ) n i 1 C n,m (ρ i )= C n,m(ρ i+1) Cn,m l,l+1 i Set T n,m (ρ i )=0,i =1,...,M, n = K,...,K 1. do n = K,...,K 1 do i =2,...,M T n,m (ρ i )=B n,m (ρ i )+C n,m (ρ i ) 1 Set T 0,m (ρ 1 )= 2lim ɛ 0 ɛ ρ1 m f m (ρ)dρ. ρ ) m+n 1 f m+n(ρ)dρ ( ) m+n 1 ρi f m+n(ρ)dρ ρ

7 334 KUI DU (3) Compute T m f(ρ i e iθj )= K 1 n= K T n,m(ρ i )e inθj using FFT, i =1,...,M, j = 1,...,N. Remark 3.1. In our implementation, we use the trapezoidal rule to approximate the integrals in step (2) of Algorithm Discretization of the R-linear integral equation (3.1). By collocating on Ω h, we obtain the discretization scheme of (3.1) as follows: (3.2) u(ρ i e iθj )+ν 1 (ρ i e iθj )T 1 u(ρ i e iθj )+ν 2 (ρ i e iθj )T 2 u(ρ i e iθj )=ν 3 (ρ i e iθj ), where (ρ i,θ j ) Ω h. By setting U(i, j) =u(ρ i e iθj ), A(i, j) =ν 1 (ρ i e iθj ), B(i, j) = ν 2 (ρ i e iθj ), and C(i, j) =ν 3 (ρ i e iθj ), we obtain the following R-linear operator equation: (3.3) U + AU = C, where the C-linear operator A is defined by AX = A (T 1 X)+B (T 2 X) and A B is the componentwise product of two matrices A and B. By FFT, AU costs O(MN log N) floating point operations. 4. Global R-linear GMRES. In this section we propose a minimal residual method to solve the R-linear operator equation (3.3). The new method using the socalled global Arnoldi algorithm (see Algorithm 2), called the global R-linear GMRES method (GRL-GMRES), is a generalization of R-linear GMRES [14] for solving a class of R-linear systems of equations. We refer the reader to [13, 12] for a further analysis of R-linear GMRES. The derivation of GRL-GMRES is similar to that of [19]. For two matrices A, B C M N, define the inner product A, B F =trace(a B). The associated norm is the well-known Frobenius norm denoted by F. We use MU =(I + Aτ)U to denote the left-hand side of (3.3), where I denotes the identity matrix whose dimension is clear from the context and τu = Ū is the conjugation operator on C M N. We call {V 1,V 2,...,V i } an F -orthonormal basis of the Krylov subspace if for j, k =1,...,i, and K i (M,V)=span{V,MV,...,M i 1 V } V j,v k F = { 0, j k, 1, j = k, span{v 1,V 2,...,V i } = K i (M,V). Here M i V is defined recursively as M(M i 1 V ). algorithm as follows. We describe the global Arnoldi

8 A SIMPLE NUMERICAL METHOD FOR CGO SOLUTIONS 335 Algorithm 2: Global Arnoldi algorithm. 1. Compute V F, and let V 1 = V/ V F. 2. for j =1, 2,...,i do W = AV j for k =1toj do h kj =trace(vk W ) W = W h kj V k end for h j+1,j = W F v j+1 = W/h j+1,j end for Proposition 4.1. The global Arnoldi algorithm constructs an F -orthonormal basis V 1,V 2,...,V i of the Krylov subspace K i (M,V). Proof. Note the shift-invariance property of the Krylov subspace. The proof follows the same arguments as those of the proof of Theorem 3.1 of [14]. Let U 0 be the initial guess and R 0 = C MU 0 be the corresponding residual. GRL-GMRES constructs the approximate solution U i U 0 + K i (M,R 0 )atstepi such that (4.1) R i F := C MU i F = min Z K i(m,r 0) R 0 MZ F. In what follows, 2 denotes the Euclidean vector norm or the associated induced matrix 2-norm. We have the following theorem. Theorem 4.2. Let Ĩi denote the i i identity matrix augmented with the row of zerosasthelastrowande 1 the first column of the identity matrix with appropriate dimension. Let H i+1,i be the upper Hessenberg matrix generated in Algorithm 2 with V = R 0. We have R0 R i F =min F e 1 Ĩis H i+1,i s. s C i 2 Proof. Lets =(s 1,...,s i ) T. From (4.1), i i R i F =min s C i R 0 V j s j s j AV j j=1 j=1 F i i =min s C i R j+1 0 F V 1 V j s j s j h kj V k j=1 j=1 k=1 R0 =min F e 1 Ĩis H i+1,i s. s C i 2 The last equality holds because V 1,...,V i+1 are F -orthonormal. The minimal problem R0 min F e 1 Ĩis H i+1,i s s C i 2 can be solved by employing the R-linear QR decomposition [14]. details of GRL-GMRES as follows. F We present the

