A nonlinear multigrid for imaging electrical conductivity and permittivity at low frequency

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1 INSTITUTE OF PHYSICS PUBLISHING INVERSE PROBLEMS Inverse Probems 17 (001) PII: S (01) A noninear mutigrid for imaging eectrica conductivity and permittivity at ow frequency Liiana Borcea Computationa and Appied Mathematics, MS 134, Rice University, 6100 Main Street, Houston, TX , USA E-mai: borcea@caam.rice.edu Received 3 May 000, in fina form 14 December 000 Abstract We propose a noninear mutigrid approach for imaging the eectrica conductivity and permittivity of a body, given partia, usuay noisy knowedge of the Neumann-to-Dirichet map at the boundary. The agorithm is a nested iteration, where the image is constructed on a sequence of grids in, starting from the coarsest grid and advancing towards the finest one. We show various numerica exampes that demonstrate the effectiveness and robustness of the agorithm and prove oca convergence. (Some figures in this artice are in coour ony in the eectronic version; see 1. Introduction We consider the inverse probem of imaging eectrica properties of a heterogeneous, isotropic materia, in a domain, given ow-frequency, aternating currents and votages at the boundary. The eectrica properties of the materia are described by the compex conductivity, or admittivity function ρ(x,ω)= σ(x) +iωε(x), (1.1) where σ(x) and ε(x) are the eectrica conductivity and permittivity, and ω is the frequency. The inverse probem, commony referred to as eectrica impedance tomography, has various appications in fieds such as medica imaging [10], underground contaminant detection [38], subsurface fow monitoring [37], nondestructive testing [39], etc. When a possibe boundary currents and votages are known, we have the Neumann-to- Dirichet (NtD) map. In theory, given perfect data (fu knowedge of the NtD map), the inverse probem has a unique answer over some arge casses of functions ρ [7,17,0,6,30,34,44]. In practice, however, one never has the fu NtD map and inversion is to be done with incompete, noisy data. Furthermore, the inverse probem is i posed [], so some reguarization strategy is needed to stabiize imaging techniques [3, 11, 14, 15, 18]. The numerica soution of the eectrica impedance tomography probem has received increasing attention in the ast decade and many imaging agorithms have been proposed, /01/ $ IOP Pubishing Ltd Printed in the UK 39

2 330 L Borcea especiay for the static (ω = 0), dc case, where the unknown is σ, the eectrica conductivity. There are basicay three casses of agorithms: (1) Noniterative, inearization (Born approximation) agorithms which assume that ρ has sma variations about a known admittivity function [, 8, 9, 1]. () Iterative, Newton-type imaging agorithms, based on either output east-squares [10, 14, 16, 46] or equation-error formuations [9, 45]. (3) Layer stripping agorithms [1, 41, 43] which cacuate ρ starting from the boundary and advancing in the interior of. A very recent and promising agorithm has been deveoped by Nachman [34] and [40]. Finay, there are statistica, Bayesian approaches, where one can reguarize the inverse probem by incorporating prior knowedge of the soution [8]. In spite of the rich iterature on eectrica impedance tomography, there are important, outstanding questions that do not have satisfactory answers, so far. For exampe, ayer stripping agorithms are unstabe and cannot be used for noisy data. The resoution and reiabiity of images of ρ have yet to be improved. The inversion agorithms have to be faster so images of ρ can be produced in rea time. Finay, there are important issues to be resoved in experiment design and accuracy of the data [10]. In this paper, we propose a noninear mutigrid approach [4, 5, 4] for the soution of the eectrica impedance tomography probem. We formuate the numerica agorithm, prove oca convergence and demonstrate its effectiveness through various numerica resuts. The formuation of the inverse probem is reguarized, output east squares, where we minimize over admittivities ρ in an admissibe set, the misfit in the votage at the boundary. The noninear equation that we sove with the mutigrid is the first-order optimaity condition that the eastsquares soution ρ must satisfy. Suppose that we have a sequence of grids that discretize. Our inversion agorithm is a nested iteration [4] which constructs images of the admittivity, sequentiay, starting from the coarsest grid and advancing towards the finest one. We emphasize that the agorithm is appied directy to the noninear probem, without any need of repeated inearization (see e.g. [4, 4] for a simiar statement). Most computations are done on the coarse grid, where the noninear equation is soved with Newton s method. On finer grids, we do a few noninear reaxation iterations with the soe purpose of smoothing the error. Then, we update the soution through a coarse-grid correction. The agorithm that we propose can appy directy to any simpy connected in R 3. For simpicity, we do a our numerica cacuations in a two-dimensiona domain. The main body of our agorithm can be found in cassica mutigrid iterature [4, 5, 4]. However, to our knowedge, a noninear mutigrid has not been appied in inversion, so far. Oder resuts, such as [33], are imited to inearization (Born approximation), and use as data the Dirichet-to-Neumann map, which seems more convenient in numerics. Nevertheess, in practice, due to noisy measurements, it is better to work with the NtD map, which is smoothing. We have found that, when using the NtD map in mutigrid inversion, the coarsegrid computations need carefu treatment, in order to get convergence, especiay at the sma reguarization parameters we are interested in. On the coarse grids, the numerica approximation of the eectric potentia φ that soves the forward, Neumann boundary vaue probem can be quite inaccurate. For inversion, we compare φ with the measured votage, at the boundary. The inverse probem is i posed, so we expect that the error in φ is highy ampified in the coarse-grid image of ρ, uness we stabiize the inversion by choosing a arge reguarization parameter. Ceary, this is undesirabe, because the images are too burry. In this paper, we introduce an idea that aows us to perform inversion with sma reguarization parameters and therefore keep a good resoution of ρ, on

