Applied Numerical Mathematics

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1 Applied Numerical Matematics 59 (2009) Contents lists available at ScienceDirect Applied Numerical Matematics Accuracy and run-time comparison for different potential approaces and iterative solvers in finite element metod based EEG source analysis S. Lew a,b,c.h.wolters c,,t.dierkes c,c.röer c,r.s.macleod a,b a Scientific Computing and Imaging Institute, University of Uta, Salt Lake City, USA b Department of Bioengineering, University of Uta, Salt Lake City, USA c Institut für Biomagnetismus und Biosignalanalyse, Westfälisce Wilelms-Universität Münster, Münster, Germany article info abstract Article istory: Received 11 January 2008 Received in revised form 4 February 2009 Accepted 25 February 2009 Available online 5 Marc 2009 Keywords: Electroencepalograpy Source reconstruction Finite element metod Dipole singularity Full subtraction potential approac Venant potential approac Partial integration potential approac Preconditioned conjugate gradient metod Algebraic multi-grid preconditioner Incomplete Colesky preconditioner Jacobi preconditioner Constrained Delaunay tetraedralization Anisotropic four-layer spere model Accuracy and run-time play an important role in medical diagnostics and researc as well as in te field of neuroscience. In Electroencepalograpy (EEG) source reconstruction, a current distribution in te uman brain is reconstructed noninvasively from measured potentials at te ead surface (te EEG inverse problem). Numerical modeling tecniques are used to simulate ead surface potentials for dipolar current sources in te uman cortex, te so-called EEG forward problem. In tis paper, te efficiency of algebraic multi-grid (AMG), incomplete Colesky (IC) and Jacobi preconditioners for te conjugate gradient (CG) metod are compared for iteratively solving te finite element (FE) metod based EEG forward problem. Te interplay of te tree solvers wit a full subtraction approac and two direct potential approaces, te Venant and te partial integration metod for te treatment of te dipole singularity is examined. Te examination is performed in a four-compartment spere model wit anisotropic skull layer, were quasi-analytical solutions allow for an exact quantification of computational speed versus numerical error. Specifically-tuned constrained Delaunay tetraedralization (CDT) FE meses lead to ig accuracies for bot te full subtraction and te direct potential approaces. Best accuracies are acieved by te full subtraction approac if te omogeneity condition is fulfilled. It is sown tat te AMG-CG acieves an order of magnitude iger computational speed tan te CG wit te standard preconditioners wit an increasing gain factor wen decreasing mes size. Our results sould broaden te application of accurate and fast ig-resolution FE volume conductor modeling in source analysis routine IMACS. Publised by Elsevier B.V. All rigts reserved. 1. Introduction Electroencepalograpy (EEG) based source reconstruction of cerebral activity (te EEG inverse problem) is an important tool bot in clinical practice and researc [35,23], and in cognitive neuroscience [2]. Metods for solving te inverse problem are based on solutions to te corresponding forward problem, i.e., te simulation of EEG potentials for a given primary source in te brain using a volume-conduction model of te uman ead. Wile te teory of tis forward problem is well establised and many numerical implementations exist, tere remain unresolved questions regarding te accuracy and * Corresponding autor. Privatdozent Dr. Carsten H. Wolters, Institut für Biomagnetismus und Biosignalanalyse, Westfälisce Wilelms-Universität Münster, Malmedyweg 15, Münster, Germany. Tel.: +49/(0) , fax: +49/(0) addresses: slew@sci.uta.edu (S. Lew), carsten.wolters@uni-muenster.de (C.H. Wolters), tomas.dierkes@uni-muenster.de (T. Dierkes), c.roeer@uni-muenster.de (C. Röer), macleod@sci.uta.edu (R.S. MacLeod). URL: ttp://biomag.uni-muenster.de (C.H. Wolters) /$ IMACS. Publised by Elsevier B.V. All rigts reserved. doi: /j.apnum

