(e.g., []). In such cases, both the grd generaton process and the soluton of the resultng lnear systems can be computatonally expensve. The lack of re

Transcription

1 A Free-Space Adaptve FMM-ased PDE Solver n Three Dmensons H. Langston L. Greengard D. orn October, Abstract We present a kernel-ndependent, adaptve fast multpole method (FMM) of arbtrary order accuracy for solvng ellptc PDEs n three dmensons wth radaton boundary condtons. The algorthm requres only a Green s functon evaluaton routne for the governng equaton and a representaton of the source dstrbuton (the rght-hand sde) that can be evaluated at arbtrary ponts. The performance of the FMM s accelerated n two ways. Frst, we construct a pecewse polynomal approxmaton of the rght-hand sde and compute far-feld expansons n the FMM from the coeffcents of ths approxmaton. Second, we precompute tables of quadratures to handle the near-feld nteractons on adaptve octree data structures, keepng the total storage requrements n check through the explotaton of symmetres. We present numercal examples for the Laplace, modfed Helmholtz and Stokes equatons. Introducton Many problems n scentfc computng call for the effcent soluton to lnear partal dfferental equatons wth constant coeffcents. On regular grds wth separable Drchlet, Neumann or perodc boundary condtons, such equatons can be solved usng fast drect methods. For free-space boundary condtons and hghly nonunform source dstrbutons defned on adaptve and/or unstructured grds, alternatve approaches are necessary. In ths paper, we descrbe a drect hgh-order adaptve solver for nhomogeneous lnear constant-coeffcent PDEs n three dmensons wth decay condtons at nfnty. A typcal case s the Posson equaton: u = g, supp(g) Ω, () where Ω s a bounded doman n R, and u(x) decays as / x at nfnty. Our solver uses a kernel-ndependent fast multpole method (FMM) [, ] whch can be appled to any ellptc PDE for whch a Green s functon evaluaton routne s provded. It can handle hghly nonunform sources n an effcent manner, usng an adaptve approxmaton of the rght-hand sde (g ()). The structure of the solver allows for natural ntegraton wth FMM-based boundary ntegral equaton technques, leadng to the constructon of an adaptve kernel-ndependent solver for nhomogeneous PDEs n complex geometres, whch wll be descrbed n a companon paper. Related work. For regular grds n separable coordnate systems (rectangles, dsks, spheres, etc) fast methods for constant-coeffcent second order PDEs are well-establshed [, ]. These methods generally rely on cyclc reducton and/or fast Fourer transforms (FFTs) to acheve nearly lnear scalng. For many problems, however, adaptve meshes are essental [,, ], and exstng solvers typcally rely on doman decomposton strateges [] or multgrd acceleraton [,,, ]. For complex geometres, unstructured grd generaton technques are often used Courant Insttute, New York Unversty, New York. Emal: {harper,greengard,dzorn}@cms.nyu.edu The work of L.G. was supported n part by the U.S. Department of Energy under contract DEFGER.

2 (e.g., []). In such cases, both the grd generaton process and the soluton of the resultng lnear systems can be computatonally expensve. The lack of regularty n the data structures adds complextes n parallelzaton as well [, ]. A more recent class of methods combnes deas from potental theory wth fnte dfference methods. In [], fast drect solvers were used on a sequence of refned grds wth boundary condtons nherted from the coarser levels. Ths results n dscontnutes at coarse-fne nterfaces whch are corrected usng a second pass through the grd herarchy. In [], the method of local correctons (MLC) [] was combned wth multgrd methods to solve the Posson equaton on a herarchy of nested grds. The authors also showed how to mpose free-space boundary condtons on the computatonal doman. The fastest free-space Posson solver for three-dmensonal problems of whch we are aware s descrbed n []. It solves local Posson problems on fne grds usng FFT-based technques and couples together the solutons on coarser grds usng the MLC. Ths approach was shown to be very effectve n parallel, wth good scalng up to processors. (A smlar two-dmensonal scheme s descrbed n []). For unstructured meshes, the precedng methods don t apply wthout sgnfcant modfcaton and most fast solvers are based on teratve methods usng multgrd or doman decomposton acceleraton [,, ]. In ths paper, we concentrate on the ntegral equaton (or more precsely the ntegral transform) vewpont. Rather than (), for example, we wll compute g(y)dy. () u(x) = π Rd x y Among the advantages of ths approach s the fact that there s no loss of precson n computng dervatves. In PDE-based methods, f frst or second dervatves of the soluton are needed, accuracy tends to degrade because of the need for numercal dfferentaton. We wll, nstead, dfferentate the kernel n () and compute dervatves from ther ntegral representaton as well. Other advantages of () are that free-space radaton condtons are automatcally satsfed, that we can obtan smple a pror error estmates, and that hgh order accuracy s straghtforward to acheve. However, the computatonal complexty of a naı ve mplementaton s hgh. Fast algorthms such as the arnes-hut method [] and the Fast Multpole Method (FMM) [,,, ], desgned for gravtatonal/coulomb nteractons are essental for effcency and scalablty. These methods fall nto the class of what are sometmes called tree codes because they separate near- and far-feld nteractons on a herarchy of spatal scales usng quadtree or octree data structures (n D and D, respectvely). We concentrate on the FMM n the present settng because t can acheve arbtrary precson at modest cost wth provable error estmates. The classcal FMM, however, s based on detaled propertes of the kernel and requres dfferent analytc tools for each case. The Helmholtz equaton was frst treated, for example, n []. A three-dmensonal verson effectve for all frequences (and addtonal references) can be found n []. The modfed Helmholtz equaton was dscussed n [, ], and the bharmonc equaton n [,, ] The Stokes equatons are somewhat exceptonal, snce they can be handled by a sequence of calls to the orgnal (Coulomb) FMM [, ]. The lterature s now qute substantal and we wll not seek to revew t here. An attractve alternatve that avods much of the detaled analytc work of these methods s the kernel-ndependent approach of [, ]. In ths approach, expansons n specal functons are replaced wth equvalent source denstes. The result s that the same numercal apparatus can be used for a varety of PDEs. The user need only supply a subroutne for the evaluaton of the relevant Green s functon. Whle the bulk of the work on FMMs over the last two decades has concentrated on partcle nteractons or the acceleraton of boundary ntegral equaton methods, there has been some work on solvng nhomogeneous PDEs. One opton s to couple the FMM wth fnte dfference methods to allow for fast solvers n complex geometry [,, ]. Whle ths s a sgnfcant mprovement n terms of range of applcablty over classcal fast solvers, these methods requre a regular volume mesh on whch s supermposed an rregular boundary. Adaptve FMMs for volume source dstrbutons n two dmensons were descrbed n [,, ]. The present paper extends these two-dmensonal schemes to three dmensons, ncorporates them nto kernel-ndependent FMMs, and ntroduces several new performance optmzatons. The result s an effcent, adaptve method that s capable of computng volume ntegrals n three dmensons for a broad varety of PDE kernels. efore turnng to the method tself, we should also note that there has been a sgnfcant body of work n the quantum chemstry communty on acceleratng volume ntegral calculatons usng the FMM, where collectons of Gaussans are typcally used to descrbe the charge dstrbuton [, ]. These are Posson problems n free-space but wth a dfferent approach to defnng the rght-hand sde.

