Substructuring Preconditioners for the Bidomain Extracellular Potential Problem

Transcription

1 Substructuring Preconditioners for the Bidomain Extraceuar Potentia Probem Mico Pennacchio 1 and Vaeria Simoncini 2,1 1 IMATI - CNR, via Ferrata, 1, Pavia, Itay mico@imaticnrit 2 Dipartimento di Matematica, Università di Boogna, Piazza di Porta S Donato, 5, Boogna, and CIRSA Ravenna, Itay vaeria@dmuniboit Summary We study the efficient soution of the inear system arising from the discretization by the mortar method of mathematica modes in eectrocardioogy We focus on the bidomain extraceuar potentia probem and on the cass of substructuring preconditioners We verify that the condition number of the preconditioned matrix ony grows poyogarithmicay with the number of degrees of freedom as predicted by the theory and vaidated by numerica tests Moreover, we discuss the roe of the conductivity tensors in buiding the preconditioner 1 Introduction A macroscopic mode accounting for the excitation process in the myocardium is the bidomain mode that yieds the foowing Reaction-Diffusion (R-D) system of equations for the intra-, extraceuar and transmembrane potentia u i, u and v = u i u: find (v(x, t), u(x, t)), x Ω, t [0, T] such that c m t v div M i v + I(v) = div M i u + I app in Ω ]0, T[ div M u = div M i v in Ω ]0, T[ (1) with M i, M e, M = M i +M e conductivity tensors modeing the cardiac fibers, I app an appied current used to initiate the process, c m the surface capacitance of the membrane The function I(v) is the transmembrane ionic current which is assumed for simpicity to depend ony on v and to be a cubic poynomia (see [7]) The genera R-D system can be more compex, incuding additiona ordinary differentia equations that govern the evoution of v These modes are computationay chaenging because of the different space and time scaes invoved; reaistic three-dimensiona simuations with uniform grids yied discrete probems with more than O(10 7 ) unknowns at every time step To improve computationa efficiency, we consider a non conforming non overapping domain decomposition, within the mortar finite eement method This aows us to concentrate the computationa work ony in regions of high

2 2 Mico Pennacchio and Vaeria Simoncini eectrica activity; in addition, the matching of different discretizations on adjacent subdomains are weaky enforced In [8, 9] we compared this technique to the cassica conforming FEM verifying its better performance In this paper, we focus on the probem of the efficient soution of the inear system arising from this discretization and here for simpicity, we concentrate on the probem with the eiptic equation of (1): for each time instant t find u(x, t), soution of: { div M u = div Mi v in Ω n T M u = n T (2) M i v on Γ Such probem is of interest in its own right, as it represents a separate mode for the bidomain extraceuar potentia [8] We consider substructuring preconditioners and we report our numerica experience on soving probem (2) Our experiments confirm the theory depicting poyogarithmic bound for the condition number of the preconditioned matrix Moreover, attention is devoted to tuning the preconditioner so as to take into account the conductivity tensor M in (2) 2 Mortar Method The computationa domain Ω is decomposed as the union of L subdomains Ω 1,, Ω L (see, eg, [10]) We set Γ n = Ω n Ω, S = Γ n and we denote by γ (i), i = 1,, 4 the i-th side of the -th domain so that Ω = 4 i=1 γ(i) Here we consider ony geometricay conforming decomposition, ie each edge γ (i) coincides with Γ n for some n The Mortar Method is appied by choosing a spitting of the skeeton S as the disjoint union of a certain number of subdomain sides γ (i), caed mortar or save sides: we fix an index set I {1,,L} {1,,4} such that S = (,i) I γ(i) The index set corresponding to trace or master sides wi be denoted by I : I {1,, L} {1,,4}, I I = and S = (,i) I γ(i) Let the spaces X and T be X = H1 (Ω ), T = H1/2 ( Ω ) with the broken norms: u 2 X = u 2 1,Ω and η 2 T = η 2 1/2, Ω For each, et aso Vh be a famiy of finite dimensiona subspaces of H1 (Ω ) C 0 ( Ω ), depending on a parameter h = h > 0, X h = L =1 V h X, T h = V h Ω and T h = L =1 T h T Then we define two composite biinear forms a X, a i X : X X R as: a X (u, φ) = φ T M u dx, a i X(u, φ) = φ T M i u dx (3) Ω Ω Since these biinear forms are not coercive on X, we consider proper subspaces of X consisting of functions satisfying a suitabe weak continuity constraint, eading to the foowing constrained approximation and trace spaces