2 S. Lew et al. / Applied Numerical Matematics 59 (2009) efficiency of contemporary approaces. In tis study, we compared a range of numerical tecniques and source representation approaces and ave sown tat careful coice of bot are critical in order to solve realistic electroencepalograpic forward (and inverse) problems. Te general approac for solving bioelectric field problems under realistic conditions is well establised. All quantitative solutions for te EEG forward problem are based on te quasi-static Maxwell equations [25]. Te primary sources are electrolytic currents witin te dendrites of te large pyramidal cells of activated neurons in te uman cortex. Even if tere are also smooter models [33], most often te primary sources are formulated as a matematical point current dipole [25,6,18]. Te finite element (FE) metod is often used for te solution of te forward problem, because it allows for a realistic representation of te complicated ead volume conductor wit its tissue conductivity inomogeneities and anisotropies [44,3,1,34,4,15,19,36,26,43,7]. To implement te point current dipole as a current source in te brain, te FE metod requires careful consideration of te singularity of te potential at te source position. One way to address te singularity is to use a subtraction approac, wic divides te total potential into an analytically known singularity potential and a singularity-free correction potential, wic can ten be approximated numerically using an FE approac [3,1,34,15,26,38,43]. For te correction potential, te existence and uniqueness for a weak solution in a zero-mean function space ave been proven and FE convergence properties are known [43]. Te subtraction FE approac as tus a sound matematical basis for point current dipole models. Anoter family of source representation metods, known as direct FE approaces to te total potential [44,1,4,36,26], are computationally less expensive, but also matematically less sound under te assumption tat a point dipole is te more realistic source model. In our previous work [38,41], we compared te projected subtraction approac [43] wit te two direct approaces using partial integration [44,1,36] and Venant [4] in regular and geometry-adapted 2 mm exaedral FE meses of a multi-layer spere model for wic quasi-analytical solutions exist [5]. Te Venant approac was found to be te overall best coice in FE source analysis practice [38,41]. It was owever speculated tat an improved numerical quadrature migt solve te accuracy problems of te projected subtraction approac for eccentric sources, so tat te igest future potential was given to an improved implementation of te FE subtraction approac [41]. It as recently been sown tat a full subtraction approac [7] using an improved numerical quadrature leads to an order of magnitude more accurate solution tan te projected subtraction approac [43], especially wen considering sources tat are close to a conductivity inomogeneity. Anoter general prerequisite for FE modeling of bioelectric fields is te generation of a mes tat represents te geometry and electric properties of te volume conductor. An effective mesing strategy will balance acceptable forward problem accuracy against reasonable computation times and memory usage. Very ig accuracies can be acieved by making use of a Constrained Delaunay Tetraedralization (CDT) [30,29] in combination wit a full subtraction approac [7]. Adaptive metods, using local refinement around te source singularity [3,34], are anoter potential utility but tey preclude te use of fast transfer matrices [36,8,40,12] and lose efficiency in solving te inverse problem (see discussion section). Solving te forward problem is rarely te ultimate goal in calculating bioelectric fields but rater a step towards solving te associated inverse problem. Tus te quest for numerical accuracy and efficiency of te forward solution requires some anticipation of te ultimate use in inverse solutions. Te longtime state-of-te-art approac as been to solve an FE equation system for eac anatomically and pysiologically meaningful dipolar source (eac source results in one FE rigt-and side (RHS) vector) [3,1,34,4,15]. Te use of standard direct (banded LU factorization for a 2D source analysis scenario [1]) or iterative (Conjugate Gradient (CG) witout preconditioning [3] or Successive OverRelaxation (SOR) [26]) FE solver tecniques limit te overall resolution of te geometric model because of teir computational cost. Te preconditioned CG metod was used wit standard preconditioners like Jacobi (Jacobi-CG) [36] or incomplete Colesky witout fill-in, IC(0)-CG [4]. One recent approac to acieve efficient computation of te FE-based forward problem is to pre-compute transfer matrices tat encapsulate te relationsip between source locations and sensor sites based only on te geometric and conductivity caracteristics of te volume conductor, i.e., tey are independent of te source. Tecniques exist to construct transfer matrices for problem formulations based on EEG [36,12] or combined EEG and MEG [8,40]. Using tis principle, for eac ead model, one only as to solve one large sparse FE system of equations for eac of te possible sensor locations in order to compute te full transfer matrix. Eac forward solution is ten reduced to multiplication of te transfer matrix by an FE RHS vector containing te source load. Exploiting te fact tat te number of sensors (currently up to about 600) is muc smaller tan te number of reasonable dipolar sources (tens of tousands), te transfer matrix approac is substantially faster tan te state-of-te-art forward approac (i.e., solving an FE equation system for eac source) and can be applied to inverse reconstruction algoritms in bot continuous and discrete source parameter space for EEG and MEG. Still, te solution of undreds of large linear FE equation systems for te construction of te transfer matrices is a major time consuming part witin FE-based source analysis. Te first goal of tis study was terefore to compare te numerical accuracy of te full subtraction approac [7] wit te two direct approaces using partial integration [44,1,36] and Venant [4] in specifically-tuned CDT meses of an anisotropic four-compartment spere model. We ten examine te interplay of te source model approaces wit tree FE solver metods: a Jacobi-CG, an incomplete Colesky CG (e.g., [27]), and an algebraic multi-grid preconditioned CG (AMG-CG), wic as already sown to be especially suited for problems wit discontinuous and anisotropic coefficients [22,32,21,9, 42].

3 1972 S. Lew et al. / Applied Numerical Matematics 59 (2009) Teory In te quasi-static approximation of te Maxwell equations, te distribution of electric potentials Φ in te ead domain Ω of conductivity σ, resulting from a primary current j p is governed by te Poisson equation wit omogeneous Neumann boundary conditions on te ead surface Γ = Ω [20,25], wic we can express as (σ Φ) = j p = J p in Ω, σ Φ,n =0 on Γ, (1) wit n te unit surface normal, and assuming a reference electrode wit given potential, i.e., Φ(x ref ) = 0. Te primary currents are modeled by a matematical dipole at position x 0 R 3 wit moment M 0 R 3 [25,6,18], J p = j p (x) = (M 0 δ(x x 0 ) ), (2) were δ is te Dirac delta distribution Finite element modeling tecniques for te potential singularity One of te key questions for all tree-dimensional EEG forward modeling tecniques is te appropriate treatment of te potential singularity introduced into te differential equation by te formulation of te matematical dipole (2). Tis study examined te interplay of FE solver metods (see Section 2.2) wit te solution accuracy in four-layer spere models applying tree singularity treatment tecniques: a full subtraction approac, a partial integration direct metod and a Venant direct metod Full subtraction approac Te subtraction approac [3,1,43,7] splits te total potential Φ into two parts, Φ = Φ 0 + Φ corr, (3) were te singularity potential, Φ 0, is defined as te solution for a dipole in an unbounded omogeneous conductor wit constant conductivity σ 0. σ 0 R 3 3 is te conductivity at te source position, wic is assumed to be constant in a nonempty subdomain Ω 0 around x 0, in te following called te omogeneity condition. Te solution of Poisson s equation under tese conditions for te singularity potential (σ 0 Φ 0 ) = j p (4) can be formed analytically for te matematical dipole (2) [7] as 1 Φ 0 (x) = 4π M 0,(σ 0 ) 1 (x x 0 ). det σ 0 (σ 0 ) 1 (5) (x x 0 ), (x x 0 ) 3/2 Subtracting (4) from (1) yields a Poisson equation for te correction potential ) (σ Φ corr ) = ((σ 0 σ ) Φ 0 in Ω, (6) wit inomogeneous Neumann boundary conditions at te surface: σ Φ corr, n = σ Φ 0, n on Γ. (7) Te advantage of (6) is tat te rigt-and side is free of any source singularity, because of te omogeneity condition te conductivity σ 0 σ is zero in Ω 0. Existence and uniqueness of te solution and FE convergence properties are sown for te correction potential in [43]. For te numerical approximation of te correction potential, we use te FE metod wit piecewise linear basis functions ϕ i. Wen projecting te correction potential into te FE space, i.e., Φ corr (x) Φ corr, (x) = N ϕ j=1 j(x)u [ j] corr, (N being te number of FE nodes), and applying variational and FE tecniques to (6) and (7), we finally arrive at a linear system [7] K u corr, = j corr,, (8) wit te stiffness matrix [i, j] K = σ ϕ j, ϕ i dx, (9) Ω for K R N N, and te rigt-and side vector j corr, R N wit entries j [i] corr, = (σ0 σ ) Φ 0, ϕ i (x) dx ϕ i (x) n(x), σ 0 Φ 0 (x) dx. (10) Ω Ω We ten seek for te coefficient vector u corr, = (u [1] corr,,...,u[n ] corr, ) RN and, using (3), compute te total potential. In [7], te teoretical reasoning and a validation in a four-compartment spere model wit anisotropic skull is given for te fact tat second order integration is necessary and sufficient for te rigt-and side integration in Eq. (10). Direct comparisons wit te projected subtraction approac from [43] ave sown tat te full subtraction approac is an order of magnitude more accurate for dipole sources close to a conductivity discontinuity [7].