3 Equatons and Kernels Gven a lnear, constant-coeffcent PDE L(u)(x) = g(x), () classcal mathematcal methods provde the correspondng Green s functon K(x, y) n free space. Ths allows for the drect computaton of the soluton to (): u(x) = K(x, y)g(y)dy, () Ω where Ω s the support of g. K(x, y), t should be noted, s n general weakly sngular. Thus, for () to yeld a useful numercal technque, we need a sutable quadrature approach. We also need a fast algorthm, snce the non-local character of the ntegral representaton would lead to an O(N ) soluton procedure, assumng g(x) s gven at N locatons and the soluton s desred at N ponts. If these problems can be overcome, a number of advantages follow. Frst, no lnear system needs to be solved; adaptvty smply means that we use an adaptve quadrature rule. Second, as ndcated n the ntroducton, dervatves can be computed wthout loss of precson. The gradent of u(x), for example, s smply u(x) = K(x, y)g(y)dy, () Ω Thrd, we have smple a pror error estmates. Lettng g (x) denote our approxmaton to g(x), and K (x, y) our approxmaton to K(x, y), we have the computed soluton u (x) = K (x, y)g (y)dy. () Ω and the error estmate e(x) = u(x) u (x) = K(x, y)g(y)dy K (x, y)g (y)dy = K(x, y)[g(y) g (y)]dy + K(x, y) K (x, y) g (y) dy Ω Ω Ω Ω C kg(y) g (y)k + kg (y)k, () where C = max x K(x, y)dy, Ω and s the tolerance of the kernel approxmaton - an easly tunable parameter n the FMM. The estmate above s much sharper than one typcally obtans when dscretzng the PDE tself, where the order of accuracy s determned by hgh dervatves of the soluton. Here, t depends only on the qualty of the approxmaton of the rght-hand sde. In partcular, a k th -order polynomal approxmaton leads to a k th -order accurate scheme wth a very small constant. (C s a bounded quantty determned by the volume of Ω wth no dependence on the data.) The prncpal drawback s that, when mplemented naı vely, the complexty of the approach s quadratc n the number of sample ponts. FMM algorthms overcome ths computatonal barrer by makng systematc use of the smoothness of dstant nteractons on a herarchy of spatal scales [,, ]. The kernel-ndependent versons of the FMM [, ] are partcularly useful n ther generalty; they make t possble to compute solutons of the form () for any (non-oscllatory) ellptc PDE, provded only a module whch evaluates the kernel. After descrbng the detals of the approach, we demonstrate ts performance for the Posson equaton (), the modfed Helmholtz equaton (), and the Stokes equatons (): u(x) = g(x), ()

4 αu(x) u(x) = g(x), () p(x) u(x) = g(x), u(x) =. () The correspondng kernels n three dmensons are gven by, πr () αr, e πr () K(x, y) = K(x, y) = and K(x, y) = πµ r r I+ r r, () respectvely. The classcal FMM s revewed brefly n secton, the kernel-ndependent method s descrbed n detal n secton, and numercal experments are presented n secton. Analytc Fast Multpole Method To establsh termnology and notaton, we summarze the structure of the orgnal two-dmensonal FMM for the Posson equaton [], followed by a descrpton of our extenson to three dmensons for a kernel-ndependent volume solver n secton. Gven a force dstrbuton g at Nsrc source locatons, we wsh to compute the nduced potentals uj at Ntrg target locatons, xj : K(xj, y)g(y)dy uj = u(xj ) = R N src K(xj, y )g(y )w, j =,..., Ntrg, () = where K(x, y) = log {x y} /π and w s a quadrature weght assocated wth source locaton y. For Nsrc Ntrg N, the FMM decreases the computatonal cost from O(N ) to O(N ) for a fxed user-prescrbed level of accuracy. It does so by ntroducng a herarchcal quadtree partton of a boundng square D, enclosng all target and source ponts, and two seres expansons for each box at each level of the herarchy. More precsely, the root of the tree s assocated wth the square D and referred to as level. The boxes (squares) at level l + are obtaned recursvely, subdvdng each box at level l nto four squares, referred to as ts chldren. For a box of dameter H, ts near feld N s defned to be the set of all boxes n D contaned nsde a box centered at of wdth H. The neghbor lst L N of a box s defned to be the set of boxes sharng a vertex wth that are elements of N. (In the nonadaptve case, L = D \ N. Fnally, the nteracton N = N. The far feld of, denoted F, s the complement of near feld: F lst of box, denoted by LI, s defned to be the chldren of s parent s neghbors that are not neghbors themselves. Thus, L I F. An example of a unformly refned D doman and quadtree structure s shown n Fgure. The depth of the tree s chosen so that the smallest boxes (leaf nodes n the tree structure) contan no more than some fxed number of ponts, say s. For smplcty, we frst consder unformly refned trees, where all leaves n the tree structure are at the same level. Let us note that the total number of boxes n the quadtree s bounded by N/s (N/s n three dmensons). Thus, f the workload per box s constant, then the net algorthm has O(N ) complexty. Two types of seres are assocated wth each box n the herarchy: A local expanson used to represent the nfluence of all sources n the far feld of. Snce ths feld s a D-harmonc functon, we can wrte t as the real part of a complex Taylor seres: " p # k uloc (x) = < ck ((x + x ) z ) k=

