Fast Nystrom¤ Methods for Parabolic Boundary Integral ...faculty.smu.edu/tausch/Papers/lecNotes.pdf · True boundary element methods that eval-uate the time convolutionnumerically

Fast Nystrom Methods for Parabolic BoundaryIntegral Equations

Johannes Tausch

Abstract Time dependence in parabolic boundary integral operators appears in formof an integral over the previous time evolution of the problem. The kernels are sin-gular only at the current time and get increasingly smooth for contributions that arefurther back in time. The thermal layer potentials can be regarded as generalizedAbel operators where the kernel is a parameter dependent surface integral operator.This special form implies that discretization methods and fast evaluation methodsmust be significantly changed from the familiar elliptic case. After a brief reviewof recent developments in the area we discuss the different options to discretizeAbel integral operators in time. These methods are combined with standard surfacequadrature rules to obtain a Nystrom method for parabolic integral equations. Themethod is explicit and we will show how a version of the fast multipole method inspace and time can be used to evaluate the time stepping scheme efficiently.

1 Introduction

The solution of parabolic problems by boundary integral techniques is a well knownalternative to the finite element or finite difference method and has a long history,beginning in the 1960’s [31, 33]. However, the application of integral equation meth-ods to realistic, three dimensional problems was hampered by the high computa-tional expense of evaluating parabolic boundary integral operators. This stems fromfact that these operators involve integrals over time in addition to integrals over theboundary surface. Furthermore, the computation of the influence matrix involves in-tegration of singular or nearly singular integrals on the boundary surface and time,adding to the overall computational cost.

Because of this computational burden, integral equation methods were performedeither in Laplace domain, or were applied to the elliptic equation that arises after

Johannes TauschDepartment of Mathematics, Southern Methodist University e-mail: [email protected]

1

2 Johannes Tausch

time discretization of the parabolic PDE. True boundary element methods that eval-uate the time convolution numerically have also been considered by several authors,for instance [11, 18, 28], but are typically limited to two dimensional problems, orfairly coarse discretizations.

The thermal single layer operator is coercive in an appropriate anisotropicSobolev space [1, 7]. Likewise, the thermal double layer operator is compact in theproper setting [7, 22]. This result is the background for the analysis of the Galerkinmethod in both space and time. Spline collocation methods for two dimensionaldomains where Fourier series techniques can be used have been discussed as well,see [21, 9]. Another discretization approach is the convolution quadrature methodfor the time discretization [27, 34]. This method is combined with a Galerkin orcollocation method for the space discretization of the Laplace transformed integraloperators. All the above methods involve the computation of multi-dimensional andpossibly weakly singular influence coefficients. The different discretization optionsfor parabolic integral equation methods and their theoretical foundations of are sur-veyed in [8].

The focus here is on Nystrom discretizations. The method is based on the ob-servation that thermal layer potentials can be viewed as generalized Abel integraloperators with smooth operator-valued kernels. The quadrature rule in time is adesingularized version of the trapezoidal rule. The time discretization results in aseries of spatial integral operators with kernel, which can be treated with standardsurface quadrature rules. The resulting scheme is explicit in time, and does not re-quire the computation of an influence matrix. The convergence analysis draws heav-ily on the compactness of the layer potentials in the space of continuous functions.Thus the arguments will, of course, break down if the the surface is not smooth. Forthe Laplace equation it is possible to show stability and convergence rates of theNystrom method on non-smooth domains [32], although it is not clear yet whethersimilar results can be obtained for the heat equation as well.

Since integral operators are non-local, discretizations lead to dense matrices andtherefore fast methods are important to handle large scale problems. This is wellknown for elliptic equations, and is true even more so for parabolic equations. Onecan distinguish three approaches to obtain efficient representations of integral op-erators: Multiscale (wavelet) discretizations, fast Fourier methods and clusteringtechniques.