3 Substructuring Preconditioners in Eectrocardioogy 3 X h = {v h X h, [v h ]λds = 0, λ M h } (4) S T h = {η T h, [η]λds = 0, λ M h }, (5) S with M h a suitaby chosen finite dimensiona mutipier space We can write the discrete probem, whose soution existence was proved in [8, Theorem 31]: Probem 1 Find u h X h such that for a φ h X h a X (u h, φ h ) = a i X (v h, φ h ) (6) We remark that Probem 1 admits a soution unique up to an additive constant reated to the reference potentia chosen In this paper we consider as reference potentia the one given by the potentia at a reference point x 0 Ω 3 Substructuring Preconditioners A key aspect of substructuring preconditioners is that they distinguish among three types of degrees of freedom: interior (corresponding to basis functions vanishing on the skeeton and supported on one sub-domain), edge and vertex degrees of freedom [4, 1] Thus, each function u X h can be written as the sum of three suitaby defined components: u = u 0 +u E +u V More specificay, et w = (w ) =1,,L X h be any discrete function, then w = w 0 + R h (w), w 0 X 0 h, (7) with w 0 Xh 0 interior function and R h(w) a discrete ifting, ie R h (w) = (Rh (w )) =1,,K, where Rh (w ) is the unique eement in Vh satisfying R h (w ) = w on Γ and Ω i,j M x i Consequenty, the spaces X h and X h can be spit as: and it can be verified that x j R h (w )v h dx = 0, v h V h (8) X h = X 0 h R h(t h ) X h = X 0 h R h(t h ) a X(w, v) = a X(w 0, v 0 ) + a X(R h (w), R h (v)) = a X(w 0, v 0 ) + s(η(w), η(v)), (9) where the discrete Stekov-Poincaré operator s : T h T h R is defined by s(ξ, η) := (M(x) Rh (ξ)) R h (η) (10) Ω

4 4 Mico Pennacchio and Vaeria Simoncini Furthermore the space of constrained skeeton functions T h can be spit as the sum of vertex and edge functions More specificay, denoting by L L =1 H1/2 ( Ω ) the space L = {(η ) =1,,L, η is inear on each edge of Ω }, then we can define the space of constrained vertex functions as T V h = P h L (11) with P h the correction operator imposing the constraint We make the (not restrictive) assumption L T h, which yieds Th V T h, and we introduce the space of constrained edge functions Th E T h defined by T E h = {η = (η ) =1,,L T h, η (A) = 0, vertex A of Ω } (12) We can easiy verify that T h = Th V T h E Then we wi consider a bock Jacobi type preconditioner s : T h T h R defined as s(η, ξ) = b V (η V, ξ V ) + b E (η E, ξ E ) (13) with bocks reated to the foowing edge and vertex goba biinear forms b E : Th E h E R such that be (η E, η E ) s(η E, η E ) b V : Th V h V such that bv (η V, η V ) s(η V, η V ) (14) 31 Matrix form In this section we derive the matrix form of the discrete Stekov-Poincaré operator s in (10) Equation (6) yieds the foowing inear system of equations: Au = b with b = A i v, (15) where A, A i are the stiffness matrices associated to the discretization of a X, a i X defined in (3) It can be shown that the matrix A is positive semidefinite and the system is consistent We reorder the vector of unknowns as: u = (u 0,u E,u V,u S ) T, with u 0,u E,u V,u S interior, edge, vertex and save nodes, respectivey From the mortar condition, it foows that the interior nodes of the mutipier sides are not associated with genuine degrees of freedom in the FEM space Indeed, the vaue of the coefficients u S corresponding to basis functions iving on save sides is uniquey determined by the remaining coefficients through the jump (mortar) condition and can be eiminated from the goba vector u, ie C S u S = C E u E C V u V u S =: Q E u E + Q V u V (16) where Q E = C 1 S C E, Q V = C 1 S C V The entries of C S, C E, C V are given by c ij = γ m [φ j ]λ i ds, λ i M h with φ j corresponding to the different noda basis functions on the save and master side and associated with the vertices Since biorthogona basis functions are empoyed, the square matrix C S is