5 n n n n n n n n Fgure : On the rght s shown a doman whch has been fully subdvded three tmes, resultng n a quadtree on the left. The root of the tree represents the entre doman at level. The non-leaf nodes at level of the quadtree represent the four boxes on the rght enclosed by the thckest black lnes. The rght mage shows the neghbor lst L N for the box (boxes marked wth n on the grd, lght gray n the tree), and LI (boxes = (D L )N. marked wth on the grd, dark gray n the tree). Here, N = L and F N I where z s the center of (vewed as a pont n the complex plane) and x = (x, x ). The error n the local expanson can be shown to be of the order O( )p so that settng p = log (/ ) s suffcent to guarantee a precson of []. A multpole expanson about z that represents the nfluence of sources nsde on boxes n the far feld F. It can be expressed as the real part of a complex Laurent seres: " # p a k u. f ar (x) = < a log(x + x z ) + (x + x z )k k= The moments of ths expanson are computed from the source dstrbuton as: f (y)((y + y ) z )k f (y)dy, ak =. a = π π k () The error n the multpole expanson s also of the order O( )p. The FMM computes the total feld at a target pont n leaf box as the sum of (a) the feld due to the source ponts contaned n the boxes of the neghbor lst L N and (b) the contrbuton from sources n the far feld F. The are computed drectly usng (), whle the contrbutons contrbutons from source ponts nsde the boxes of L N from F are obtaned by evaluatng the local expanson of box at the target. The essental task of the FMM s the constructon of the local expansons n a herarchcal manner. Ths takes place n two steps. The upward pass. Ths pass begns at the fnest level of the tree data structure, convertng force values at source ponts to multpole expanson coeffcents for each leaf box; ths computaton s carred out by the source-to-multpole (SM) operator, an p s matrx, where s s the number of source ponts n. The multpole coeffcents for coarser level boxes are obtaned recursvely, by mappng coeffcents of multpole expansons wth respect to chldren s centers to the multpole expanson wth respect to s center. Ths map, the multpole-to-multpole (MM) operator, s lnear and gven by a p p matrx for each chld. The downward pass. Ths pass starts at the coarsest level of the tree. For each box, the local expanson of the far feld s obtaned by frst shftng the local expanson of s parent to the center of. The mappng whch carres ths out s a p p matrx referred to as the local-to-local (LL) translaton operator. We then need to add the contrbutons from the multpole expansons centered at each of the boxes n s nteracton lst L I. It s straghtforward to check that these contrbutons are exactly the dfference between the far feld of and the far feld of s parent. For each box n L I, one converts ts multpole expanson to a local expanson centered n. Ths mappng from the vector of multpole coeffcents ak to the vector of local expanson coeffcents ck s referred to as the multpole-to-local (ML)

6 translaton operator. It s also lnear and gven by a p p matrx. It s easy to see that the work per box n both upward and downward passes s constant. At the end of the downward pass, local expansons are avalable n each leaf node. These can then be evaluated at each target pont. We refer to the evaluaton of the local potental as the local to target (LT) translaton operator; f the number of target ponts n a box s t, then the LT operator s gven by a p t matrx. To summarze, the FMM uses SM, M M, M L, LL and LT lnear operators, each of whch s represented by a matrx. For the M M and LL operators, each matrx s determned unquely by the relatve poston of a box and ts parent - there are such matrces for quadtrees and for octrees. For M L operators, each matrx corresponds to the relatve poston of a box n the nteracton lst - there are such matrces for quadtrees and for octrees. These numbers can be consderably reduced by takng advantage of symmetres, a topc we wll return to later. The SM and LT operators depend on source and target pont locatons, and can be dfferent for each box. Algorthm outlnes the basc FMM, omttng the techncal detals. For fxed s and p, the computaton s constanttme per box, leadng to an O(N ) method overall. Fgure llustrates the data flow nvolved n the M M, M L and LL operators. Algorthm Non-Adaptve Analytc FMM STEP - CONSTRUCT TREE T AND LISTS buld T such that each leaf contans at most s ponts for each box n preoder traversal of T do buld lst of nearest neghbors, L N and nteracton lst, LI end for STEP - UPWARD PASS for each box n postoder traversal of T do f s a leaf box then Construct multpole expanson αk, { k p} from all source ponts and forces usng SM operator else Construct multpole expanson αk, { k p} from each of s chldren usng the M M operator end f end for STEP - DOWNWARD PASS for each box n preoder traversal of T do Compute the contrbuton to s local expanson from ts parent s local expanson usng the LL operator Compute the contrbuton to s local expanson from L I lst usng the M L translaton operator end for for each leaf box n T do Compute the potental at each target locaton from s local expanson usng the LT operaton Compute the potental at each target locaton from L N usng drect calculatons end for P LL MM ML C Fgure : oxes used by MM, LL and ML operators. For the Laplace kernel, /r, n three dmensons, far-feld expansons are represented usng sphercal harmoncs [] rather than Laurent seres n the analytc FMM. Sgnfcant speedups can be obtaned by usng plane-wave representatons as well []. We turn now to the kernel-ndependent approach [, ] n order to desgn a volume ntegral FMM n three dmensons that can handle a broad class of PDEs.