4 S. Lew et al. / Applied Numerical Matematics 59 (2009) Te partial integration direct approac Multiplying bot sides of Eq. (1) by a linear FE basis function ϕ i and integrating over te ead domain leads to a partial integration direct approac for te total potential [1,36,17] expressed as (σ Φ)ϕ i dx = j p ϕ i dx. Ω Ω Integration by parts, applied to bot sides of te above equation, yields σ Φ, ϕ i dx + σ Φ,n ϕ i dγ = j p, ϕ i dx + j p, n ϕ i dγ. Ω Γ Ω Using te omogeneous Neumann boundary condition from Eq. (1) and te fact tat te current density vanises on te ead surface, we arrive at σ Φ, ϕ i dx = j p, ϕ i dx (2) = M 0, ϕ i (x 0 ). Ω Ω Setting Φ(x) Φ (x) = N ϕ j=1 j(x)u [ j], leads to te linear system K u = j PI,, wit te same stiffness matrix as in (9) and te rigt-and side vector j PI, R N wit entries { j [i] PI, = M0, ϕ i (x 0 ) if i NodesOfEle(x 0 ), (12) 0 oterwise. Te function NodesOfEle(x 0 ) determines te set of nodes of te element wic contains te dipole at position x 0.Note tat wile te rigt-and side vector (10) is fully populated, j PI, as only NodesOfEle non-zero entries. Here, denotes te number of elements in te set NodesOfEle. For te linear basis functions ϕ i considered ere, te rigt-and side (12) and tus te computed solution for te total potential in (11) will be constant for all x 0 witin a finite element Te Venant direct approac Te Venant potential approac [4] follows te principle of Saint Venant and is made up from monopolar loads on all neigboring FE nodes so tat te dipolar moment is fulfilled and te source load is as regular as possible. In tis case, variational and FE tecniques yield te linear system K u = j Venant, (13) wit te same stiffness matrix as in (9). Te rigt-and side vector j Venant, R N as only C non-zero entries, if C is te number of neigboring FE nodes to tat FE node wic is closest to te location of te dipole [4] FE solver metods Te solution of undreds of large scale systems of Eqs. (8), (11) or (13) wit te same symmetric positive definite (SPD) stiffness matrix (9) is te major time consuming task of te inverse source localization process. Te spectral condition of te SPD matrix K is equal to κ 2 (K ) = λ max λ min wit λ max te largest and λ min te smallest eigenvalues, respectively, of K [11, 2.10]. Te condition number beaves asymptotically as O( 2 ) and condition numbers of more tan 10 7 ave been computed for FE problems in EEG source analysis [42]. Large condition numbers are te reason for slow convergence of common iterative solvers [11,24] and any effective solution approac as to minimize te effects of tis poor conditioning. Te Preconditioned Conjugate Gradient (PCG) iterative solver sown in Algoritm 1 (see, e.g., [27,11,24]) can provide efficient procedures for suc problems. Note tat, in teory, te convergence speed of te PCG is independent of te rigt-and side j of te linear equation system [11, 3.4]. Te goal of a preconditioner, C R N N,istereductionofκ 2 (C 1 K ) for te preconditioned equation system C 1 K u = C 1 j. Furter requirements are tat it is ceap wit regard to aritmetic and memory costs to solve linear systems C w = r wit r te residual and w te residual for te preconditioned system. Teorem 2.1 (Error estimate for PCG metod). Let K and C be positive definite. If u denotes te exact solution of te equation system, ten te k t iterate of te PCG metod u k fulfills te following energy norm estimate u k u K c k 2 u c 2k κ 2 (C 1 u K, c := K ) 1. κ 2 (C 1 K ) + 1 Γ (11)

5 1974 S. Lew et al. / Applied Numerical Matematics 59 (2009) r = r 0 = j K u Solve C w = r s = w γ 0 = γ = γ OLD = w, r ( ( r ) wile γ /γ C = = r 0 C 1 v = K s α = γ / s, v u = u + αs r = r αv Solve C w = r γ = w, r β = γ /γ OLD, γ OLD = γ s = w + βs end wile ( K e 1 C K e 0 C 1 ) 2 = ( e i K C 1 K e 0 K C 1 K ) 2 > accuracy 2 ) do Algoritm 1. PCG : (K, u, j, C, accuracy) (u ). Proof. Hackbusc [11, Teorem ]. As indicated in Algoritm 1, te PCG metod is stopped after te kt iteration if te relative error, i.e., e k = uk u in te controllable K C 1 K -energynormisbelowagivenaccuracy. Besides te scalar products, te sum of vectors and te multiplication of a vector wit a scalar value, te most important steps in Algoritm 1 are te multiplication of te sparse stiffness matrix K wit a vector and te preconditioning step C w = r. We used te compact row storage format for sparse matrices so tat bot te storage of te stiffness matrix and te matrix-vector multiplication are of complexity O(N ) (see [27, Capter 4.6.4], [24,10]). In te following, te tree different preconditioners for te CG metod are presented and teir relative performances evaluated in Section Jacobi preconditioning or scaling Te simplest preconditioner is te scaling or Jacobi-preconditioning ([24, p. 265f], [27, p. 257f]), were C := D 2, ( D := DIAG K [11],..., ) K [N N ]. Wen splitting te Jacobi-preconditioner between left and rigt (row and column scaling), one as to solve K v = D 1 j wit K = D 1 K D tr and u = D tr v. Row and column scaling preserves symmetry, so tat te scaled matrix K is again SPD wit unit diagonal entries. Te scaling may lead to a first substantial condition improvement, because diagonal entries in K of FE nodes from inside te skull are muc smaller tan from outside (because of a jump in conductivity at eac internal and external boundary) and because it can be sown, tat te smallest (largest) eigenvalue of a symmetric matrix is at most (at least) as large as te smallest (largest) diagonal element, so tat te condition number is at least as large as te quotient of maximal and minimal diagonal element [27, p. 258]. Teorem 2.2. Let K be SPD and C := D 2 te Jacobi-preconditioner. Assume tat eac row of K does not contain more tan d non-zero entries. Ten, for all diagonal matrices D 1,itis ( ) ( κ 2 C 1 K dκ2 D 1 ) K, i.e., te cosen diagonal preconditioner is close to te optimal one. Proof. Hackbusc [11, Teorem 8.3.3] Incomplete Colesky preconditioning Te SPD stiffness matrix K can be decomposed into a left triangular matrix L and its transpose using te Coleskydecomposition, K = L L tr [27, p. 209f]. Neverteless, because of a large fill-in, C := L L tr would not be appropriate as a preconditioner. Te Incomplete Colesky (IC) preconditioner witout fill-in, IC(0), is defined as C := L 0 L tr 0 were L 0 is te Colesky-decomposition of te scaled stiffness matrix K wic is restricted to te same non-zero-pattern as te lower triangular part of K. For incomplete factorizations, te preconditioning operation C w = r in Algoritm 1 is solved by a forward-back sweep, wic can efficiently use te compact row storage format for te sparse matrix L 0 as described in detail in [27, Capter 4.6.4]. Te existence of IC(0) is not necessarily guaranteed for general SPD matrices. Terefore, a reduction of non-diagonal stiffness matrix entries as to be carried out in certain applications before IC(0) computation is