3 A noninear mutigrid for imaging eectrica conductivity and permittivity at ow frequency 331 a grid eves. The idea amounts to taking a specia proection of the excitation currents, on the coarse grid. To get this proection, we formuate an additiona, coarse-grid, east-squares probem, which we sove once, at the beginning of the nested iteration. The computationa cost of the coarse-grid soution is sma and, as demonstrated by our numerica resuts, the coarse-grid images are often good approximations of the admittivity in. This is important in the mutigrid iteration, for at east two reasons: good starting vaues in the nested iteration and effective coarse-grid corrections. The resut is often a nice image on the fine grid, given at a ow computationa cost. The paper is organized as foows: in section, we formuate the inverse probem and derive the noninear equation that is soved with the mutigrid. In section 3, we describe the agorithm. In section 4, we introduce the ideas and numerica agorithm for coarse-grid imaging. In section 5, we present various numerica resuts that demonstrate the robustness and effectiveness of the noninear mutigrid approach for the soution of the eectrica impedance tomography probem. Finay, in section 6, we prove oca convergence. Our proof is based on the known resuts in [4] and it assumes that the inverse probem is propery reguarized.. Formuation of the probem In this section, we give the reguarized, output east-squares formuation of the inverse probem. The noninear equation that we sove with the mutigrid is the first-order optimaity condition that the east-squares soution must satisfy. We consider a simpy connected domain R containing an inhomogeneous, isotropic materia with admittivity ρ(x,ω)= σ(x) +iωε(x), satisfying rea [ρ(x)] = σ(x) m L > 0, ρ(x) =[σ (x) + ω ε (.1) (x)] 1 mu, for a x and some finite, positive constants m L and m U. Suppose that, at the boundary, we appy a norma excitation current I(x,ω). The eectric potentia φ(x,ω)satisfies [ρ(x,ω) φ(x,ω)] = 0 in, (.) ρ(x,ω) φ(x,ω) n(x) = I(x,ω) on, where n(x) is the outer norma to. The Neumann boundary vaue probem (.) is derived from Maxwe s equations, in the imit of sma ω, as shown in [10]. We assume a fixed frequency and, to simpify notation, we henceforth drop ω from the arguments of ρ, φ and I. It is we known (see e.g. [19]) that for current densities I(x) satisfying I(s)ds = 0, (.) has a soution φ(x) H 1 ( ), that is unique up to an additive constant. We fix this constant by φ(s)ds = V(s)ds, (.3) where V is the measured potentia at the boundary. In the inverse probem, ρ(x) is unknown and it is to be found from simutaneous measurements of currents and potentias at the boundary. Thus, for a given I(x), we overspecify (.) by requiring that, at the true admittivity function ρ(x), the potentia satisfy φ(x) = V(x) for x. (.4) When a possibe excitations I(x) and measurements V(x) are avaiabe, we know the NtD map ρ : H 1 ( ) H 1 ( ), ρ I(x) = V(x), for x. (.5)

4 33 L Borcea In practice, however, we ony have partia knowedge about ρ. That is, we have a set of currents {I (x)} N exp =1 and corresponding votage measurements {V (x p )} N exp =1. Here, index represents the experiment, or current excitation I(x) = I (x) in (.), and x p, p = 1,...,N B, denote points of measurement of the resuting potentia V at the boundary. The inverse probem is: find ρ(x) in the interior of, given boundary data {I (x)} N exp =1 and {V (x p )} N exp =1, for p = 1,...,N B. We consider the output east-squares formuation: N exp Minimize over ρ ϒ, functiona J(ρ) = R (ρ) V L ( ), (.6) where we take some interpoation between the points of measurement of potentia V. We are concerned with finding ρ in the interior of so we assume that we know the admittivity at the boundary. The set ϒ of admissibe ρ is defined by ϒ ={ρ L ( ), ρ satisfies (.1) and, at the boundary, ρ(x) = ρ(x), given for x }. (.7) In (.6), R is the map R : ϒ L ( ), R (ρ) = ρ I = trace of φ at the boundary, (.8) where φ is the soution of =1 [ρ(x) φ (x)] = 0 in, ρ(x) φ (x) n(x) = I (x) on, φ (s) ds = V (s) ds. We have the foowing theoretica resuts. (.9) Lemma 1. R (ρ) is a continuous map. That is, for an arbitrary perturbation δρ such that ρ + δρ ϒ, there exists a finite, positive constant c such that R (ρ + δρ) R (ρ) L ( ) c δρ L ( ). Moreover, if ρ,δρ C k,1 ( ), where is the cosure of and k is stricty positive, we have R (ρ + δρ) R (ρ) L ( ) c δρ L ( ). Lemma. R (ρ) is Fréchet differentiabe, with Fréchet derivative DR (ρ)δρ = δφ, where δφ soves the inearized probem [ρ(x) δφ (x)] = [δρ(x) φ (x)] in, δφ (x) n(x) = 0 on, δφ (s) ds = 0. (.10) Proofs of emmas 1 and are given in [14]. The Fréchet derivative operator defined in emma is compact (see e.g. [11]), and so probem (.6) is i posed. We consider the reguarization min J α(ρ), where J α (ρ) = J(ρ)+ α ρ ρ ϒ L ( ), (.11) and α>0is a sma reguarization parameter. Lemma 3. For a α>0, the functiona J α has at east one minimizer ρ α in the set ϒ H 1 ( ).

5 A noninear mutigrid for imaging eectrica conductivity and permittivity at ow frequency 333 Proof. Choose a sequence {ρ i } ϒsuch that J α (ρ i ) Jα = inf{j α(ρ), where ρ ϒ}. Due to the reguarization term α ρ L ( ) in (.11), for i sufficienty arge, we have ρ i H 1 ( ). Moreover, ρ i (x) ρ(x) H0 1 ( ), where we take the harmonic extension, inside, of the known ρ at the boundary. The compact embedding H0 1( ) L ( ) [1] shows us that there is a subsequence, denoted by {ρ ik ρ}, that converges to ρ α ρ, strongy in L ( ). Finay, emma 1 impies that J α is a continuous functiona in L ( ) so ρ i ρ α means J α (ρ i ) J α (ρ α ) = Jα. Thus, ρ α ϒ H 1 ( ) is a minimizer of J α (ρ) and emma 3 is proved. By taking the first variation of functiona J α (ρ), we find that, at a minimum, ρ(x) satisfies the first-order optimaity condition N exp α ρ(x) + [DR (ρ)] [R (ρ) V ](x) = 0, x, (.1) =1 where [DR (ρ)] is the L adoint of the Fréchet derivative DR defined in emma. Note that (.1) is vaid if the minimizer is in the interior of ϒ. Otherwise, one has additiona terms in (.1). For exampe, it may be necessary that the strict positivity of the rea part of ρ be enforced, in which case one must add a penaty term in (.11), by means of a Lagrange mutipier [3]. Such extra compications are not addressed in this paper, where we assume that our soution is in the interior of ϒ and we sove the noninear equation (.1) with a mutigrid approach. First, we observe that, in (.1), terms invoving the adoint operators [DR (ρ)] can be written in more expicit form as shown in emma 4. Lemma 4. For any function χ L ( ), such that χ (s) ds = 0, we have [DR (ρ)] χ (x) = φ (x) ψ (x), (.13) where φ is the compex conugate of φ, the soution of (.9), and ψ soves the adoint equation [ρ(x) ψ (x)] = 0 in, ρ(x) ψ (x) n(x) = χ (x) on, ψ (s) ds = 0. (.14) The proof of emma 4 is simiar to that given in [14,35] for the particuar case of the rea-vaued function ρ in L ( ). From (.10) and (.14), we have {δφ (x) [ρ(x) ψ (x)] ψ (x) [ρ(x) δφ (x)]} dx = ψ (x) [δρ(x) φ (x)]dx, where the overbar indicates the compex conugate. Integrating by parts and using emma, we obtain {δφ (s)ρ(s) ψ (s) n(s) ψ (s)ρ(s) δφ (s) n(s)} ds = [DR (ρ)δρ]χ (s) ds = δρ(x)[dr (ρ)] χ (x) dx = δρ(x) φ (x) ψ (x) dx. This hods for arbitrary perturbations δρ, such that ρ + δρ ϒ and (.13) foows.