7 D Kernel-Independent FMM olume Integral Solver Our algorthm follows the overall structure of the FMM algorthm descrbed above. Gven an octree T for our threedmensonal doman D, let { }, =... M be the set of leaf boxes. For a sngle-layer kernel K, we compute the ntegral () at some pont x as u(x) = M K[, g ](x), () = R where we use notaton K[, g ](x) for K(x, y)g(y)dy, and g represents the restrcton of the source dstrbuton to the box. As n the analytc FMM, the contrbutons to u(x) from boxes nearest to x are calculated drectly as near-feld computatons whle all other contrbutons are calculated usng the SM, M M, M L, LL and LT translaton operators n the upward and downward passes of the FMM algorthm. The prncpal dfference between the approach of ths paper and the analytc FMM s that we use sampled equvalent denstes nstead of seres coeffcents, as n [, ]. Ths requres only a black-box kernel evaluaton routne and allows for a kernel-ndependent mplementaton, rather than specalzed translaton operators for each kernel. As n [], we use polynomal bass functons to approxmate the source dstrbuton g on each leaf box. More precsely, we assume that the nput source s gven on each leaf box by a polynomal g of degree k + wth coeffcents γ : g = Nk γj βj ` (x c ), () j= where βj are polynomal bass functons, ` s the depth of the box (` = at the root of T ), and c s ts center. (We use monomals for low-order accuracy and tensor-product Chebyshev polynomals for hgher-order accuracy.) The number of coeffcents s Nk = r k(k + )(k + )/ where r s the dmenson of the kernel. We descrbe an nterpolaton scheme to convert a set of source values defned on a grd of sample ponts to a polynomal representaton n Secton.. The output of our algorthm s ether pont values of the potental at target ponts or a polynomal approxmaton of the potental on each leaf box (whch can then be evaluated at arbtrary locatons). To smplfy the exposton, we present our algorthm frst for a unformly refned octree of depth ` and then dscuss the changes necessary for the adaptve octree case separately. We begn by explanng our use of equvalent densty representatons for g and γ.. Equvalent Denstes The kernel-ndependent approach to translaton operators s based on the followng dea. For kernel K, suppose we have a (generalzed) source dstrbuton gs n a volume Ωs (t can be concentrated entrely on the surface Γs of Ωs ). Let Γt denote an auxlary surface n the exteror of Γs that encloses Ωs, and let Γcheck denote yet another auxlary surface n the exteror of Γt. Fnally, let Ω denotes the exteror of Γcheck. We wll compute a charge densty φt on Γt such that the potentals K[Ωs, gs ] and K[Γt, φt ] concde n Ω. Ths s always possble f the exteror Drchlet on Γt has a unque soluton and the exteror feld can be represented n terms of a sngle layer potental. Remark For some problems, such as the Helmholtz equaton, a combnaton of sngle and double layer sources may be requred because of non-physcal resonances n the sngle layer representaton but t s generally suffcent for non-oscllatory kernels (cf. [] for the Posson equaton, [] for the Stokes equatons). Snce our goal s to use K[Γt, φt ] to represent the far-feld nstead of a multpole expanson, we let Γcheck approxmate the outer boundary of the neghbor lst L N. We then solve the Fredholm ntegral equaton of the frst type for φt : K[Γt, φt ](x) = K[Ωs, gs ](x), for all x Γcheck. () Havng matched the feld on Γcheck, the felds wll match n the exteror Ω (wth precse estmates dependng on the specfc kernel). We refer to Γt as an equvalent surface wth equvalent densty φt, and Γcheck as a check surface. In the case when the orgnal densty s concentrated on the surface Γs, then () can be wrtten as

8 K[Γt, φt ](x) = K[Γs, φs ](x), for all x xt. () Equatons () and () form the foundaton for the dervaton of our translaton operators. Just as the equvalent denstes wll be used to replace multpole expansons n the FMM, a dfferent equvalent densty wll be used to replace the local expanson. We wll match the feld created by charges outsde the near neghbors of a box by a dscertzed layer potental defned on a surface enclosng the box. The number of samples used to represent the equvalent densty s the analog of the number of expanson terms. The equvalent and check surfaces are cubc surfaces, unformly sampled at p locatons. For a requested FMM precson, f mm = np, p = np (np ) (the nature of these ponts s dscussed n secton.). In dscretzed form, () can be wrtten as KΓt,xt φt = KΓs,xt φs, () Ka,b j where φs and φt are vectors of pont-sampled denstes, and Ka,b are matrces wth entres gven by = K(a, bj ) for sample ponts a and aj on surfaces a and b. If φs s known, and we solve for φt, () s a dscretzaton of a Fredholm equaton of the frst knd. For large p the resultng lnear system s poorly condtoned, so we must use regularzaton methods to nvert of KΓt,xt. Tkhonov regularzaton [] replaces (K) wth (αi + (K) K) K, where the regularzaton parameter α s chosen to mnmze the error n matrx nverson. We dscuss the choce of α and the resultng accuracy n secton.. Kernel nvarance and matrx precomputaton. For all equatons we consder, the kernels are nvarant wth respect to rgd transformatons: for scalar kernels, K(T x, T y) = K(x, y) for any rgd transformaton T, and for matrx kernels such as those used for the Stokes equatons, K(T x, T y) = T K(x, y)t T. Hence, all matrces K need to be computed only once for each class of pars of equvalent surfaces, closed wth respect to rgd transformatons. As we defne equvalent surfaces relatve to boxes, these classes typcally correspond to adjacency relatonshps between boxes. Furthermore, many (but not all) kernels are homogeneous: for any postve c, K(cx, cy) = cr K(x, y) for some r =. We wll refer to r as a scalng exponent. In such cases, the number of classes of equvalent surface pars requrng separate matrces can be further reduced. Smlarly, for SM and near-feld calculatons, kernel nvarance can be used to precompute translaton coeffcents. We consder optmzatons due to nvarance for each translaton operator n the next sectons. To smplfy formulas, we assume scalar kernels n our presentaton, although our mplementaton can handle matrx kernels.. Upward Pass For the upward pass, we defne the source-to-multpole and multpole-to-multpole operators. The analog of sources n the analytcal multpole algorthm n our case are polynomals approxmatng the force on a leaf box. The analog of multpole expansons are upward equvalent denstes. For consstency wth analytc FMM, we use S and M n operator names to denote these quanttes. Source to Multpole (SM) translatons. For each leaf box, we choose y,u, the upward equvalent surface, to be a box of radus ( + δ)r and x,u, the upward check surface, to be a box of radus ( δ)r. oth surfaces are centered at c, the center of and algned wth ; δ s chosen to satsfy δ. y choosng δ to be small, x,u and y,u are well-separated, ensurng smooth kernels for evaluatng equvalent denstes [] (we use δ =. n practce). Equaton () for upward equvalent densty φ,u n ths case becomes K[y,u, φ,u ](x) = K[, g ](x), for all x x,u. () Wth polynomal coeffcents γ of g, the rght-hand sde of () s approxmated by K[, g ](x) Nk γj Fj (x), j= where ()