6 S. Lew et al. / Applied Numerical Matematics 59 (2009) possible [27, p. 266]. If te scaled stiffness matrix is decomposed by means of K = E + Id +E tr,wite R N N its strict lower triangular part, te reduction can be formulated as K = Id ς ( ) E + E tr. For sufficiently large ς R + 0, te existence of IC(0) is guaranteed, but wit increasing ς, te preconditioning effect decreases. Note tat for certain special cases, a condition improvement to O( 1 ) can be proven as, e.g., wen using a modified ILU ω -preconditioning wit ω = 1 (in te symmetric case, te ILU 0 is equal to te IC(0)) for diagonally dominant symmetric matrices arising from a 5-point discretization of a two-dimensional Poisson equation (Hackbusc [11, Teorem and Remarks ,17]) Algebraic multi-grid preconditioning Te above preconditioning metods ave te disadvantage tat te convergence rate, i.e., te factor by wic te error is reduced in eac iteration, is still dependent on te mes size. Wit decreasing mes size and tus increasing order of te equation system, te convergence rate tends to 1 from below, so tat te number of iterations needed to acieve a given accuracy increases. For te Geometric Multi-Grid (GMG), an -independent convergence rate ρ < 1 and an -independent condition number as been proven in [11, Lemma , Teorem ] as ( ) κ 2 C 1 K 1 1 ρ, m (15) wit C te preconditioner resulting from m steps of te GMG metod. As sown in [13,11,32], a robust metod wic provides a small convergence rate for a wide class of real-life problems is given by exploiting te MG-metod as a preconditioner for te CG metod. Wit MG(m)-CG, we denote te MG-preconditioned CG metod wit m te number of MG iterations for te CG preconditioning step. Te GMG(m)-CG can improve te convergence rate to ρ/4, if ρ is assumed to be small, as sown in [11, ]. In contrast to GMG, in wic a grid ierarcy is required explicitly, Algebraic MG (AMG) is able to construct a matrix ierarcy and corresponding transfer operators based only on te entries in K (see, e.g., [22,32,21,9]). It is well known tat te classical AMG metod is robust for M-matrices and, wit regard to our application, tat small positive off-diagonal entries are admissible [22,32,21]. Te metod is especially well suited for our problem wit discontinuous and anisotropic coefficients, in wic an optimal tuning of te GMG is difficult ([22, 4.1,4.6.4], [32, 4.1]). Stand-alone AMG is ardly ever optimal as tere may be some very specific error components wic are reduced wit significantly less efficiency, causing a few eigenvalues of te AMG iteration matrix to be muc closer to 1 tan te remaining ones [32, 3.3]. In suc a case, acceleration by means of using AMG as a basis for te CG metod eliminates tese particular frequencies very efficiently. As in GMG, te basic idea in AMG is to reduce ig and low frequency components of te error by te efficient interplay of smooting and coarse grid correction, respectively. In analogy to GMG, te denotation coarse grids will be used, altoug tese are purely virtual and do not ave to be constructed explicitly as FE meses. Te diagonal entry of te it row of K is considered as being related to a grid point in ω (te index set of nodes), and an off-diagonal entry is related to an edge in an FE grid. A description of AMG is now given for a symmetric two grid metod, were is related to te fine grid and H to te coarse grid. Eac AMG algoritm consists of te following components: (a) Coarsening: define te splitting ω = ω C ω F of ω into sets of coarse and fine grid nodes ω C and ω F, respectively. (b) Transfer operators: prolongation P,H : R N H R N and its adjoint as te restriction R H, := P tr,h. (16) (c) Definition of te coarse matrix by Galerkin s metod, i.e., K H := R H, K P,H. (17) Because of (b), K H R N H N H is again SPD. (d) Appropriate smooter for te considered problem class: In order to acieve a symmetric metod, e.g., a forward Gauss Seidel metod for pre-smooting and te adjoint, a backward Gauss Seidel metod for post-smooting ([11, 4.8.3, ,2], [22, 4.4]). Coarsening. Te coarsening process as te task of reducing te number of nodes suc tat N H = ω C < N = ω.te grid points ω can be split into two disjoint subsets ω C (coarse grid nodes) and ω F (fine grid nodes), i.e., ω = ω C ω F and ω C ω F = suc tat tere are (almost) no direct connections between any two coarse grid nodes and suc tat te resulting number of coarse grid nodes is as large as possible [32, p. 12]. Instead of considering all connections between nodes as being of te same rank, te following sets are introduced Ne i = { j [ij] K ζ K [i,i] }, i j, (18) S i = { j Ne i [ij] K > coarse(i, j, K ) }, S i,t = { j Ne i j i S }, (19) (14)