6 334 L Borcea We use emma 4 with χ = R (ρ) V and rewrite equation (.1) as N exp α ρ(x) φ (x; ρ) ψ (x; ρ) = 0, x, (.15) =1 where we emphasize that φ and ψ depend nonineary on the unknown admittivity function ρ. Moreover, equation (.15) is couped with equations (.9) and (.14), satisfied by potentias φ and ψ, for each ρ and excitation I, = 1,...,N exp. In what foows, we focus attention on the noninear mutigrid numerica soution of (.15). 3. Noninear mutigrid soution In this section, we describe the noninear mutigrid agorithm for soving equation (.15), with Dirichet boundary conditions ρ(x) = ρ(x) on. (3.1) We write (.15) and (3.1) in abstract form F(α, ρ) = 0, (3.) where the unknown is ρ and α is the reguarization parameter Discretization In a mutigrid approach, equation (3.) is discretized on a sequence {G } K =0 of grids, where G 0 is the coarsest grid, which we take with uniform spacing h 0. We define grid G as G ={(x i,y ) = (i h,h ) R,i, Z} interior, where h = h 0, = 0, 1,...,K. (3.3) Note that the uniform discretization (3.3) hods for the interior of. Depending on the shape of, near the boundary, the grid points may be irreguar, such as (s h,th ), where s and t may not be integers. We ca the set of discrete boundary points G, and define the set G = G G, which contains a discretization points at eve, in. For simpicity, et us take a unit square domain, such that (x i,y ) = (i h,h ), for i, = 0,...,n, are the grid points of G. The finite-difference discretization of (3.) is F (α,ρ () ) = 0, (3.4) where α and ρ () are the reguarization parameter and the restriction of ρ on grid G, respectivey. We use a nine-point, finite-difference scheme [F (α,ρ () )] i = α h (ρ () i+1,+1 + ρ() i 1,+1 + ρ() i+1, 1 + ρ() i 1, 1 +4ρ() i+1, +4ρ() i 1, Nexp +4ρ () i,+1 +4ρ() i, 1 0ρ() i, ) [( ) φk (x i+1,y ) φ k (x i 1,y ) h k=1 ( ) ψk (x i+1,y ) ψ k (x i 1,y ) h ( )( )] φk (x i,y +1 ) φ + k (x i,y 1 ) ψk (x i,y +1 ) ψ k (x i,y 1 ) = 0, (3.5) h h where ρ () i, = ρ() (x i,y ), for a (x i,y ) G.

7 A noninear mutigrid for imaging eectrica conductivity and permittivity at ow frequency 335 A numerica approximation of φ k (x i,y ) and ψ k (x i,y ) is obtained by soving equations (.9) and (.14) with a piecewise inear, finite-eement method (see e.g. [7]). Let T be a trianguation of with vertices given by the grid points in G. In each triange of T, we take the admittivity ρ as a piecewise inear interpoation of the vaues of ρ () at grid points (x i,y ) G. We define the vector of unknowns ( () ) φ φ () (interior) = φ () C (n +1), (3.6) (boundary) where φ () (interior) C (n 1) contains the vaues of φ () at the interior nodes and φ () (boundary) C 4n is the vector of boundary potentias. The finite-eement discretization of (.9) eads to the inear system of equations A () (ρ () )φ () = b () (I ), (3.7) where A () is an (n +1) (n +1), compex matrix that depends ineary on ρ () and b () is an (n +1) -dimensiona compex vector that depends ineary on the excitation current I. Due to the we-posedness of (.9), A () is invertibe and the soution φ () exists and is unique. Simiary, the finite-eement soution of (.14) eads to A () (ρ () )Ψ () = c () (ρ () ), (3.8) where A () is the same as in (3.7), except that it depends on the compex conugate of ρ (). Note that the right-hand side in (3.8) depends nonineary on ρ (), due to the adoint excitation current χ = φ V defined at the boundary. In the mutigrid, one must transfer functions from one grid to the next. This is done through proongation and restriction operators. For ρ () and ρ ( 1) given on grids G and G 1, we define the proongation P and restriction R by ρ () = Pρ ( 1), ρ ( 1) = Rρ (), = 1,...,K. (3.9) Since our differentia equation is of second order, it suffices to take P as a piecewise inear interpoation [4]. We choose the nine-point proongation P and the fu weighted restriction R, which is the adoint of P [4]. We concude with the observation that our finite-difference discretizations F 1 and F, on adacent grids G 1 and G, are independent. In other words, we can seek a soution of (3.4) on G 1, independent of the fact that we actuay wish to sove (3.4) on G,G +1,...,G K. This is essentia in a mutigrid, where coarse-grid computations are needed to correct the soutions on the finer grids. Another possibe discretization of (3.) is given by the conforming finite-eement (Gaerkin) approach. Then, F 1 (ρ ( 1) ) = RF (Pρ ( 1) ), as shown in [4]. This discretization has the advantage of a perfect reative consistency (F 1 RF P = 0), which is important in coarse-grid correction [4]. However, the Gaerkin approach has the maor disadvantage that F 1 depends on F. This means that the evauation of F 1 and its Jacobian, on coarse grid G 1, requires the cacuation of F (Pρ ( 1) ) and its Jacobian, on the finer grid, G. This is ceary too expensive and eaves us with the ony choice of a finite-difference discretization, such as (3.5).