Wavelets lead to asymptotically optimal algorithms in very general settings, werefer to the survey [10]. Recently, they have attracted considerable interest for solv-ing parabolic PDEs because of their good scaling properties in high dimensions [30].On the other hand, it appears that very little has been done in the context of parabolicboundary integral equations. The author is only aware of the preprint [3] that derivessome optimal convergence results.

Fourier techniques suggest themselves because of the convolutional nature of theheat kernel. The underlying principle is that convolutions turn to multiplications inFourier domain. The first paper in this direction is [15], which considers expandingthe Green’s function for the heat equation a bounded domain by a Fourier series. Inthe more natural setting of an unbounded domain the kernel appears as a continuous

Fast Nystrom Methods for Parabolic Boundary Integral Equations 3

Fourier transform which requires special care to obtain discrete spectral approxima-tions [14]. An application of this approach to evaluate thermal potentials is reportedin [24].

Clustering techniques, which include the fast multipole method, H-matrices, andadaptive cross approximations, have proved to be extremely successful to solve el-liptic problems with complicated geometries.

The fast Gauss transform [16] can be seen as a precursor of clustering meth-ods for parabolic integral equations, because the heat kernel with frozen time is aGaussian in space. In [35] this algorithm is applied to handle the elliptic problemthat arises after time discretization of the heat equation. The original version ofthe fast Gauss transform is based on Hermite expansions of the heat kernel. Sincethe spatial variables of the kernel separate, the translation operator appear in tensorproduct form, which can be exploited to reduce the computational cost associatedwith translation operators [36]. Recently, more efficient approximations have beenconsidered, in particular exponential expansions which result in diagonal translationoperators [17] and Chebyshev expansions [41], which allow global approximationsof the Gaussian.

We will concentrate on space-time clustering methods for discretizations of ther-mal layer potentials, which was first introduced in [38]. The algorithm relies on ahierarchic subdivision of space and time and uses a truncated Chebyshev expansionof the heat kernel to evaluate interactions of well separated spatio-temporal clus-ters efficiently. Here we will present the method together with some backgroundmaterial on the discretization and fast evaluation of Abel integral operators and onthe approximation theory of the heat kernel. We will also include some new unpub-lished developments including results obtained with an improved implementation ofour method.

2 Heat Potentials as Abel Integral Operators

Green’s formula relates the Dirichlet and Neumann data on the boundary surface S.For the heat equation ut = ∆u with homogeneous initial conditions it can be statedas

±12u(x, t) = K u(x, t)−V

∂u∂n (x, t), x ∈ S, t > 0. (1)

Here, the plus sign applies for an exterior and the minus sign for an interior problem.The single- and double layer potentials are given by

V g(x, t) =

t∫

0

∫

S

G(x− y, t − τ)g(y,τ)ds(y)dτ ,

K g(x, t) =

t∫

0

∫

S

∂∂ny

G(x− y, t − τ)g(y,τ)ds(y)dτ ,

4 Johannes Tausch

respectively. The Green’s function of the heat equation in d spatial dimensions is

G(r,δ ) =1

(4πδ )d2

exp(

−|r|24δ

)

.

Green’s formula will include additional volume integral operators if nontrivial ini-tial conditions and source terms are present. Moreover, many problems of physicalinterest deal with geometries that evolve in time, in which case the normal velocityof the surface must be added to the normal derivative of the solution, see, e.g. [4].In principle, it is possible to extend the methodology discussed below in these moregeneral settings. However, in the interest of conciseness, we will keep the focus onthe numerical solution of problems that are governed by (1).

For δ > 0, the Green’s function is a Gaussian in space that becomes increasinglypeaked as δ gets smaller. In the limit as δ → 0, it converges to the delta function.This follows from the well-known Poisson-Weierstrass integral, which states that

limδ→0

∫

Rd

G(x− y,δ )φ(y)dy = φ(x).