7 1976 S. Lew et al. / Applied Numerical Matematics 59 (2009) ω C, ω F wile ω C ω F ω do i Pick(ω \ (ω C ω F )) if S i,t + Si,T ω F =0 ten ω F ω \ ω C else ω C ω C {i} ω F ω F (S i,t \ ω C ) end if end wile Algoritm 2. COARSE : ({S i,t }, ω ) (ω C, ω F ). if CoarseGrid ten u DirectSolve(K u = j ) else u ν F times smoot Forward(K, u, j ) d = K u j d H = P tr,h d w H = 0 w H = MG(K H, w H, d H ) w = P,H w H u = u w u ν B times smoot Backward(K, u, j ) end if Algoritm 3. V-cycle MG : (K, u, j, ν F, ν B ) (u ). were Ne i is te index set of neigbors (a pre-selection is carried out by te tresold-parameter ζ R+ 0 ), Si denotes te index set of nodes wit a strong connection from node i and S i,t is related to te index set of nodes wit a strong connection to node i. In addition, coarse(i, j, K ) is an appropriate cut-off (coarsening) function, e.g., { coarse(i, j, K ) := α max K [ij] }, (20) j, j i wit α [0, 1] (see, e.g., [22, 4.6.1]). Wit tose definitions a splitting into coarse and fine grid nodes can be acieved. For our application, a modified splitting algoritm is used [22, 4.6] as sown in Algoritm 2. Terein, te function i ( Pick ω \ (ω C ω F ) ) returns a node i for wic te number S i,t + Si,T ω F is maximal. Note tat tissue conductivity inomogeneity and anisotropy are taken into account witin te coarsening algoritm. Prolongation. To acieve prolongation, te operator P,H : V H V as to be defined correctly. Te form tat turned out to be te most efficient for te presented application was proposed in [14] and is given by 1 i = j ω C, P [ij],h = 1/ S i,t ω C i ω F, j S i,t ω C, (21) 0 else. After te proper definition of te prolongation and coarse grid operators, it is possible to create in a recursive way a matrix ierarcy and an associated multi-grid cycle, sown in Algoritm 3. Terein, te variable CoarseGrid denotes te level at wic a direct solver is applied. For an m-v (ν F, ν B )-cycle AMG preconditioned CG metod, te operation Solve C w = r in Algoritm 1 is realized by m calls of MG(K, w, r, ν F, ν B ). Specific performance optimizations for te AMG preconditioner used in tis study are described in detail in [10, Section 2]. 3. Metods 3.1. Validation platform Te numerical examinations of te teory presented above were carried out in a four-layer spere model wit anisotropic skull compartment wose parameterization is sown in Table 1. For te coice of tese parameters, we closely followed [12,15]. Forward solutions were computed for dipoles of 1 nam amplitude located on te y axis at depts of 0% to 98.7% (in 1 mm steps) of te brain compartment (78 mm radius) using bot radial (directed away from te center of te model) and tangential (directed parallel to te scalp surface) dipole orientations. Eccentricity is defined ere as te percent ratio of te distance between te source location and te model midpoint divided by te radius of te inner spere (78 mm). Te most

8 S. Lew et al. / Applied Numerical Matematics 59 (2009) Table 1 Parameterization of te anisotropic four-layer spere model. Medium Scalp Skull CSF Brain Outer sell radius (mm) Tangential conductivity (S/m) Radial conductivity (S/m) eccentric source position considered was tus only 1 mm below te CSF compartment. To acieve error measures wic were independent of te specific coice of te sensor configuration, we distributed 748 electrodes in a regular fasion over te outer spere surface. All simulations ran on a Linux-PC wit an Intel Pentium 4 processor (3.2 GHz) using te SimBio software environment [31] in wic te algebraic multi-grid solver package PEBBLES was embedded [9,42,39,10] Analytical solution in an anisotropic multilayer spere model De Munck and Peters [5] derived series expansion formulas for a matematical dipole in a multi-layer spere model, denoted ere as te analytical solution. Te model consists of S sells wit radii r S < r S 1 < < r 1 and constant radial, σ rad (r) = σ rad R +, and constant tangential conductivity, σ tang (r) = σ tang R +, witin eac layer r j j j+1 < r < r j.itis assumed tat te source at position x 0 wit radial coordinate r 0 R is in a more interior layer tan te measurement electrode at position x e R 3 wit radial coordinate r e = r 1 R. Te sperical armonics expansion for te matematical dipole (2) is expressed in terms of te gradient of te monopole potential to te source point. Using an asymptotic approximation and an addition subtraction metod to speed up te series convergence yields φ ana (x 0, x e ) = 1 x e M, S 0 + (S 1 cos ω 0e S 0 ) x 0 4π r e r 0 wit ω 0e te angular distance between source and electrode, and wit and S 0 = F 0 r 0 Λ (1 2Λ cos ω 0e + Λ 2 ) + 1 3/2 r 0 n=1 { (2n + 1)Rn (r 0, r e ) F 0 Λ n} P n (cos ω 0e) (22) Λ cos ω 0e Λ 2 S 1 = F 1 (1 2Λ cos ω 0e + Λ 2 ) + { (2n + 1)R 3/2 n (r 0, r e ) F 1 nλ n} P n (cos ω 0e ). (23) n=1 Te coefficients R n and teir derivatives, R n, are computed analytically and te derivative of te Legendre polynomials, P n, are determined by means of a recursion formula. We refer to [5] for te derivation of te above series of differences and for te definition of F 0, F 1 and Λ. Here, it is only important tat te latter terms are independent of n and tat tey can be computed from te given radii and conductivities of layers between source and electrode and of te radial coordinate of te source. Te computations of te series (22) and (23) are stopped after te kt term if te following criterion is fulfilled t k /t 0 υ, t k := (2k + 1)R k F 1kΛ k. (24) In te following simulations, a value of 10 6 was cosen for υ in (24). Using te asymptotic expansion, no more tan 30 terms were needed for te series computation at eac electrode Tetraedral mes generation Te FE meses of te four-layer spere model were generated by te software TetGen [28] wic used a Constrained Delaunay Tetraedralization (CDT) approac [30,29]. Tis mesing procedure starts wit te preparation of a suitable boundary discretization of te model in wic for eac of te layers and for a given triangle edge lengt, nodes are distributed in a regular fasion and connected troug triangles. Tis yields a valid triangular surface mes for eac of te layers. Meses of different layers are not intersecting eac oter. Te CDT approac is ten used to construct a tetraedralization conforming to te surface meses. It first builds a Delaunay tetraedralization starting wit te vertices of te surface meses. Te CDT ten uses a local degeneracy removal algoritm combining vertex perturbation and vertex insertion to construct a new set of vertices wic includes te input set of surface vertices. In a last step, a fast facet recovery algoritm is used to construct te CDT [30,29]. Tis approac is combined wit two furter constraints to te size and sape of te tetraedra. Te first constraint is important for te generation of quality tetraedra. If R denotes te radius of te unique circumspere of a tetraedron and L its sortest edge lengt, te so-called radius-edge ratio of te tetraedron can be defined as radius-edge-ratio = R/L. (25)