8 336 L Borcea 3.. The noninear mutigrid agorithm For numerica computations, we transform the compex probem (3.4) into a rea system of equations, with rea unknowns. We define rea ρ () 1,1 σ () rea ρ () 1,1 1, σ () 1,.. u = rea ρ () n 1,n 1 = σ () imag ρ () n 1,n 1 R (n 1), (3.10) 1,1 ωε () 1,1.. imag ρ () n 1,n 1 ωε () n 1,n 1 and rewrite (3.4) as L (u ) = rea [F ] 1,1. rea [F ] n 1,n 1 imag [F ] 1,1. imag [F ] n 1,n 1 = 0. (3.11) To sove (3.11), starting with the coarsest grid G 0 and advancing towards the finest grid G K, we use the noninear nested iteration [4] agorithm sketched beow: Choose α 0. Cacuate u 0 L 1 0 (0). For = 1,...,K, u = Pu 1. (3.1) Choose α and set α k = α, for k = 0, 1,..., 1. ca NMGM(, u, 0) M times. End. In (3.1), K and M are defined by the user. In our impementation, the index K of the finest grid is and M = 3. On the coarsest grid G 0, we sove L 0 (u 0 ) = 0 with Newton s method [13]. The initia guess of ρ (0) or, equivaenty, u 0, is the restriction on G 0 of the harmonic extension in, of the known ρ at. Procedure NMGM (, u, g ) performs the noninear mutigrid iteration for soving L (u ) = g. (3.13) The important eements of NMGM are a smoothing or reaxation process and a coarse-grid correction. The purpose of smoothing on grid G is not to sove (3.13), but rather to reduce the high-frequency components in the error u L 1 (g ). This aows us to approximate the error on the coarse grid G 1, where soution is cheaper. Finay, given the coarse-grid soution, we update u on G, through the coarse-grid correction. Let us describe the smoothing (reaxation) process denoted by u S (ν) (u, g ), (3.14) where index ν stands for the number of reaxation sweeps. We choose a bock noninear Gauss Seide smoother, where the bocks of unknowns are defined as foows: consider a grid point (x i,y ) G and define the set N i ={(x k,y p ) G such that k i 1 and p 1}.

9 A noninear mutigrid for imaging eectrica conductivity and permittivity at ow frequency 337 Note that N i is the set of cosest neighbours of (x i,y ), in the interior of. Moreover, N i contains the grid points of the nine-point stenci of discretization (3.5). A bock of unknowns in our Gauss Seide scheme consists of the components of u that correspond to σ () (x) and ωε () (x), at points x N i. Let us denote by I m the set of indices of components of u that beong to such a bock, and suppose that we have a tota of M bocks. The noninear Gauss Seide reaxation scheme is For m = 1,...,M, u,r = u,r, for a r I m, where u,r are soutions of the noninear system of equations L,r ({u,p,p/ I m }, {u,p,p I m}) = g,r, r I m. End. (3.15) In [4, 4], it is shown that it is not necessary to sove the noninear system of equations in (3.15). Instead, we ust take one Newton step toward its soution. As emphasized in [4], this is not reated to any goba inearization of our probem. We ust inearize a few equations in (3.13), with respect to a few unknowns, for the soe purpose of smoothing the error. At sma α, making a fu Newton step might not be satisfactory [13]. Instead, we use a ine search approach, where we take a fraction λ m of the Newton step. The ine search parameter λ m is cacuated such that the sum of square residuas at grid points x G, satisfying x (x i,y ) kh, does not increase after updating u,p, for a p I m. In our numerica experiments, we take k = 3. We have impemented two versions of procedure NMGM. The first version is the fu approximation storage (FAS) agorithm introduced in [5]. Procedure NMGM(, u, g ) : If = 0, u = L 1 0 (g ). Ese, do ν 1 pre-smoothing iterations u S (ν 1) (u, g ). v = Ru, d = L 1 (v) + R[g L (u )], w = v. Ca NMGM( 1, w, d) M times. Do coarse-grid correction u = u + P(w v). Do ν post-smoothing iterations u S (ν ) (u, g ). End. (3.16) The terms pre- and post-smoothing refer to the noninear, bock Gauss Seide iterations performed before and after coarse-grid correction, respectivey. Parameters ν 1, ν and M are defined by the user. In our impementation, ν 1 =, ν = 0 and M =. The second version of NMGM, introduced in [4], is a sight modification of (3.16), which uses coarse-grid soutions

10 338 L Borcea u 1 cacuated in advance. Suppose that u k and g k = L k (u k ), for k = 0, 1,..., 1, are known. Procedure NMGM(, u, g ) : If = 0, u = L 1 0 (g ). Ese, do ν 1 pre-smoothing iterations u S (ν 1) (u, g ), d = g 1 + τ R[g L (u )], (3.17) w = u 1. Ca NMGM( 1, w, d) M times. Do coarse-grid correction u = u + 1 τ P(w u 1). Do ν post-smoothing iterations u S (ν ) (u, g ). End. Version (3.17) is more convenient for anaysis, due to the extra parameter τ that can be chosen in such a way that d, needed for the coarse-grid correction, is sufficienty sma (see section 6..1). Nevertheess, for our probem, numerica experiments show that, with a proper τ, (3.16) and (3.17) have a simiar performance. We concude this section with an expanation of the coarse-grid correction. The anaysis of versions (3.16) and (3.17) is simiar, so we consider the FAS agorithm (3.16). After ν 1 pre-smoothing iterations, the residua on grid G is given by r = g L (u ), (3.18) and the error is δu = L 1 (g ) L 1 (g r ) = L 1 (L 1 (g ))r +O( r ), (3.19) where L (L 1 (g )) is the Jacobian of L, cacuated at the soution of (3.13). The computation of δu from (3.19) is too expensive. However, δu is smooth (due to pre-smoothing) and thus it can be approximated by δũ on the coarser grid G 1. Agorithm (3.16) gives δũ = P(w v) = P[L 1 1 (d) Ru ], (3.0) where L 1 1 (d) = L 1 1 (L 1(Ru ) + Rr ) = Ru + L 1 1 (Ru )Rr +O( r ). (3.1) Hence, the coarse-grid correction is δũ = PL 1 1 (Ru )Rr +O( r ), (3.) an approximation of (3.19) Acceeration of coarse-grid correction in the FAS agorithm After ν 1 reaxation sweeps, procedure NMGM, given by (3.16), takes u and residua r = g L (u ), and proects them on G 1, for the purpose of coarse-grid correction. Since the nested iteration agorithm (3.1) makes M repeated cas of NMGM, wehavea sequence of admittivities u. At step = 1,,..., M, et us denote by ũ the admittivity obtained after ν 1 pre-smoothing iterations. We can speed up the convergence of (3.16) by using the minima residua smoothing (MRS) acceeration technique [47, 48], where we proect on