5 Substructuring Preconditioners in Eectrocardioogy 5 diagona and easiy invertibe (cf [11]) The reduction in (16) may be written in matrix form as u = Q u I u E with Q = 0 I E I u V (17) V 0 Q E Q V where Q is a goba switching matrix The resuting reduced system is thus given by Ãu M = b (18) with à = QT AQ and b = Q T b We note that the (1,1) bock in à is cheapy invertibe therefore, the Schur compement of the system reative to the (1,1) bock can readiy be obtained, yieding the further reduced system ( ) ( ) ue be S = u V b V The Schur compement S represents the matrix form of the Stekov-Poincaré operator s(, ) To obtain the matrix form of s(, ) we consider the space L of inear functions, used in the spitting of the trace space (11) Then, we introduce an interpoation map denoted by RH T (say piecewise interpoation) from the noda vaue on V (vertices) onto a nodes of S The matrix R H can be viewed as the weighted (( ) restriction ) map from S onto V By defining the IE square matrix J = R O H, with I E the n E n E identity matrix, we can derive the new Schur compement matrix S, after the vertex correction, ) ( SE S = J T SEV SJ = S EV T S V (19) 32 The preconditioner We describe a generaization of a known optima preconditioner for S, and some more computationay effective variants The matrix discretization of the form s yieds the foowing (bock-jacobi type) diagona preconditioner ( ) PE 0 P =, 0 P V where P E, P V are the matrix counterparts of the biinear forms b E and b V in (14), respectivey It can be verified that the preconditioned matrix P 1 S satisfies the theory deveoped in [1, 3] so that ( )) 2 H cond(p 1 S) (1 + og (20) h

6 6 Mico Pennacchio and Vaeria Simoncini with H size of the subdomains and h finest meshsize of the finite eement spaces used Moreover, if an auxiiary coarse mesh is chosen for the vertex bock with mesh size δ > h as studied in [3], then a simiar estimate can be obtained but with a factor ( 1 + og ( )) H 3 h The next three variants make the preconditioner above computationay more appeaing with no essentia oss of optimaity This goa is achieved by repacing either or both the edge and vertex bocks P E and P V with more convenient approximations In their construction, we were inspired by a simiar approach first proposed in [4, 1, 3] for eiptic probems For ater considerations, we reca here an important bound for the condition number of the preconditioned matrix, expressed in terms of the preconditioning quaity of the two diagona bocks More precisey, et P = diag(p 1, P 2 ) be a Jacobitype preconditioner, and et µ M = max{λ max (P1 1 min{λ min (P1 1 S E ), λ min (P2 1 S V )} Then cond(p 1 S) 1 + γ 1 γ S E ), λ max (P 1 2 S V )}, µ m = µ M µ m γ 1, (21) where 1 + γ is the argest eigenvaue of the preconditioned matrix obtained by using the bock diagona of S as preconditioner P (see, eg, [6]) Foowing [1, 4] a simpe approach consists in dropping a coupings between different edges and between edges and vertex points: P E is repaced by its bock diagona part with one bock for each mortar This simpification provides our first variant, ( ) P diag P 1 = E 0 0 P V Assembing the edge and vertex bock preconditioner with such a choice coud be quite expensive A more efficient preconditioner may be obtained by approximating the edge bock P E of P as P (R) E = αr where R is the square root of the stiffness matrix associated on each edge to the discretization of the operator d 2 /dx 2 with homogeneous Dirichet conditions at the extrema [4, 5, 10] The choice of the parameter α is discussed beow Thus our second variant is ( ) (R) P P 2 = E 0 0 P V It can be easiy verified that (cf, eg, [4, 5]) c 1 v T SE v v T Rv c 2 (1 + og ( )) 2 H v T SE v (22) h