9 1978 S. Lew et al. / Applied Numerical Matematics 59 (2009) Table 2 Te six tetraedral models used for te solver time comparison and accuracy tests. Te table sows te number of nodes and elements of eac mes and factor indicates te ratio of te number of nodes of te most igly resolved to bot oter models witin eac group. Additionally, te cosen radius-edge-ratio (see Eq. (25)), te average edge lengt of te four triangular surface meses, te corresponding volume constraint (see Eq. (26)) and te compartments were te volume constraint was not applied are indicated. Group 1 Group 2 Model tet503k tet125k tet33k tet508k tet128k tet32k nodes 503, ,624 32, , ,847 31,627 elements 3,068, , ,307 3,175, , ,060 factor radius-edge-ratio edge (in mm) volume (in mm 3 ) no volume constraint in brain brain brain / / / Te radius-edge ratio can distinguis almost all badly-saped tetraedra except one type of tetraedra, so-called slivers. A sliver is a very flat tetraedron wic as no small edges, but can ave arbitrarily large diedral angles (close to π ). For tis reason, an additional mes smooting and optimization step is required to remove te slivers and improve te overall mes quality. A second constraint can be used to restrict te volume of te generated tetraedra in a certain compartment. We follow te formula for regular tetraedra: volume = 2/12 edge 3. (26) Table 2 sows te number of nodes and elements of te six tetraedra models used for te solver run-time comparison and accuracy tests. factor indicates te ratio of te number of nodes of te most igly resolved to bot oter models witin eac group. Additionally, te table contains te cosen radius-edge-ratio (see Eq. (25)), te average edge lengt of te four triangular surface meses, te corresponding volume constraints (see Eq. (26)) for te tetraedra and te compartments were te volume constraint is not applied. Te most igly resolved meses tet503k and tet508k of bot groups ad approximately te same resolution, wile te oters were cosen to ave a factor of 4 coarser resolution wit regard to te number of nodes. Te meses of group 1 concentrated te nodes in te outer tree compartments because no volume constraint was applied for te inner brain compartment, wile te nodes in te meses of group 2 were distributed in a regular way trougout all four compartments. Te meses of group 1 were tus preferentially beneficial to te full subtraction approac, since te entries of te volume integral in Eq. (10) are zero ((σ (x) σ 0 ) = 0 for all x in te brain compartment) so tat a coarse resolution can be expected to ave no impact on te overall numerical accuracy, but will reduce te computational cost. In contrast, te meses of group 2 were beneficial to bot direct potential approaces. Fig. 1 sows samples from te six tetraedra models tat were generated using te parameterizations from Table Error criteria We compared numerical solutions wit analytical solutions using tree common error criteria [16,3,34,15,26]. Te relative (Euclidean) error (RE) is defined as RE := φ num φ ana 2 φ ana 2, were φ ana,φ num R m denote te analytical and te numerical solution vectors, respectively, at te m = 748 measurement electrodes. We furtermore defined ( ) RE(%) := 100 RE, maxre(%) := max RE(%) j (27) j were j is te source eccentricity. In order to better distinguis between te topograpy (driven primarily by canges in dipole location and orientation) and te magnitude error (indicating canges in source strengt), Meijs et al. [16] introduced te relative difference measure (RDM) and te magnification factor (MAG), respectively. For te RDM, we can sow tat RDM := 1 1 φ ana φ num φ ana 2 φ num = ( 2 1 cos (φ ana,φ num ) ). (28) 2 2 It terefore olds tat 0 RDM 2, so tat we can furtermore define ( ) RDM(%) := 100 RDM/2, maxrdm(%) := max RDM(%) j. (29) j Te MAG is defined as MAG := φ num 2 / φ ana 2

10 S. Lew et al. / Applied Numerical Matematics 59 (2009) Fig. 1. Cross-sections of te six tetraedral meses of te four compartment spere model. Te corresponding parameterizations of te models are sown in Table 2. Visualization was done using te software TetView [28]. so tat error minimum is at MAG = 1 and we terefore defined MAG(%) = 1 MAG 100, maxmag(%) := max j ( MAG(%) j ). (30) Wit maxre(%) k we denote te maximal relative error in percent over all source eccentricities for an accuracy level of accuracy = 10 k. Te so-called plateau-entry for an iterative solver is ten defined as te first k at wic te condition maxre(%) k maxre(%) k+1 /maxre(%) k+1 < 0.05 (31) is true FEM and solver parameter settings Te parameters of te Venant approac were cosen as proposed in [4]: Te maximal dipole order (n 0 in [4, Eq. (22)]) and te scaling reference lengt (a ref in [4, Eq. (23)]) were set to n 0 = 2 and a ref = 20.0 mm, respectively. Since te cosen mes size was a large factor smaller tan te reference lengt, te second order term (( x r ki )2 in [4, Eq. (23)]) was small and te model focused on fulfilling te dipole moments of te zeros and first order. Te exponent of te source weigting matrix (n s in [4, Eq. (25)]) was fixed to n s = 1 and te regularization parameter (λ D in [4, Eq. (25)]) was cosen as λ D = Tese settings effect a spatial concentration of te monopole loads in te dipole axis around te dipole location. Te initial solution guess for all solvers was a zero potential vector. For te IC(0), ς = 0wascosenfor(14).Forte AMG-CG, te 1-V (1, 1)-cycle AMG-preconditioner was used wit α = 0.01 for (20). Te factorization in Algoritm 3 was carried out wenever te size of te coarsest grid (coarsegrid) in te preconditioner-setup was below 1000 and te coarse system was solved using a Colesky-factorization. Te setup times for te preconditioners were neglected in all calculations of computational cost because tis step must be performed only once per ead model. Te evaluation wit regard to relative solver accuracy in Algoritm 1 was limited to te discrete set of accuracy levels accuracy = 10 k wit k {0,...,9}. 4. Results 4.1. Numerical error versus potential approac In a first study, we compared te numerical accuracy of te full subtraction approac (Section 2.1.1) wit te two direct metods: Venant (Section 2.1.3) and partial integration (Section 2.1.2). Fig. 2 sows te RE(%) for te different source eccentricities for te two finest models tet503k of group 1 (left) and tet508k of group 2 (rigt) (see Fig. 1 and Table 2) wit regard to te full subtraction (top row), te Venant (middle row) and te partial integration approac (bottom row).