9 336 KUI DU Algorithm 3: Global R-linear GMRES. 1. Compute R 0 = C MU 0,whereU 0 is the initial guess. 2. for j =1, 2,..., Generate {V 1,V 2,...} and H i+1,i by Algorithm 2 with V = R 0 Solve the problem min s C i R 0 F e 1 Ĩis H i+1,i s 2 for s =(s 1,...,s i ) T Set U i = U 0 + i j=1 V js j and R i = C MU i Exit if satisfied end for A related C-linear operator equation to the R-linear operator equation (3.3) is (4.2) W (Aτ) 2 W = C. If (4.2) is solved, then U can be readily obtained by U = W AW. The C-linear operator equation (4.2) can be solved by the global GMRES method (G-GMRES) [19]. We compare GRL-GMRES and G-GMRES in the following theorem. Theorem 4.3. Let R i and Ri C be the ith residual of GRL-GMRES applied to (3.3) and the ith residual of G-GMRES applied to (4.2), respectively. If we further assume that R 0 = R0 C, then we have R 2i F R C i F. Proof. By the shift-invariance property [31] of Krylov subspaces, we have R C i F = min Z K i(i (Aτ ) 2,R 0) R 0 (I (Aτ) 2 )Z F = min Z K i((aτ ) 2,R 0) R 0 (I (Aτ) 2 )Z F = min Z K i((aτ ) 2,R 0) R 0 M(I Aτ)Z F = min V (I Aτ )K i((aτ ) 2,R 0) R 0 MV F The inequality follows from min Z K 2i(M,R 0) R 0 MZ F = R 2i F. (I Aτ)K i ((Aτ) 2,R 0 ) K 2i (Aτ,R 0 )=K 2i (M,R 0 ). Remark 4.4. The assumption R 0 = R0 C in Theorem 4.3 is attainable by setting the zero matrix as the initial guess. For this case we have R 0 = R0 C = C. Remark 4.5. Note that G-GMRES applied to (4.2) requires implementing the operator Aτ twice every iteration, and for U an extra implementation is required. Therefore, by Theorem 4.3, GRL-GMRES requires fewer implementations of the operator Aτ. 5. Numerical tests. In this section we report numerical results of two examples. Throughout, the computation is performed in MATLAB 2008a on a MacBook Pro with 2.26G CPU and 4GB memory.

10 A SIMPLE NUMERICAL METHOD FOR CGO SOLUTIONS Example 1. This is an artificial example. We consider the R-linear integral equation (5.1) u(ζ) (T 1 + T 2 )u(ζ) =ζ 3 ζ + ζ( ζ 4 1) ζ 4 1, 20 for which the exact solution is u(ζ) =ζ 3 ζ. See [9, 15] for a proof. In Table 1, we present errors and the corresponding error orders in the maximum norm defined by Max Error = max u(ρ i e iθj ) U(i, j). i,j Note that there is only one mode, namely, 2nd mode, in the Fourier expansion of the exact solution u. Therefore, the error will arise only from the approximate evaluation of the one-dimensional integrals in step (2) of Algorithm 1 if N/ We set N = 8. The corresponding error orders are determined using the formula Error Order = log(e 2h/e h ), log 2 where h =1/(M 1) and e h is the error corresponding to the M 8 grid of the unit disk Ω. We observed second-order convergence rate in the maximum norm. Table 1 Numerical results for Example 1. Grid M N Max Error Error Order O(h δ ) E E E E Example 2. In this example we consider the R-linear integral equation (2.7). First, we reproduce the numerical results in Figures 5 and 6 of [3] to verify our codes. In [3], the scattering transform τ (k) considered by Astala and Päivärinta [5] is computed from the formula (5.2) τ (k) = 1 2π C (ω ω )dζ 1 dζ 2, where ω is computed using μ and ω using μ. By Proposition 2.1, the scattering transform can be directly computed from (5.3) τ (k) = 1 (u u )dζ 1 dζ 2, 2π Ω where u corresponds to μ and u to μ. The integral in (5.3) can be approximated by the formula (5.4) Ω (u u )dζ 1 dζ 2 2π (M 1)N M N i=1 j=1 i 1 M 1 (U(i, j) U (i, j)).