9 Fj (x) = βj ` (y c ) K(x, y)dy, for x x,u, () and γj s the j th coeffcent of γ, βj s the jth of Nk bass polynomals. y translaton nvarance, Fj (x) depends only on the choce of βj and tree level ` of. To evaluate the ntegrals n expressons for Fj (x ), we use adaptve Gaussan quadrature []. The Nystro m dscretzaton of () at p sample ponts on y,u yelds (): K[y,u, φ,u ](x ) = Nk Fj (x )γj, for x x,u, () j= or n matrx form,u K = F SM φ SM γ,,u () F SM where φ s the vector of samples of equvalent densty of sze p, represents the matrx of precomputed weghts Fj (x ) of sze p Nk, and K s the matrx wth entres K(x, y j ), =... p, j =... p. SM Solvng for φ,u, φ,u = (K FSM γ = T () SM ) SM γ. Snce for a unformly refned tree all leaves are at the same level, the matrx T SM depends only on the level ` due to translaton nvarance, so only one matrx needs to be computed. Fgure (a) llustrates the computaton of φ,u from γ. Multpole to Multpole (MM) translatons. MM translaton operators convert the sampled equvalent densty representaton of a feld at a chld box C to a sampled equvalent densty for the parent box, shown n fgure (b). The upward equvalent surfaces yc,u, y,u, and the upward check surface x,u are defned n the same way as for SM translatons. For chld C of, we use () wth φs = φc,u, φt = φ,u : K[y,u, φ,u ](x) = K[yC,u, φc,u ](x), for all x x,u, () C leadng to the dscretzed equaton for at level ` n T,u K, = M M φ C,u KC,. M M φ C Smlar to the SM computatons, these systems are solved as, C, C,u φ,u = (KM M ) KC, = TM M φc,u. M M φ C () C For any two chldren C and C, there s a rotaton R mappng C to C ; therefore, for kernels nvarant wth respect to translatons and rotatons, only one matrx TC, M M needs to be computed per level, wth the contrbuton to φ,u from any other chld obtaned by composng ths matrx wth an approprate permutaton of φc,u. For homogeneous kernels, only one matrx needs to be stored at a sngle level `. Indeed, for a box at depth `, the matrx K, M M has entres wj K(y, xj ), where wj s the quadrature weght of sample at xj. If the scalng factor for a matrx K s r, K(y, xj ) = (/)r` K(y, x j ) where y and x j are samples correspondng to y and xj based on a normalzed box = [, ] at level ` =. The quadrature weghts scale as area; that s, they need to be multpled C, by /` such that matrx entres scale as /(r+)`. As K, M M and KM M scale n the same way, the normalzed C, matrx TM M does not depend on scale. For nhomogeneous kernels, at most one matrx per level needs to be stored.

10 x,u (KSM ) x,u (K, ) M M y,u y,u yc,u FSM a) KC, M M b) Upward Equvalent Surface Upward Check Surface y,u yp,d y,d y,d K, M L y,d x,d x,d KP, LL x,g K, LT, (KLL ), ) LL (K c) d) Downward/Upward Equvalent Surface e) Downward Equvalent Surface Downward Check Surface Fgure : Kernel-ndependent FMM translaton operators. Top: upward pass SM and M M translatons; ottom: downward pass M L, LL, LT translatons.. Downward Pass In the downward pass, we compute the analog of local expansons n the analytc FMM, the downward equvalent denstes. We defne the kernel-ndependent versons of M L operators (for boxes n the nteracton lst L I ), LL operators (for translatng a parent s local expanson), and LT operators for fnal evaluaton at target locatons. These translaton operators are llustrated n Fgure (c-e). Multpole to Local (ML) translatons. The M L operator translates an upward equvalent densty φ,u, approx,d matng the feld of sources nsde a box L for a box, approxmatng I, to a downward equvalent densty φ the nfluence of these far-feld sources nsde. In ths case, we seek to have dentcal potentals nsde the box. To satsfy the condtons for check and equvalent surfaces, should be enclosed by the check surface x,d, whch, n turn, s enclosed by the downward equvalent surface y,d, not overlappng y,u. Ths s acheved by swappng upward equvalent and check surfaces to obtan downward equvalent and check surfaces: y,d = x,u and x,d = y,u. Equaton () takes the form

11 K[y,d, φ,d ](x) = K[y,u, φ,u ](x), for all x x,d. (),d Downward equvalent denstes are also dscretzed at p unformly spaced samples on y. The rght-hand sde of () s computed and stored as a downward check potental, u,d at x,d, and φ,d s recovered after the LL contrbuton s added. For a box at depth `:,u K,. () u,d M L φ M L = L I As x,d and y,u are chosen at equspaced locatons on the boundares of and, we can effcently evaluate u wth FFTs by treatng y,u and x,d as Cartesan grds wth zero densty n the nteror, resultng n a D convoluton. Ths results n O(p/ ) sample and target locatons nstead of p. The FFT and ts nverse are computed once per box at a cost of O(p/ log (p)). There are = possble locatons for L I relatve to (at most for any partcular ); however, usng translaton and rotaton nvarance of the kernel, each relatve poston of can be assgned to one of equvalence classes, as dscussed n secton. For a homogeneous kernel, we compute and store matrces K, M L, one for each class of from the nteracton of the normalzed box = [, ], wth K, lst L I M L obtaned by scalng as n the M M case. For an nhomogeneous kernel, at most matrces are needed for each possble level of T (the actual number s smaller than the maxmum number, due to boundary effects at the coarse levels of the tree).,d Local to Local (LL) translatons. Contrbutons of the far-feld boxes outsde the nteracton lst L I are captured through the local feld computed for s parent box P. We translate φp,d at yp,d to φ,d at y,d usng the equaton K[y,d, φ,d ](x) = K[yP,d, φp,d ](x), for all x x,d. () As for the M L operator, we compute the rght-hand sde as a contrbuton to the check potental u,d, so the dscretzed verson of () for at depth ` becomes: P, P,d u,d. LL = KLL φ (),d φ,d = (K, u,d LL ) M L + ull. () Then, φ,d s calculated as C, The precomputaton of matrx KP, LL s completely analogous to KM M, wth parent and chld swapped. As n, the M M computatons, the nverted operator, (KLL ) s precomputed once for homogeneous kernels as a p p at ` = wth normalzed box = [, ] and scaled as necessary; for nhomogeneous kernels, at most one matrx s stored for each possble level of T. Local to Grd Target (LT) translaton. At the end of the downward pass, we evaluate the potental at n grd locatons, x,g wthn each leaf box. LT translaton operators map samples of the downward equvalent denstes to potental values at the grd locatons. For every box at depth `, φ,d accounts for all contrbutons from F whle drect near-feld calculatons dscussed n detal n the next secton below account for the contrbutons from N. The far-feld potental s computed by evaluaton at x x,g : u(x) = K[y,d, φ,d ](x). That s,,,d u,g = KLT φ. (), For a unformly-refned tree, all leaves are at the same level, so we precompute and store one n p matrx KLT.