11 A noninear mutigrid for imaging eectrica conductivity and permittivity at ow frequency 339 G 1 the inear combination βũ + (1 β)ũ 1. The acceeration parameter β is chosen such that the new admittivity minimizes the Eucidian norm of the residua. Suppose that, in nested iteration (3.1), we have reached eve, where 1 K. At this eve, for = 1,..., M, we ca procedure NMGM. To acceerate convergence, we modify (3.16) as Procedure NMGM(, u, g ): If = 0, u = L 1 0 (g ). Ese, u S (ν 1) (u, g ), r = g L (u ). If =, ũ = u. If >1, given β, a minimizer of g L (βũ + (1 β)ũ 1 ), ũ = βũ + (1 β)ũ 1, u =ũ, (3.3) r = g L (u ), end. End. v = Ru, d = L 1 (v) + Rr, w = v. Ca NMGM( 1, w, d) M times. u = u + P(w v), u S (ν ) (u, g ). End. In our impementation, we sove the noninear, one-dimensiona minimization probem for β, with the MATLAB function fminbnd. The MRS acceeration is done ony at the highest eve, reached in iteration (3.1), where g = 0 hods. On grids G, where = 1,..., 1, we are soving L (u ) = g, for the purpose of correcting the soution on G and, the right-hand side g on G, given by a vector d such as in (3.3), changes at each iteration. This means that the smoothed u from the previous iteration has nothing to do with u of the current iteration and the MRS acceeration technique does not appy to the intermediate grids. Note however that, for coarsegrid correction, procedure NMGM is caed M times, with the same right-hand side d. Hence, we can define shorter sequences of admittivities on the intermediate grids and use the MRS technique for acceeration. In our numerica experiments, we take M =, so we have no intermediate-grid acceeration. 4. The coarse-grid soution As shown in [4] and section 6, an essentia requirement for a successfu mutigrid iteration is that the approximation property L 1 (g ) PL 1 1 (Rg ) = O(h p ) hod, for some p>0.

12 340 L Borcea In particuar, this means that ρ ( 1), the soution of (3.4) on G 1 must be cose to ρ (), the soution on G. In section 3.1, we describe the piecewise inear, finite-eement method used to cacuate φ and ψ and, impicity, the noninear part of F or L. The accuracy of φ and ψ is O(h ), provided that the excitation current I at is smooth [7]. On the coarse grids, especiay on G 0, where the grid spacing can be quite arge, we expect a significant error in the cacuation of both φ ψ and the misfit φ V at the boundary. Due to the i-posedness of the inverse probem, the error in the boundary misfit φ V is highy ampified in the image of the admittivity in the interior of. To obtain a reasonabe ρ () on a coarse grid, we need a arge reguarization parameter which eads to unsatisfactory, burry images. One possibe way of ameiorating this probem is to use a higher-order numerica scheme for the soution of (.9) and (.14), on the coarse grids. In the finite-eement setting, this means interpoating φ, ψ and ρ () by higher-degree poynomias. For exampe, suppose that, on the trianguation T 0, we take a piecewise quadratic interpoation of φ and the admittivity. Then, the vector of unknowns in (3.7) contains vaues of φ at (x i,y ) G 0 as we as at additiona points, such as (x i + h 0,y ), (x i,y + h 0 ), (x i + h 0,y + h 0 ). Simiary, a piecewise quadratic interpoation of the admittivity causes matrix A (0) in (3.7) to depend not ony on ρ (0), the restriction of ρ at (x i,y ) G 0 but aso on ρ(x i + h 0,y ), ρ(x i,y + h 0 ) and ρ(x i + h 0,y + h 0 ). Hence, the discretization (3.4) at eve = 0 becomes dependent on the discretization on the next grid, G 1, and the computationa cost on G 0 is too high. A high-order finite-difference scheme coud be used to sove (.9) and (.14), as we, but the accuracy of the resut woud be highy dependent on the smoothness of I (x). Finay, it is most convenient to have the same numerica method for the soution of (.9) and (.14) on a grid eves. In this section, we introduce an idea that aows us to get a good approximation of ρ on the coarse grids, without changing the numerica scheme for the soution of (.9) and (.14) on coarse grids and, especiay, without increasing α and thus compromising resoution A coarse-grid east-squares probem We assume that we know the NtD map at some or a boundary points of the finest grid G K, and we denote by V the vector of measured potentias V (x), for x G K. Then, we expect that V Λ K (ρ (K) )I, (4.1) where Λ K is the discrete NtD map for the fine grid and ρ (K) is the restriction of the true admittivity on grid G K. The idea coarse-grid image is the restriction of ρ on G 0. Therefore, on grid G 0, we formuate the east-squares probem N exp min ρ (0) R 0 K Λ K (P K 0 ρ (0) )I R 0 K V + α 0 h0 ρ (0) h 0,0, (4.) =1 where h0 ρ (0) h0,0 is the discrete version of the L norm of ρ. By soving (4.), we ook for a soution ρ (0) that is cose to the restriction of the true admittivity ρ on G 0. The restriction operator in (4.) is R 0 K = R R K 1 K, (4.3) where R 1 takes functions defined at G into functions defined at G 1. We consider the weighted restriction d 1 (t) = R 1 d = 1 4 [d (t h ) +d (t) + d (t + h )],

13 A noninear mutigrid for imaging eectrica conductivity and permittivity at ow frequency 341 for d given at G and t G 1. Simiary, P K 0 = P K K 1...P 1 0, (4.4) where P 1 is the nine-point proongation defined in section 3.1. Proposition 1. The foowing factorization hods: R 0 K Λ K (P K 0 ρ (0) )I = Λ 0 (ρ (0) )I (0) (ρ (0) ), (4.5) where matrix Λ 0 is the coarse-grid NtD map and I (0) is a vector of currents defined at G 0. The definition of Λ 0 and I (0) depends on the numerica soution of (.9), on G 0. We choose the piecewise inear, finite-eement approach described in section 3.1, where the boundary current is I (0) L ( ), the inear spine interpoation of the noda vaues in vector I (0). Proof. Consider trianguation T 0 of, with vertices given by the grid points in G 0. Let r (0) (x) be the piecewise inear interpoation of ρ (0), given on G 0. The finite-eement soution φ (0) (x) of (.9), with admittivity r (0) (x) and excitation I (0), satisfies r (0) (x) φ (0) (x) ϑ(x) dx = ϑ(t)i (0) (t) dt, (4.6) for a piecewise inear functions ϑ(t) defined on T 0. The discretization of (4.6) eads to a inear system of equations, simiar to (3.7), which we write in bock form as ( ) ( B (0) (ρ (0) ) C (0) (ρ (0) ) φ (0) ) ( ) (interior) 0 D (0) (ρ (0) ) E (0) (ρ (0) = ) φ (0) Q I (0), (4.7) (boundary) where φ (0) (interior) C (n 0 1) contains the vaues of φ (0) at the interior nodes and φ (0) (boundary) C 4n 0 is the vector of boundary potentias. Matrices B (0), C (0), D (0) and E (0) depend ineary on ρ (0), whereas Q R 4n 0 4n 0 is a constant, nonsinguar matrix. Now, consider the Dirichet probem r (0) (x) ϕ (0) (x) ϑ(x) dx = 0, ϕ (0) = inear spine interpoation of R 0 K Λ K (P K 0 ρ (0) )I, (4.8) for a piecewise inear functions ϑ(t) defined on T 0, that vanish at. We wish to cacuate such that I (0) φ (0) (boundary) = R 0 KΛ K (P K 0 ρ (0) )I = ϕ (0) (boundary). (4.9) Since φ (0) and ϕ (0) sove the same equation, (4.9) impies φ (0) (interior) = ϕ(0) (interior), the unique soution of (4.8). Finay, I (0) is uniquey defined by I (0) (ρ (0) ) = Q 1 (D (0) (ρ (0) )E (0) (ρ (0) )) ( (0) ) ϕ (interior) ϕ (0) (boundary) (4.10) and proposition 1 is proved. In ight of proposition 1, we rewrite the coarse-grid minimization probem (4.) as N exp min Λ 0 (ρ (0) )I (0) ρ (0) (ρ (0) ) R 0 K V + h 0 ρ (0) h 0,0. (4.11) =1 It appears that a numerica soution of (4.11) is very expensive because the coarse-grid current (ρ (0) ) depends on ρ (0) and, to cacuate it, we must sove equation (.9), on the fine grid I (0)