11 1980 S. Lew et al. / Applied Numerical Matematics 59 (2009) Fig. 2. RE(%) versus source eccentricity for te two most igly resolved models tet503k of group 1 (left) and tet508k of group 2 (rigt) using te full subtraction (top row), te Venant (middle row) and te partial integration (bottom row) potential approaces. Te necessary accuracy in Algoritm 1 for te plateau-entry (31) of te AMG-CG is indicated for bot source orientation scenarios. Note tat te y-axis is differently scaled.

12 S. Lew et al. / Applied Numerical Matematics 59 (2009) Fig. 3. maxre(%), maxrdm(%) and maxmag(%) accuracies for te full subtraction, te Venant and te partial integration approac for all six tetraedra models (see Fig. 1 and Table 2) and bot tangential (left) and radial (rigt) source orientation scenarios at te AMG-CG plateau-entry (31).

13 1982 S. Lew et al. / Applied Numerical Matematics 59 (2009) Fig. 4. maxre(%) versus PCG solver accuracy (see Algoritm 1 and Section 3.5) for models tet503k of group 1 (left column) and tet508k of group 2 (rigt column) for te AMG-CG (top row), te IC(0)-CG (middle row) and te Jacobi-CG (bottom row). Source orientations and potential approaces can be distinguised by teir specific labels. Te plot is in log log scale.

14 S. Lew et al. / Applied Numerical Matematics 59 (2009) Te results were computed wit te AMG-CG and te necessary accuracy in Algoritm 1 for te plateau-entry (31) is indicated for bot source orientation scenarios. In Fig. 3, te maximal RE, RDM and MAG errors over all source eccentricities at te AMG-CG plateau-entry (31) are sown for all tetraedra models, bot source orientation scenarios and te tree dipole modeling approaces. Fig. 2 clearly presents te advantages of te full subtraction approac wose error curves are smoot, wile Venant and partial integration sow an oscillating beavior. Wit RDM and MAG errors below 1% over all source eccentricities and for bot orientation scenarios (see Fig. 3), te full subtraction approac performs best for all source eccentricities for model tet503k (its mes resolution was sufficiently ig and te FE nodes were concentrated in te compartments CSF, skull and skin), were bot direct approaces sowed oscillations wit a relatively ig magnitude. As te results for model tet508k sow, te oscillation magnitudes for te direct approaces could be strongly reduced by means of distributing te FE nodes in a regular way over all four compartments, ence decreasing te mes size in te brain compartment. Neverteless, even for model tet508k, te full subtraction approac was te most accurate metod for nearly all source eccentricities. It was only outperformed by partial integration for te source wic was only 1 mm below te CSF compartment. As bot Figs. 2 and 3 sow, te partial integration approac performed well if te mes was sufficiently fine in te brain compartment. Te oscillation magnitudes of te Venant approac were generally even sligtly smaller tan for te partial integration approac, wit only one exception (te result for te radial source 1 mm below te CSF compartment, sown in te middle row of Fig. 2). Te main reason for te outlier was tat for te source 1 mm below te CSF, monopoles were positioned in te CSF compartment, wic ad a strong effect on te MAG for te radially oriented source Numerical error versus PCG accuracy Fig. 4 sows te numerical error maxre(%) versus te PCG solver accuracy from Algoritm 1 for te discrete set of accuracy levels from 10 0 to Results for te ig-resolution model tet503k of group 1 are sown in te left and from te ig-resolution model tet508k of group 2 in te rigt column for te AMG-CG (top row), te IC(0)-CG (middle row) and te Jacobi-CG (bottom row). Te PCG accuracy measures te error in te solution vector of te FE linear equation system (8) (correction potential), (11) and (13) (total potential). For te full subtraction approac, maxre(%) was tus not equal to 100 for accuracy = 10 0 because φ num is equal to te analytically computed singularity potential Φ 0 from Eq. (5). Because te PCG accuracy is measured in te K C 1 K -energy norm, te plateau-entry (31) differs for different preconditioners C. As sown in Fig. 4 for te ig-resolution models and as collected in Table 3 for all six tetraedra models, te maximally needed k (for a PCG accuracy of accuracy = 10 k ) decreased wen te preconditioning quality increased (except for te radial source orientation in model tet503k, see Fig. 4). Furtermore, as Table 3 sows, a iger PCG accuracy was needed for te plateau-entry wen te mes resolution increased Numerical error versus solver time In a last study, we compared solver wall-clock time versus numerical accuracy for te tree different CG preconditioners AMG, IC(0) and Jacobi. Table 3 Maximally needed k {0,...,9} for a PCG accuracy = 10 k for te plateau-entry (31) over all tree potential approaces. tangential source Group 1 radial source solver AMG-CG IC(0)-CG Jacobi-CG AMG-CG IC(0)-CG Jacobi-CG tet503k tet125k tet33k Group 2 tangential source radial source solver AMG-CG IC(0)-CG Jacobi-CG AMG-CG IC(0)-CG Jacobi-CG tet508k tet128k tet32k Table 4 Average setup time (sec.) for te AMG- and te IC(0)-preconditioner. Group 1 Group 2 preconditioner tet503k tet125k tet33k tet508k tet128k tet32k AMG IC(0)