7 Substructuring Preconditioners in Eectrocardioogy 7 where c 1, c 2 are independent of H, h but may depend on the coefficients of M(x) Since P V = S V, in (21) we obtain µ M = max{λ max ((αr) 1 SE ), 1} ( ( max{α 1 c og H )) 2 h, 1} Anaogousy, µm min{α 1 c 1, 1} Therefore, the determination of µ M, µ m is infuenced by the magnitude of c 1, c 2 and of α The anisotropic conductivity tensor M = M i +M e is given as M s = M s (x) = σt si + (σs σs t )aat, s = i, e, where a = a(x) is the unit vector tangent to the cardiac fiber at a point x Ω, I is the identity matrix and σ s, σs t for s = i, e are the conductivity coefficients aong and across fiber, in the (i) and (e) media, assumed constant with σ s > σt s > 0 As aready mentioned, the magnitude of c 1, c 2 depends on the conductivity coefficients Therefore, to minimize the bound on the condition number in (21), it is standard practice to seect α of the same order of magnitude as the conductivity coefficients [5] In our case, by choosing α as M 2σ t +σ =: α we optimize the upper bound µ M with respect to the conductivity coefficients Since α 1, this vaue of α usuay aso eads to the ower estimate µ m = 1 Numerica experiments vaidated this choice Our third variant copes with the aready mentioned fact that buiding P V becomes expensive when grid refinements are required Various choices have been discussed in the iterature [10]; for instance, in [3] the vertex preconditioner was chosen as the vertex bock of the Schur compement matrix on a fixed auxiiary coarse mesh, independent of the space discretization We thus approximate P V with the matrix P V c obtained with a fixed coarse mesh, yieding ( ) (R) P P 3 = E 0 0 P V c This variant eads to very moderate (cose to unit) vaues of λ min (P 1 S V c V ) and λ max (P 1 S V c V ) to achieve an estimate in (21) Therefore, we maintained the seection of α as discussed above In Tabe 1 we report numerica experiments with the preconditioners P 1, P 2 and P 3 for the Schur compement system associated with the matrix in (19) The resuts are in cose agreement with the theory: the condition number of the preconditioned matrix grows at most poyogarithmicay with the number of degrees of freedom per subdomain, as indicated by (20) The coumns with α = 1 refer to such parameter seection in P (R) E This corresponds to discarding information on the conductivity tensor M in (2) The worse convergence vaidates our choice and shows the importance of an appropriate choice of the parameter Note that the seected vaue of α was more effective than other choices of simiar magnitude Indeed choosing α = σ and α = σ t for N 2 = 256, n = 5, 10, 20, 40 (fourth row in the tabe) and P 3, we obtained 29, 31, 33, 34 and 42, 45, 48, 49, respectivey In summary, our experiments demonstrate that the proposed variants aow us to imit the computationa cost (the cost of forming P (R) E and P V c is much ower than that for their origina counterparts), with basicay no oss in convergence rate, for the appropriate scaing factor

8 8 Mico Pennacchio and Vaeria Simoncini Tabe 1 Number of conjugate gradient iterations needed to reduce the residua of a factor 10 5 with the preconditioners P 1, P 2, P 3 and P 2, P 3 with α = 1 K = N 2 : # of subdomains n 2 : # of eements per subdomain Symbo * : the preconditioner P 1 coud not be buit due to memory constraints P 1 P 2 P 2 α = 1 P 3 P 3 α = 1 N 2 \n * * * References 1 Achdou, Y, Maday, Y, Widund, O: Substructuring preconditioners for the mortar method in dimension two SIAM J Numer Ana, 36, (1999) 2 Bernardi, C, Maday, Y, Patera, AT: Domain decomposition by the mortar eement method In: Kaper, H et a (eds) Asymptotic and Numerica Method for Partia Differentia Equations with Critica Parameters Dordrecht: Reide, (1993) 3 Bertouzza, S, Pennacchio, M: Preconditioning the mortar method by substructuring: the high order case App Num Ana Comp Math, 1, (2004) 4 Brambe, J H, Pasciak, JE, Schatz, A H: The construction of preconditioners for eiptic probems by substructuring, I Math Comp, 47, (1986) 5 Chan, T F, and Mathew, T P, Domain Decomposition Agorithms, In: Acta Numerica 1994, Cambridge University Press, (1994) 6 Y Notay, Agebraic mutigrid and agebraic mutieve methods: a theoretica comparison Numer Lin Ag App, 12, (2005) 7 Pennacchio, M and Simoncini, V: Efficient agebraic soution of reaction diffusion systems for the cardiac excitation process J Comput App Math, 145 (1): (2002) 8 Pennacchio, M : The mortar finite eement method for the cardiac bidomain mode of extraceuar potentia, J Sci Comput, 20, n2, pag (2004) 9 Pennacchio, M: The Mortar Finite Eement Method for Cardiac Reaction- Diffusion Modes, Computers in Cardioogy 2004; IEEE Proc, 31: (2004) 10 Tosei, A, Widund, O Domain decomposition methods agorithms and theory, Springer Series in Computationa Mathematics, voume 34 Springer-Verag, Berin (2005) 11 Wohmuth, B: Discretization Methods and Iterative Sovers Based on Domain Decomposition, Lecture Notes in Computationa Science and Engineering, voume 17, Springer (2001)