11 338 KUI DU σ 1 ( ζ ) σ 2 ( ζ ) ζ t(k) k Fig. 2. Left: the profile of σ 1. Right: the corresponding scattering transform t(k) ζ t(k) k Fig. 3. Left: the profile of σ 2. Right: the corresponding scattering transform t(k). The following formula gives a connection between the scattering transforms t(k) considered by Nachman [27] and τ (k) for sufficiently smooth conductivity σ: (5.5) t(k) = 4πikτ (k). Let and σ 1 (ζ) = { 1.1, ζ < 0.5, 1 otherwise, σ 2 (ζ) = 2, ζ < 0.3, σ 3 (ζ) = 1, ) ζ > 0.7, 1+φ otherwise, ( 10 ζ 3 4 { 2, ζ < 0.5, 1 otherwise, where φ(ρ) =1 10ρ 3 +15ρ 4 6ρ 5. See Figures 2 4 for the profiles of σ 1, σ 2,andσ 3 and the corresponding scattering transform t(k). Next, we consider a piecewise continuous conductivity σ(ζ) in the unit disk, σ(ζ) = 3, ζ 0.5i < 0.25, 0.4, ζ i < 0.25 or ζ i < 0.25, 1 otherwise. See Figures 5 and 6 for plots of some solution ω(ζ,k) in the unit disk. In Table 2, we present the CPU time for solving (2.7) for different grids and k. It takes less than

12 A SIMPLE NUMERICAL METHOD FOR CGO SOLUTIONS σ 3 ( ζ ) ζ t(k) k Fig. 4. Left: the profile of σ 3. Right: the corresponding scattering transform t(k). Fig. 5. Real and imaginary parts of ω(ζ,1) for the piecewise continuous conductivity σ. Fig. 6. Real and imaginary parts of ω(ζ, (7 + 7i)/ 2) for the piecewise continuous conductivity σ. 10 seconds on the grid and less than 40 seconds on the grid to compute ω. However, for a similar problem, with a processing capability similar to that of the laptop used here, it takes roughly 1 minute on the uniform grid and 7 minutes on the uniform grid to compute ω using the numerical method in [3]. 6. Concluding remarks. We have proposed a simple numerical method for complex geometrical optics (CGO) solutions to the conductivity equation. A new