15 1984 S. Lew et al. / Applied Numerical Matematics 59 (2009) Fig. 5. Solver time versus maxre(%) for models tet503k and tet33k of group 1 for tangentially and radially oriented sources for te potential approaces full subtraction (left), Venant (middle), and partial integration (rigt). Results are presented for te tree different CG preconditioners AMG, IC(0) and Jacobi. Eac marker represents a PCG accuracy = 10 k level and te largest examined level is indicated. Te x-axis is in log scale. Te average setup time for te AMG- and te IC(0)-preconditioner is given in Table 4. In te following, te time for te setup of te preconditioner was not included, because tis step was carried out only once per ead model. In Fig. 5, te solver time is sown versus te maxre(%) for different levels of PCG accuracy for models tet503k and tet33k of group 1. Te largest examined PCG accuracy level 10 k is indicated in te figure. Please note tat tis level does not necessarily correspond to te plateau-entry level. In most cases results are presented up to one level iger.

16 S. Lew et al. / Applied Numerical Matematics 59 (2009) Table 5 Average solver time (sec.) and iteration count (iter) over all source eccentricities, source orientations and potential approaces for plateau-entry (31). For all tetraedra models of groups 1 and 2, results are presented for te tree different CG preconditioners AMG, IC(0) and Jacobi. Te gain factor indicates teperformancegainofteamg-cgrelativetotejakobi-cg. Group 1 Group 2 tet503k tet125k tet33k tet508k tet128k tet32k solver time iter time iter time iter time iter time iter time iter AMG-CG IC(0)-CG Jacobi-CG gain factor For all tetraedra models of groups 1 and 2, average solver times and iteration counts over all source eccentricities, source orientations and potential approaces for a plateau-entry (31) are collected in Table 5. Bot Fig. 5 and Table 5 clearly sow te superiority of te AMG preconditioner. In all cases, even for te low-resolution grids tet33k and tet32k, te AMG-CG was te fastest solver, followed by te IC(0)-CG and te Jacobi-CG. Te main result of Table 5 is te so-called gain factor, wic is defined ere as te result (solver time or iteration count) for te Jacobi-CG divided by te result for te AMG-CG. Te gain factors clearly sowed tat te iger te mes-resolution, i.e., te iger te condition number of te corresponding FE stiffness matrix, te larger te difference in performance between AMG-CG, IC(0)-CG, and Jacobi-CG. An increasing mes-resolution led to a strong increase in te number of iterations of IC(0)-CG (factor of 3.2 between tet503k and tet33k and 4.1 between tet508k and tet32k) and Jacobi-CG (factor of 3.0 between tet503k and tet33k and 3.6 between tet508k and tet32k), wile te number of AMG-CG iterations was only sligtly increasing (factor of 1.9 between tet503k and tet33k and 1.8 between tet508k and tet32k). Tis clearly sows te stronger -dependence of te IC(0) and Jacobi preconditioners. 5. Discussion Te goals of tis tecnical study of finite element (FE) based solution tecniques for te electroencepalograpic forward problem were twofold. Te first aim was to compare tree efficient iterative FE solver tecniques under realistic conditions tat still allowed quasi-analytical solutions. Te second aim was to evaluate tree different numerical formulations of te current dipole, wic is te bioelectric source most commonly used to represent neural electrical activity. A major motivation of suc studies is te special need to acieve ig accuracy and efficiency wit FE based approaces for tis problem. Te many advantages of tis approac are often indered by te unacceptable computation costs of implementing it so tat improved efficiency will provide substantial progress to te field. Wen using te K C 1 K -energy norm stopping criterion for te PCG algoritm applied on meses wit up to 500K nodes, a relative solver accuracy of 10 6 for AMG-CG, 10 7 for IC(0)-CG and 10 8 for Jacobi-CG was necessary and sufficient to fall below te discretization error. Te AMG-CG acieved an order of magnitude iger computational speed tan te CG wit te standard preconditioners wit an increasing gain factor wit decreasing mes size. Te increasing gain factor sows tat te convergence rate of te Jacobi- and IC(0)-preconditioning metods are muc more dependent on te mes size tan te AMG-CG, as discussed in detail in Section However, wile for te geometric multi-grid, an -independent convergence rate and an -independent condition number can be proven [11, Lemma , Teorem ], te AMG-CG was not optimal in our application wit a sligt -dependence sown by a sligtly increasing iteration count wit increasing mes resolution. Suc a result ad to be expected because te source analysis stiffness matrix was not an M-matrix and te prolongation operator of te presented AMG-CG was tuned for speed and not for an optimal beavior wit regard to te iteration count. A discrete armonic extension as proposed in [22] improved te interpolation properties, but te application of tis prolongation operator is more expensive, wic decreased te overall run-time performance in our application. We generated two groups of Constrained Delaunay tetraedralization (CDT) FE meses [30,29], tuned for te specific needs of te different potential approaces. In group 1, for te full subtraction approac [7], FE nodes were concentrated in te CSF, skull and skin, wile te brain compartment was mesed as coarsely as possible. Group 2 was tuned for te needs of bot direct potential approaces [44,4,1,36], wic profit more from a regular distribution of FE nodes over all four compartments and especially a iger resolution at te source positions. Wit regard to te numerical error, in te tuned FE meses wit about 500K nodes we acieved ig accuracies in te range of a few percent maximal relative error (maxre) over all source eccentricities for bot te full subtraction and te two direct potential approaces. Wit a maximal relative difference measure (maxrdm) and a maximal magnification factor (maxmag) of less tan 1% over all source eccentricities for sources up to 1 mm below te CSF compartment (model tet503k, maximal examined eccentricity of 98.7%), te full subtraction approac performed consistently better tan bot direct approaces. We found tat te instability of te full subtraction approac wit regard to te RE for ig source eccentricities was mainly a magnitude instability (MAG), not a topograpic one (RDM). Our results clearly illustrate te advantages of te full subtraction approac as long as te omogeneity condition is sufficiently fulfilled, i.e., as long as te distance of te source to te next conductivity inomogeneity is large enoug or te resolution of te FE mes at te nearest