12 . Near-Feld Interactons After far-feld contrbutons are computed, the fnal step s to compute near-feld nteractons for leaf boxes. Ths s often the most expensve step n the computaton, and t s essental to optmze ths part of the algorthm. For a box, our approach can be thought of as two steps (n the actual computaton, the steps can be combned nto one). U U Frst, for a box U LN wth volume densty g, approxmated by γ, we evaluate the potental nsde of,g,g from U on an n grd of samples x on, to obtan a vector u. Then, we compute an nth -order polynomal approxmaton υ to the potental u,g. The resultng polynomal can be evaluated at arbtrary locatons n the box, f necessary. For the frst step, we can use precomputed matrces whch depend only on relatve box poston. Smlar to the case of SM translatons, the contrbuton to the soluton u,g n from the force g U on U s u,g (x) = K[U, f ](x) = Nk γju FjU, (x), () j= FjU, (x) = βj ` (y cu ) K(x, y)dy, for x x,g, () U where cu s the center of box U. We evaluate u,g on a unformly spaced or Chebyshev grd of ponts x,g n, =... n (larger n requre Chebyshev ponts as dscussed n secton.). In matrx form () becomes u,g = F U, γ U. () For a unformly subdvded octree, each leaf box has at most same-level neghbors U L N (ncludng tself). As all leaf boxes are on the same level, we need at most one matrx of sze n Nk per possble relatve neghbor poston. Usng symmetres, we can reduce precomputaton and storage to matrces (we descrbe symmetres n secton ). As n the SM computatons, adaptve Gaussan quadrature [] s used to precompute the weghts for these matrces.. Polynomal approxmaton of the soluton Addng the downward pass and near-feld contrbutons for a box yelds u,g, a vector of potental values on a unform or Chebyshev n grd x,g. The fnal step n our algorthm s to obtan a local polynomal approxmaton to the soluton from these grd values. We compute a polynomal approxmaton υ to u,g usng a least-squares ft, mnmzng n = ku,g (x ) Nn υj βj (x c )k for x x,g, () j= where Nn = n(n+)(n+)/, βj are polynomal bass functons, and c s s center. For n, we use monomal bass functons {xa y b z c, a + b + c n }. For n >, nstead of usng a unform grd of ponts x, we use Chebyshev ponts, and Chebyshev bass {Ta (x)tb (y)tc (z), a + b + c n }. In secton. we nvestgate the accuracy of regularly-spaced ponts and Chebyshev ponts for n =,,. If Γ s the n Nn matrx of bass functons βj (` (x c )), () leads to the equaton υ = Γ(+) u,g, where (+) Γ s the pseudonverse of Γ. We note that Γ(+) needs to be precomputed only once; that s, t does not depend on the kernel and s scale-nvarant n all cases: as Γj = βj (` (x c )) = βj (x j ) where x j are unformly spaced or Chebyshev grd ponts n = [, ], computng υ does not depend on the center c of box and only requres a scalng of Γ(+) by s depth `. If desred, we can evaluate the soluton at an arbtrary pont xt as

13 u(xt ) = Nn υj βj (xt c ). () j= Typcally, we assume that the order of the approxmaton γ of the force g on a box s equal to the order of approxmaton of υ. That s, we choose k = n where forces can be represented on the same k grd on whch we compute u,g. However, as source and target locatons need not be the same, k and n can be chosen as dfferent values.. Polynomal force approxmaton from grd samples In our descrpton of the upward pass, we have assumed that the rght-hand sde s already gven as a polynomal. If the force s avalable n another form (e.g., as samples on an AMR grd or polynomals on an unstructured fnte element grd), we can resample t adaptvely, to obtan k th order approxmatons wth desred error on leaf boxes, then convert t to a polynomal representaton. The only requrement on the nput force n ths case s that we can evaluate t at our grd locatons wth k th order accuracy or better, whch s nontrval only n the case of forces gven by samples at scattered ponts. If the values of the force f (x) are known at k unformly sampled or Chebyshev ponts on a leaf wth center c, an approxmaton to the force s constructed as g (x) Nk γj βj (x x ) j= for Nk = k(k + )(k + )/. As for the evaluaton at arbtrary target locatons, we use the monomal bass for k <, and Chebyshev bass for k >.. Non-Unform Source Dstrbutons and Adaptve FMM We have thus far assumed that all leaves n T are at the same level. Adaptve refnement of the octree results n leaf boxes at dfferent levels for nonunform source dstrbutons. Ths leads to several addtonal types of nteractons between boxes that need to be taken nto account. For arbtrary adaptve octrees, the number of relatve postons of boxes one needs to consder can become very large. In order to avod storng large number of precomputed matrces, we consder level-restrcted refnement: we requre adjacent leaf boxes be wthn one level of each other, a common restrcton n tree codes and structured grds. Many fast approaches exst to convert arbtrary octrees to ones satsfyng ths constrant []; we currently use a straghtforward sequental algorthm (secton.). We begn by ntroducng the notaton for these lsts and then dscuss how ths affects the SM and near-feld nteracton computatons. Lsts for adaptve FMM. For adaptvely refned trees, we defne several lsts n addon to the neghbor lst L N and nteracton lst L used n the unform case. Our defntons and notaton follow [,, ]. I For a leaf box, we defne the U and W lsts. The U-lst, L U, s the set of other leaves adjacent (at arbtrary levels) to, ncludng tself; LU concdes wth the neghbor lst LN for the unformly refned case. The W-lst, L W, s the set of descendants of s neghbors, not adjacent to, but whose parents are adjacent to. Ths lst contans boxes at fner levels than for whch s n ther far range, but whch are n the near range of the parent of. For any W L (conversely, F ). W, W s at a fner level than and W N For leaf and non-leaf boxes, we defne and lsts.