14 34 L Borcea G K, for admittivity P K 0 ρ (0). However, proposition shows that the dependence of I (0) on ρ (0) is much weaker than that of Λ 0. Hence, we can sove (4.11) with ony a few updates of I (0). Proposition. Consider a sma perturbation δρ (0) of ρ (0), such that δρ (0) G0 = 0. The coarse-grid boundary vaues of the potentia are perturbed by δ[λ 0 (ρ (0) )I (0) (ρ (0) )] = δλ 0 (ρ (0) )I (0) (ρ (0) ) + Λ 0 (ρ (0) )δi (0) (ρ (0) ), where δλ 0 (ρ (0) )I (0) (ρ (0) ) Λ 0 (ρ (0) )δi (0) (ρ (0) ). (4.1) Proof. As we did in the proof of proposition 1, we denote by r (0) (x) and δr (0) (x) the piecewise inear interpoation of ρ (0) and δρ (0), respectivey. We have δλ 0 (ρ (0) )I (0) (ρ (0) ) = δφ (0) G0 +O( δρ (0) ), (4.13) where δφ (0) is the finite-eement soution of [r (0) (x) δφ (0) (x)] = [δr (0) (x) φ (0) (x)] in, δφ (0) (x) n(x) = 0 on, (4.14) δφ (0) (s) ds = 0, and φ (0) where δ φ (0) soves (4.6). Moreover, Λ 0 (ρ (0) )δi (0) (ρ (0) ) = δ φ (0) G0 +O( δρ (0) ), (4.15) is the finite-eement soution of [r (0) (x) δ φ (0) (x)] = 0 in, r (0) (x) δ φ (0) (x) n(x) = δi (0) (x) on, δ φ (0) (s) ds = 0. (4.16) Current density δi (0) is the inear spine interpoation of the noda vaues in vector δi (0),given by ( (0) ) ϕ δi (0) (ρ (0) ) = Q [(δd 1 (0) (ρ (0) )δe (0) (ρ (0) (interior) )) +(D (0) (ρ (0) )E (0) (ρ (0) )) ϕ (0) (boundary) ( δϕ (0) (interior) δϕ (0) (boundary) )], (4.17) where ϕ (0) and δϕ (0) are the finite-eement soutions of (4.8) and [r (0) (x) δϕ (0) (x)] = [δr (0) (x) ϕ (0) (x)] in, δϕ (0) = inear spine interpoation of R 0 K δλ K (P K 0 ρ (0) (4.18) )I, respectivey. On the finest grid G K, we denote by r (K) (x) and δr (K) (x) the piecewise inear interpoation of P K 0 ρ (0) and P K 0 δρ (0), on trianguation T K. Suppose that φ (K) is the fine-grid, finiteeement soution of (.9), with admittivity r (K) (x) and known excitation current I (x). We have δλ K (P K 0 ρ (0) )I = δφ (K) GK +O( δρ (0) ), (4.19)

15 A noninear mutigrid for imaging eectrica conductivity and permittivity at ow frequency 343 where δφ (K) is the finite-eement soution of [r (K) (x)δ φ (K) (x)] = [δr (K) (x) φ (K) (x)] in, δφ (K) (4.0) n(x) = 0 on, on grid G K. The discretization of (4.0) eads to a inear system of equations which we write in bock form as ( δb (K) (P K 0 ρ (0) ) δc (K) (P K 0 ρ (0) ) δd (K) (P K 0 ρ (0) ) δe (K) (P K 0 ρ (0) ) + ) ( φ (K) (interior) φ (K) (boundary) ( ) ( B (K) (P K 0 ρ (0) ) C (K) (P K 0 ρ (0) ) δφ (K) (interior) D (K) (P K 0 ρ (0) ) E (K) (P K 0 ρ (0) ) δφ (K) (boundary) ) ) = 0. (4.1) Here, matrices B (K), C (K), D (K) and E (K) depend ineary on the admittivity, and they are simiar to B (0), C (0), D (0) and E (0) in (4.7). At ast, we note that, by construction, at G 0, ϕ (0) is the restriction of φ (K). Moreover, the corresponds to an admittivity P K 0 ρ (0) that is a inear interpoation fine-grid potentia φ (K) of ρ (0). On both grids G 0 and G K, we are discretizing the same eiptic operator and the discretization is consistent. This impies that the perturbation current in (4.17) is approximatey ( (K) ) φ (interior) δi (0) (ρ (0) ) Q 1 R 0 K [(δd (K) (P K 0 ρ (0) )δe (K) (P K 0 ρ (0) )) +(D (K) (P K 0 ρ (0) )E (K) (P K 0 ρ (0) )) ( δφ (K) (interior) δφ (K) (boundary) φ (K) (boundary) )] = 0 and so Λ 0 (ρ (0) )δi (0) (ρ (0) ) O( δρ (0) ) is much smaer than δλ 0 (ρ (0) )I (0) (ρ (0) ) = O( δρ (0) ). To iustrate the statement of proposition, we show next a numerica comparison of δλ 0 I (0) and Λ 0 δi (0). Consider an admittivity ρ (0) with rea and imaginary parts shown in figures 1 and. We perturb ρ (0) by δρ (0), where 0.1 if (x i,y ) = (4 h 0, 3 h 0 ) δρ (0) (x i,y ) = 0.1i if (x i,y ) = (3 h 0, 5 h 0 ) 0 otherwise. The rea parts of δλ 0 I (0) and Λ 0 δi (0) are shown in figures 3 and 4, where we take two different current excitations I on the fine grid. The comparison between the imaginary parts is simiar to that in figures 3 and 4. As predicted by proposition, perturbation δλ 0 I (0) is indeed much stronger than Λ 0 δi (0). 4.. Numerica agorithm for coarse-grid soution Nested iteration (3.1) begins with the coarse-grid admittivity u 0 L 1 0 (0), which we cacuate as foows: We start with the initia guess ρ (0) given by the restriction on G 0 of the harmonic extension inside, of the known admittivity at the boundary. For this ρ (0),we cacuate the coarse-grid currents I (0), for = 1,...,N exp, as expained in proposition 1. Given the sensitivity anaysis of proposition, we update ρ (0) a coupe of times, without