13 340 KUI DU Table 2 CPU time (seconds) for computing ω for different grids and k. Grid M N k =1 k =4 k =7 k =10 k = minimal residual method has been proposed to solve the resulting R-linear equation. We have shown that the new CGO solver outperforms the one given in [3] in a real life application. Acknowledgments. The author thanks Olavi Nevanlinna, who carefully read the early version and gave valuable advice on the content and direction of this paper. The author also thanks Kari Astala for the discussion about the regularity of u in (3.1), Huiyuan Li for helpful discussions about spectral-collocation methods, and Marko Huhtanen and Allan Perämäki for helpful discussions about this work and providing their R-linear GMRES MATLAB codes. REFERENCES [1] K. Astala, private communication, Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland, [2] K. Astala, T. Iwaniec, and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton Math. Ser. 48, Princeton University Press, Princeton, NJ, [3] K. Astala, J. L. Mueller, L. Päivärinta, and S. Siltanen, Numerical computation of complex geometrical optics solutions to the conductivity equation, Appl. Comput. Harmon. Anal., 29 (2010), pp [4] K. Astala and L. Päivärinta, A boundary integral equation for Calderón s inverse conductivity problem, Collect. Math., Vol. Extra (2006), pp [5] K. Astala and L. Päivärinta, Calderón s inverse conductivity problem in the plane, Ann. of Math. (2), 163 (2006), pp [6] A.-P. Calderón, On an inverse boundary value problem, in Seminar on Numerical Analysis and Its Applications to Continuum Physics (Rio de Janeiro, 1980), Soc. Brasil. Mat., Rio de Janeiro, 1980, pp [7] M. Cheney, D. Isaacson, and J. C. Newell, Electrical impedance tomography, SIAMRev., 41 (1999), pp [8] H. D. Cornean, K. Knudsen, and S. Siltanen, Towards a d-bar reconstruction method for three-dimensional EIT, J. Inverse Ill-Posed Probl., 14 (2006), pp [9] P. Daripa, A fast algorithm to solve nonhomogeneous Cauchy Riemann equations in the complex plane, SIAM J. Sci. Statist. Comput., 13 (1992), pp [10] P. Daripa, A fast algorithm to solve the Beltrami equation with applications to quasiconformal mappings, J. Comput. Phys., 106 (1993), pp [11] P. Daripa and D. Mashat, Singular integral transforms and fast numerical algorithms, Numer. Algorithms, 18 (1998), pp [12] K. Du, Global R-Linear GMRES for Solving a Class of R-Linear Matrix Equations, manuscript, [13] K. Du and O. Nevanlinna, AnoteonR-linear GMRES for solving a class of R-linear systems, Numer. Linear Algebra Appl., to appear. [14] T. Eirola, M. Huhtanen, and J. von Pfaler, Solution methods for R-linear problems in C n, SIAM J. Matrix Anal. Appl., 25 (2004), pp [15] D. Gaidashev and D. Khmelev, On numerical algorithms for the solution of a Beltrami equation, SIAM J. Numer. Anal., 46 (2008), pp [16] M. Hanke and M. Brühl, Recent progress in electrical impedance tomography, InverseProblems, 19 (2003), pp. S65 S90. [17] M. Huhtanen and A. Perämäki, Numerical solution of the R-linear Beltrami equation, Math. Comp., to appear.

14 A SIMPLE NUMERICAL METHOD FOR CGO SOLUTIONS 341 [18] D. Isaacson, J. L. Mueller, J. C. Newell, and S. Siltanen, Imaging cardiac activity by the D-bar method for electrical impedance tomography, Physiol. Meas., 27 (2006), pp. S43 S50. [19] K. Jbilou, A. Messaoudi, and H. Sadok, Global FOM and GMRES algorithms for matrix equations, Appl. Numer. Math., 31 (1999), pp [20] K. Knudsen, The Calderón problem with partial data for less smooth conductivities, Comm. Partial Differential Equations, 31 (2006), pp [21] K. Knudsen, M. Lassas, J. L. Mueller, and S. Siltanen, D-bar method for electrical impedance tomography with discontinuous conductivities, SIAM J. Appl. Math., 67 (2007), pp [22] K. Knudsen, M. Lassas, J. L. Mueller, and S. Siltanen, Reconstructions of piecewise constant conductivities by the D-bar method for electrical impedance tomography, J.Phys. Conf. Ser., 124 (2008), [23] K. Knudsen, M. Lassas, J. L. Mueller, and S. Siltanen, Regularized D-bar method for the inverse conductivity problem, Inverse Probl. Imaging, 3 (2009), pp [24] K. Knudsen, J. L. Mueller, and S. Siltanen, Numerical solution method for the dbarequation in the plane, J. Comput. Phys., 198 (2004), pp [25] K. Knudsen and A. Tamasan, Reconstruction of less regular conductivities in the plane, Comm. Partial Differential Equations, 29 (2004), pp [26] J. L. Mueller and S. Siltanen, Direct reconstructions of conductivities from boundary measurements, SIAM J. Sci. Comput., 24 (2003), pp [27] A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math. (2), 143 (1996), pp [28] Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7 (1986), pp [29] S. Siltanen, J. L. Mueller, and D. Isaacson, An implementation of the reconstruction algorithm of A. Nachman for the 2D inverse conductivity problem, InverseProblems,16 (2000), pp [30] S. Siltanen, J. L. Mueller, and D. Isaacson, Reconstruction of high contrast 2-D conductivities by the algorithm of A. Nachman, in Radon Transforms and Tomography (South Hadley, MA, 2000), Contemp. Math. 278, AMS, Providence, RI, 2001, pp [31] V. Simoncini and D. B. Szyld, Recent computational developments in Krylov subspace methods for linear systems, Numer. Linear Algebra Appl., 14 (2007), pp [32] G. Uhlmann, Electrical impedance tomography and Calderón s problem, InverseProblems, 25 (2009),