14 The -lst, L, s the set of s parent s neghbor s chldren, not-adjacent to. L = LI for unformly-refned trees, and f one completes an adaptve tree T by addng all mssng boxes, on non-empty levels to a unformly u refned tree T u, then L n T s a subset of LI n T. A The -lst, L, s the set of boxes A such that LW. In a more geometrc manner, one can defne as the set of leaf boxes on levels coarser than, overlappng a box n the nteracton lst of n T u but not overlappng the neghbors of n T u. A The followng observatons can be made about these lsts: LA U f and only f A LU, L f and only f A A L, LW f and only f A L. A fragment of an adaptvely refned level-restrcted tree, wth U,,W and lsts for a box s shown n Fgure. U U U U U U U U U W W W W W Fgure : A d quadtree resultng from a non-unform source dstrbuton. For box, correspondngly marked boxes represent the boxes n the U,, W, and lsts of. For level-restrcted trees, boxes n W and lsts can have only a lmted number of possble postons as can be seen from the the followng lemma. Lemma. For a level-restrcted tree T n whch all neghborng leaf boxes are wthn one level of each other n the octree, for a box,, all boxes n L W and L must also be wthn one level of. Proof Assume for a box that there exsts a box W L W such that level(w ) level(). That mples that for W s parent, PW, level(pw ) level(), further mplyng that for some descendant D of PW, D L U and level(d) level(), volatng our W tree-level restrcton. So, W must be wthn one level of. Snce W L W mples L, ths also means that all boxes L must also be wthn one level of. Possble postons of boxes n U,, W and lsts n a level-restrcted tree are shown n Fgure. oxes n L U and L are treated exactly n the same way as boxes n LN and LI n the unform case: for LU, the near-feld nteracton operators are used, and for L, M L operators are used. For some leaf box wth parent P, f W L W, then W F ; therefore, W s contrbuton to s not W accounted for through P. At the same tme, snce s n F, W s contrbuton to s potental can be computed by evaluatng ts upward densty potental (the analog of multpole expansons) at target locatons n ; hence, usng notaton analogous to other operators, M T operators need to be defned. For L but F P., N Thus, we need to evaluate contrbutons of sources from drectly, but can apply them to the downward densty of ; that s, we need to defne an SL operator. As explaned below, for the local low-order polynomal representatons to the force dstrbutons, t may be preferable to use near-feld computatons mappng polynomal coeffcents from boxes n L and LW to potental values at target locatons n. To summarze, for adaptve FMM, n addton to M M, M L, LL and LT already defned, two addtonal operators, M T and SL need to be defned. Further, as leaves of the tree now may exst at arbtrary levels, and boxes U L U may be on levels dfferent from, both SM and near-feld (ST ) computatons need to be modfed. We begn by descrbng adaptatons to the SM and ST operators and follow wth a dscusson of the new M T and SL operators.

15 same level U W fner level U coarser level U Fgure : Possble box postons for dfferent lsts n a level-restrcted trees n d. The confguratons n d are analogous. SM operators for the adaptve case. For a unformly-refned doman, all leaves n the octree structure are on the same level, so only one matrx, T SM needs to be computed for a box at leaf level `. In the adaptve case, however, leaf boxes can be located at multple levels. For homogeneous kernels, we store a sngle matrx T SM, scalng for level ` n a smlar fashon as was done for the M M and LL operators. Let be the the normalzed box [, ] at ` = and r the scalng exponent of the kernel K. For a box wth center c, mappng a pont x at level ` to x s gven by x = ` (x c ). Changng the varables n the formula for Fj` n (), for x x,u, x x,u we obtan Fj (x ) = (r+)` (r+)` K[, βj ](x ) = Fj (x ). ` r` Smlarly, K(x, yj ) = K(x, yj ). (r+)` r` In matrx form, F FSM and K KSM. Solvng for φ,u, () becomes SM = SM = φ,u = (KSM ) FSM γ = T SM γ, () where T SM s precomputed and stored. (For nhomogeneous kernels, we store one matrx per level contanng leaf boxes). Neghbor lst nteractons for adaptve trees. In secton., for a box wth neghbor U, the assumpton that all U leaves are at the same level allows us to use () wth precomputed matrces FU, ST and U s coeffcents γ to evaluate,g U s contrbuton to s potental at the grd ponts x. For a level-restrcted tree, there are two dfferences: leaves may exst at any level, and the adjacent boxes U L U may contan boxes one level fner or coarser than. For homogeneous kernels, the need to compute separate matrces FU, ST on dfferent levels can be elmnated as done above for the SM operator. One needs to compute matrces only for pars of boxes (, U ) wth scaled to = [, ] (U s scaled to the approprate locaton as well). Gven a box at level ` and an adjacent box U, for x = ` (x c ), () becomes FjU, (x) = ` K[, βj ](x ) = (r+)` FjU, (x ),

16 or n matrx form, u,g = U FU, ST γ (r+)` U FU, = ST γ, () where r s the kernel scalng exponent. As dscussed earler, there are possble same-level neghbors, and due to tree-level restrctons, there are fnelevel neghbors (one level deeper n the tree) and coarse-level neghbors (one level hgher n the tree), all consttutng the possble locatons for boxes n L U. As shown n secton, usng symmetres of relatve postons of U and, we only precompute and store matrces of sze n Nk. For nhomogeneous kernels, ths set of matrces s precomputed for each level for whch leaf boxes exst. MT and SL operators. As explaned above, for a leaf box and W L W, we need an operator that evaluates the potental represented by φw,u, the upward equvalent densty of W, at the target grd locatons on : u,g (x) = K[yW,u, φw,u ](x) for x x,g, or n matrx form, W,u u,g = KW,, M T φ () KW, M T where the operators are precomputed and stored. Smlar to all prevous cases, for homogeneous kernels, W, KM T can be computed for the normalzed box only and scaled as necessary. For leaf and non-leaf boxes, the lst L contans leaf boxes, for whch contrbutons to are computed by evaluatng contrbuton of g, represented by coeffcents γ, on s downward check surface: u,d (x) = K[, g ](x) Nk γj Fj (x) for x x,d, j= or n matrx form, u,d = F, SL γ. () For a box there are possble locatons for W L W ; however, due to symmetres, only locatons are dstnct up to translaton and rotaton, so only KW, matrces of sze n p are stored. Also, due to the nverse M T, relatonshp between L and LW, the number of symmetry classes s the same; that s, matrces FSL of sze p Nk need to be precomputed for each level for whch leaf boxes exst. As n all other cases, for homogeneous kernels, the matrces need to be precomputed only for one level ` = and scaled as necessary. Symmetry classes are dscussed n. We use one addtonal optmzaton whch apples n cases when the order of local polynomal approxmatons of the force s low compared to the order of approxmaton used for upward and downward check denstes. In such cases, the sze of M T and SL matrces may actually be larger than the sze of the matrces needed for drect computaton,g of contrbutons from coeffcents on boxes W L on. Assumng we W or L to the target grd locatons x W, have an homogeneous kernel, f W LW s a leaf for a box, we can replace KM T wth an ST operator FW, ST, constructed exactly n the same way as for boxes n the neghbor lst L U. Smlarly, f s a leaf, for a box L,,, we can replace FSL wth FST. In other words, for low order polynomal approxmatons of the force, one treats leaf W boxes n the same way as adjacent boxes of and for leaf boxes, L s also treated as adjacent boxes. W, Specfcally, FW, s of sze n N whle K s of sze n p, so when a box W s a leaf and Nk < p, we k M T ST, use the faster ST translatons. Smlarly, F, s of sze n N and F s of sze p Nk, so for leaves, we k ST SL, use FST when n < p. Agan, symmetres result n matrces needed of each type (secton ). For nhomogeneous kernels, we must compute and store these matrces for each necessary level `.