16 344 L Borcea Figure 1. Rea part of admittivity ρ (0). Figure. Imaginary part of ρ (0). x x Figure 3. Rea part of the perturbation of the boundary potentia due to δρ (0). Experiment 1. Figure 4. Rea part of the perturbation of the boundary potentia due to δρ (0). Experiment. changing I (0). Then, we recacuate the coarse-grid currents, make a coupe of updates on ρ (0) and so on. The agorithm that we impemented is Get ρ (0) and, impicity, u 0, the initia guess. Set reguarization parameter α 0 to a sufficienty arge vaue. For k = 1,,... Cacuate coarse-grid currents I (0) (ρ (0) ), = 1,...,N exp, as expained in proposition 1. Sove L 0 (α 0, u 0 ) = 0. In the evauation of L 0 (α 0, u 0 ), (4.) use φ (0) given by (4.7). To cacuate the adoint potentia ψ (0), use the adoint current φ (0) G0 R 0 K V. Decrease α 0. End. Foragivenα 0 and fixed coarse-grid currents, we sove L 0 (α 0, u 0 ) = 0 with a Newton-type method, impemented in the pubic domain software hybr []. Note that we start the coarse-

17 A noninear mutigrid for imaging eectrica conductivity and permittivity at ow frequency 345 grid iteration with a arge reguarization parameter α 0 and we reduce it as we go aong. The motivation for this iterative reguarization approach is that, in the beginning, we expect to have a poor guess of the soution. This means that, in factorization (4.5), we can have coarse-grid currents that are very different from I (0) (ρ (0) ), where ρ (0) is the restriction of the true ρ, on G 0. Hence, as discussed at the beginning of section 4, we need a arge α 0 to stabiize the coarse-grid inversion. As we iterate, we expect ρ (0) to get coser to ρ (0), so we decrease the reguarization parameter. In our numerica experiments, α 0 is reduced by a factor of 1.5 at each iteration. We terminate iteration (4.) over k by asking that the reative change in u 0 at two consecutive iterations is ess than or equa to Numerica agorithm (4.) is used ony once, at the beginning of nested iteration (3.1). Then, we interpoate u 0 on G, for = 1,...,K 1, and cacuate the currents on G,by proceeding as in proposition 1. These currents are needed in the cacuation of φ () and ψ (), as we as the evauation of L (u ) on G. As the admittivity on the intermediate grid eves is updated, it may be usefu to recacuate the coarse-grid currents from time to time. However, in a the numerica tests that we tried, we found that such additiona current corrections were not reay necessary. That is, as stated in proposition, the coarse-grid currents change very sowy with the admittivity and this aows us to get good coarse-grid images with a sma number of fine-grid function evauations. In fact, in a our experiments, after getting u 0,we fix the currents on G, where = 0, 1,...,K 1, for the entire duration of the mutigrid iteration. 5. Numerica resuts In a numerica cacuations, we consider a unit square domain and a sequence of three, uniform grids, with spacing h = 1, where = 0, 1,. For inversion, we take N +3 exp = 7 excitation currents I, that satisfy I (s) ds = 0, and I (x) = 0 for x in the vicinity of the corners of. Given any I, consider the restriction R 0 K, which is simpe inection from to G 0. We have chosen currents I such that the coarse-grid currents in sequence { R 0 K I } 7 =1 are ineary independent. That is, we have a compete set of currents on G 0 \{(0, 0), (0, 1), (1, 0), (1, 1)}. Note that the soe purpose of mentioning R 0 K I is to expain the geometrica configuration of the current excitations I. Restrictions R 0 K I are not used in the cacuations. Instead, the coarse-grid currents are obtained from I,as expained in proposition 1. For any excitation I, the resuting votage V is given at a boundary nodes in G, except the corners of. We have synthetic data, where we cacuate potentia φ by soving forward probem (.9), for a I, = 1,...,7, on the fine grid G. The forward probem sover is the piecewise inear, finite-eement numerica scheme described in section 3.1. Data V are the restriction of φ on G, to which we usuay add random, Gaussian, mutipicative noise. In the mutigrid procedure NMGM defined by (3.16) or (3.17), we take ν 1 = presmoothing iterations and no post-smoothing (ν = 0). For coarse-grid correction, we ca NMGM, recursivey, M = times. In a experiments that we tried, choice τ = 0.1 in version (3.17) proved successfu. In fact, we found that versions (3.16) and (3.17) of NMGM behaved quite simiary. However, we prefer the FAS version (3.16), because it can be acceerated, as shown in (3.3). A numerica resuts of this section are obtained with procedure NMGM, given by (3.3). We iustrate the performance of our agorithm by reconstructing the true admittivity function ρ(x) with rea and imaginary parts shown in figures 5 and 6.

18 346 L Borcea Figure 5. Rea part of true ρ(x). Figure 6. Imaginary part of true ρ(x). Figure 7. Rea part of coarse-grid image ρ (0) cacuated with agorithm (4.). Noiseess data. Figure 8. Imaginary part of coarse-grid image ρ (0) cacuated with agorithm (4.). Noiseess data Noiseess data First, we consider noiseess data. The coarse-grid image ρ (0) shown in figures 7 and 8 is obtained with agorithm (4.). At the boundary, the true admittivity is ρ(x) = and its harmonic extension inside is a constant, equa to, as we. Hence, we start agorithm (4.) with initia guess ρ (0) (x) =, for a x G 0. The starting vaue of α 0 in (4.) is 10 6 and, at each iteration, α 0 is reduced by a factor of 1.5. For α 0 [10 10, ], the image ρ (0) remained practicay unchanged, so we stopped the iteration when α 0 became smaer than Figures 7 and 8 demonstrate a very good reconstruction of the coarse grid restriction of the true admittivity. For comparison, in figures 9 and 10, we show the coarse-grid image cacuated directy with the given data currents {I } 7 =1, and a reguarization parameter α 0 = That is, to evauate L 0 (u 0 ), we sove equation (.9) on G 0, where, instead of cacuating the coarse-grid currents as in proposition 1, we simpy take the given data {I } 7 =1, known at G. The resuting image is ceary far inferior to that given by agorithm (4.). The two conductive peaks of the rea part of the admittivity cannot be distinguished. Instead, figure 9 shows a burry, arge conductive region, where the two peaks of rea(ρ) are merged together. Furthermore, figure 10 does not show the arge imaginary part of ρ in the centre of. It may appear that the burry coarse-grid

19 A noninear mutigrid for imaging eectrica conductivity and permittivity at ow frequency 347 Figure 9. Rea part of coarse-grid image ρ (0). No use of ideas in section 4. Figure 10. Imaginary part of coarse grid image ρ (0).No use of ideas in section Figure 11. Rea part of fine-grid image ρ (). Noiseess data. Figure 1. Rea part of fine-grid image ρ (). Noiseess data. image in figures 9 and 10 might be improved by reducing α 0. However, our experiments show quite the opposite. For exampe, α 0 = 10 8 gives a ρ (0) that is highy osciatory, and its rea part even becomes negative at some grid points in G 0. Such difficuties have been observed in a our numerica experiments and they are, in fact, what we expect. As expained in section 4, the inaccurate coarse-grid soution of equation (.9) and the i-posedness of the inverse probem force us to take a arge reguarization parameter in order to get a reasonabe (athough burry) coarse-grid image. In contrast, the ideas of section 4, impemented in agorithm (4.), ead to a much improved image on G 0, as shown in figures 7 and 8. The reguarization parameter on the intermediate grid G 1 is α 1 = Finay, the image on G is shown in figures The reguarization parameter on G is α = Note that we have done reconstructions with various reguarization parameters and found that α < give osciatory images of ρ (), whereas α > 10 8 give a ρ () that is too smooth. The progress of nested iteration agorithm (3.1), with NMGM given by (3.3) is as foows: Initia guess ρ(x) = in. Initia residua on fine grid: L (u ) = e.

20 348 L Borcea Figure 13. Imaginary part of fine-grid image ρ (). Noiseess data. Figure 14. Imaginary part of fine-grid image ρ (). Noiseess data. Sove coarse-grid probem and begin nested iteration (3.1): Leve = 1 L 1 (u 1 ) L 1 (u 1 ) Starting residua: e e 5 After two smoothing iterations: e e 6 After coarse-grid correction: e e 6 After two smoothing iterations: e e 7 After coarse-grid correction: e e 7 Leve = L (u ) L (u ) Starting residua: e e 5 After two smoothing iterations: e e 5 After coarse-grid correction: e e 6 After two smoothing iterations: e e 6 After coarse-grid correction: e e 6 After two smoothing iterations: e e 6 After coarse-grid correction: e e Noisy data To test the performance of our imaging agorithm for noisy data, we add random, Gaussian, mutipicative noise to V, for = 1,...,7. First, we take a.5% noise eve and obtain the image shown in figures 15 and 16. This reconstruction is done with a reguarization parameter α = For the reconstructed image, boundary misfit 7 =1 φ G V is at the noise eve. Note that the image ρ () is very cose to the noiseess one shown in figures The same is true for the coarse- and intermediate-grid images. The good quaity of the coarse-grid image in the experiments shown so far might ead us to the question: why bother to cacuate ρ () with the mutigrid and not stop at ρ (0)? The advantage of doing fine-grid reconstruction is shown in the next experiment, where the noise eve is 4.5%. The coarse-grid image shown in figures 17 and 18 is not as good as before and the smaer conductive region is mispaced. Nevertheess, this is corrected in the fine-grid

21 A noninear mutigrid for imaging eectrica conductivity and permittivity at ow frequency 349 Figure 15. Rea part of fine-grid image ρ (). Noise eve =.5%. Figure 16. Imaginary part of fine-grid image ρ (). Noise eve =.5%. Figure 17. Rea part of coarse-grid image ρ (0). Noise eve = 4.5%. Figure 18. Imaginary part of coarse grid image ρ (0). Noise eve = 4.5%. image shown in figures 19 and 0. The reguarization parameter on G is α = 10 9 and, at the end, the boundary misfit 7 =1 φ G V is at the noise eve. Ceary, the images in figures 19 and 0 are not as good as the noiseess ones, but we can sti distinguish the important features of the true admittivity. Moreover, the recovered image has a magnitude that remains cose to the actua vaue of the true admittivity in Summary of numerica resuts The numerica experiments presented in this section demonstrate that the noninear, nested iteration, mutigrid agorithm (3.1) can be effective in inversion. The constructed images of admittivity function ρ have the correct geometrica features as we as a good contrast. Moreover, the agorithm is stabe in the presence of noisy data. A key part in the success of the mutigrid iteration is payed by the coarse-grid soution. We have showed that agorithm (4.), described in section 4., can give coarse-grid images ρ (0) that are good approximations of the true admittivity. This is of great importance in the mutigrid iteration (3.1) for two reasons: (1) We have a good starting admittivity in the nested iteration or, equivaenty, a sma starting residua. For exampe, in the case of noiseess data, before having the coarse-grid image,

22 350 L Borcea Figure 19. Rea part of fine-grid image ρ (). Noise eve = 4.5%. Figure 0. Imaginary part of fine-grid image ρ (). Noise eve =4.5%. the residua is After the coarse-grid soutions, the residua on G drops by two orders of magnitude. () The coarse-grid corrections are effective in speeding up convergence. The combination of these two attributes aows us to construct fine-grid images with ust three cas of mutigrid procedure NMGM, on grid G, where computations are expensive. In contrast, if the ideas of section 4 are not used in the coarse-grid cacuations, we have found that, to get mutigrid convergence, we need arge reguarization parameters. This is ceary undesirabe because the images have poor resoution. 6. Convergence anaysis In this section, we estabish the oca convergence of the noninear mutigrid agorithm (3.1), for the compex Dirichet probem (.15) and (3.1) written in abstract form as (3.). To fix ideas, et us suppose that R. Our proof is based on the theory presented in [4] and it assumes that (3.) is sufficienty reguarized to achieve oca convergence Definitions and reguarity estimates We proceed as in section 3. and write (3.) or, equivaenty, (.15), (3.1), in the rea form L(α, u) = 0, (6.1) for some reguarization parameter α. For given admittivity ρ (or u), potentias φ and ψ entering the noninear part of equation (6.1) are soutions of (.9) and (.14), respectivey. Suppose that in (.9) and (.14) we have a ρ H ( ), with stricty positive rea part. By Soboev inequaities [3], ρ L q ( ), for any q< and, given a smooth boundary and an excitation current I H 1 ( ), equation (.9) has a unique soution φ H ( ) (see emma 9.1 in [31]). Simiary, equation (.14) has a unique soution ψ H ( ). Hence, by Soboev inequaities [3], φ and ψ are in L 4 ( ) and φ ψ L ( ). In ight of this resut, we suppose that equation (6.1) has a soution u, or equivaenty, ρ with ρ L ( ). Given that such a soution ρ H ( ) exists, we show that the mutigrid agorithm proposed in this paper is ocay convergent. In what foows, we sha need the divided difference DL defined as L(α, u + δu) L(α, u) = DL(u, δu)δu, (6.)