17 . Pseudocode and Complexty for Kernel-Independent FMM olume Solver. Pseudocode. In the followng pseudocode, we assume that a tree-level restrcted octree, T already exsts and that for each box,, we are gven the polynomal approxmaton, γ to the force g (we dscuss how to balance a tree whch does not satsfy the tree-level restrcton constrant n secton. and how to construct γ from g n secton.). For clarty of presentaton, n ths pseudocode we do not nclude the optmzaton replacng M T and SL operators wth drect (ST) computatons when ths s more effcent as dscussed above. Algorthm Kernel-Independent olume FMM STEP - UILD LISTS for each box n preoder traversal of T do buld L U, LW, L, and L (secton.) end for STEP - UPWARD PASS (secton.) for each box n postoder traversal of T do f s a leaf box then Convert local force approxmatons to upward denstes: φ,u := T SM γ () else Translate chldren s C,upward denstes to parent s upward densty: φ,u := TM M φc,u () C end f end for STEP - DOWNWARD PASS (secton.) for each non-root box n preoder traversal of T do Add potentals due to parent downward densty, U and boxes to get the downward check potental P,d u,d := KP, + LL φ,u K, + M L φ L F, (), (), () SL γ L Translate the check potental to the downward densty:,d φ,d := (K, u () LL ) f s a leaf box then Compute potentals from adjacent and W boxes to the potental at grd locatons: u,g := U L U U FU, ST γ + W,u KW, (), () M T φ W L W Add the potental from the far feld: u,g := u,g + FLT φ,d end f end for Computatonal complexty and storage requrements. We analyze the complexty of our algorthm for a nonadaptve tree n whch all leaves are at the same level. The analyss for the adaptve FMM s smlar but slghtly more complcated. We begn by assumng that there are ` levels n the octree T. For a unform tree, ths mples we have M` = ` leaves and Mt = (`+ )/ total boxes n T. If we are usng a k th order polynomal approxmaton to the force dstrbuton at each leaf, we further assume there are approxmately N = M` n total target ponts and C = M` Nk total coeffcents. Further, let p be the number of coeffcents sought n the multpole expanson, affectng the sze of the equvalent denstes and surfaces; as dscussed earler, for a desred level of precson f mm = np n the expanson, p = np (np ). In table, we ndcate the computatonal complexty of each step of the FMM algorthm as well as the amount of precomputaton and storage used for operators (ndcated n parentheses) at each step. Further, we assume a homogeneous kernel wth a sngle degree-of-freedom n the source and target drectons for the storage complexty. Hence, storage complexty wll scale lnearly for nhomogeneous or matrx kernels. For non-unform source dstrbutons, we store addtonal operators for the near-feld nteractons (Table ).

18 Operator Computatonal Complexty Storage SM : T SM O(Cp) pnk M M : TC, M M O((Mt M` )p ) p M L: K, M L O(Mt p/ log (p) + Mt p/ ) p/, LL: KP, LL, (KLL ) O(Mt p ) p LT : K, LT O(N p) pn Near Interacton: FU, ST O(N Nk ) Nk n Table : Computatonal complexty and storage requrements for non-adaptve algorthm Operator Storage U -lst (same-level, fne and coarse): FU, ST Nk n W, W -lst: FW, ST, KM T n (Nk + p), -lst: F, ST, FSL Nk (n + p) Table : Storage requrements for addtonal matrces used n the adaptve algorthm Symmetres for precomputed nteracton operators For each of the lsts L U,L,LW and L, we may need to precompute translaton matrces for denstes or polynomal coeffcents. The number of dfferent relatve postons of the box and a box n one of these lsts can be large, and precomputng all possble matrces may requre sgnfcant tme and substantal storage. Performance can also be affected due to the need for random access of large amounts of precomputed data. The number of matrces we need to precompute can be substantally reduced f we take nto account symmetres; that s, many box postons are equvalent n the sense that there s a rgd transformaton T, mappng a box to and the box to tself. We store a sngle matrx for a representatve box for each symmetry class, obtanng matrces for all elements of the class by applyng a transformaton T to the matrx for the representatve box. For every lst type {U,, W, }, we defne a set of possble box postons P os() and a set of symmetry classes whch form a partton of P os(). For each class, we defne a reference box, and for each box poston n P os(), we need an effcent way to determne ts class and a transformaton T () : R R mappng t to the reference box. For all lsts, the symmetres are related to the transformatons of space whch map a grd of cubes to tself. Referrng to a grd of sze N N N grd as an N grd, we consder grds of szes to (we dscuss whch lsts correspond to whch cubes n more detal below). efore consderng ndvdual lsts, we classfy all symmetres of such grds. Grd symmetres. The cubes on the grd N are ndexed by (, j, k) where each ndex takes values M...,,... M for odd N = M + and M...,... M for even N. We skp ndex for even grds to ensure that the coordnates of the cube centers and cube ndces are transformed by symmetres of the cube n the same way. If the cube sze s, then the cube center coordnates are exactly the ndces (, j, k) for odd N and dffer by ±/ for even N, dependng on the ndex sgn. Each N grd can be parttoned nto M (for even N ) or M + (for odd N ) layers, wth layer l consstng of cubes (, j, k) wth max(, j, k) = l. Layer conssts of one cube and exsts only for odd N, and layer M conssts of the