When the Green’s formula is applied to a problem in three dimensions, the layer po-tentials involve integrals over two dimensional manifolds. To apply the the Poisson-Weierstrass integral in this case one must factor (4πδ )−

12 out to obtain the two-

dimensional Green’s function. This motivates us to the time-dependent surface in-tegral operators

[V (δ )φ ](x) =1

4πδ

∫

S

exp(

−|x− y|24δ

)

φ(y)ds(y), (2)

[K(δ )φ ](x) =1

4πδ

∫

S

∂∂ny

exp(

−|x− y|24δ

)

φ(y)ds(y), (3)

and thus the single and double layer potentials appear in the form

V g(t) =1√4π

t∫

0

1√t − τ

V (t,τ)g(τ)dτ , (4)

K g(t) =1√4π

t∫

0

1√t − τ

K(t,τ)g(τ)dτ . (5)

This representation clearly exposes the (t − τ)−12 - singularity because V (t − τ) and

K(t − τ) can be regarded as smooth operator-valued functions on the triangle

∆T = {(t,τ) : 0 ≤ τ ≤ t ≤ T},


where T is the last time of interest. This observation is made more precise the fol-lowing version of the Poisson-Weierstrass integral formula. The proof can be foundin [39].

Theorem 1. If S is smooth and φ ∈ C2p(S), then the functions Vφ and Kφ aresmooth functions in S× (0,∞). Furthermore, the derivatives ∂ m

δ Vφ and ∂ mδ Kφ have

continuous extensions to S× [0,∞) when m ≤ p. For x ∈ S and q ≤ p the expansions

[V (δ )φ ](x) = φ(x)+δv1(x)+ · · ·+δ q−1vq−1(x)+δ qvq(x,δ )

[K(δ )φ ](x) = H(x)φ(x)+δw1(x)+ · · ·+δ q−1wq−1(x)+δ qwq(x,δ )

hold, where H(x) is the mean curvature of S, vm,wm ∈ C2(p−m)(S) and vq, wq ∈C2(p−q)(S× [0,T ]).

Here and in the following we write g(t) := g(·, t) to denote the function of the spa-tial variable with frozen time variable. The representation (4) and (5) shows thatthe thermal layer potentials may be regarded as generalized Abel integral opera-tors time where kernels are smooth layer potentials. This view is helpful for theirdiscretization and the design of fast methods for their evaluation.

3 Time dependent Integral operators

A generalized Abel integral operator has the form

Kg(t) =

t∫

0

k(t,τ)√t − τ

g(τ)dτ (6)

where the kernel k(·, ·) is a smooth function on the triangle ∆T and k(t, t) ≥ k0 > 0.We call this operator causal because the potential Kg(t) at time t depends only

on the function g(τ) for times τ ≤ t. Abel integral operators arise in the context ofsingular Volterra integral equations

λg(t)+Kg(t) = f (t), t ∈ [0,T ]. (7)

If λ 6= 0 the equation is of the second kind, otherwise of the first kind.We have seen that integral equations of the heat equation are also of this form,

where the kernel in (6) is a time-dependent boundary integral operator. Many ofthe ideas to treat the time dependence in parabolic integral equations can be betterexposed by considering operators that only depend on time, which is what we willdo in the following two sections.

The remainder of this section will briefly review the various options of discretiz-ing (6). It is not our intention to give a complete account of the vast literature in thisfield as it can be found, e.g., in the monographs [25, 5]. Instead, we will focus our

6 Johannes Tausch

attention on how these methods can be combined with the fast evaluation methodsintroduced in section 4.

While Abel integral operators are causal, discretization methods may or maynot preserve this property. In the discrete case, causality means that the operator isreplaced by a lower triangular matrix and solving (7) amounts to a simple forwardsubstitution.

In the case of equations in time the computational cost of nontrivial super diag-onals may not be significant. On the other hand, the complexity of the analogousscheme for a parabolic integral equation will be significantly increased, thereforewe will look for schemes that preserve causality in order to apply them to parabolicequations.

3.1 Projection Methods

In a projection method the solution of the integral equation is approximated by afunction gh selected from a finite dimensional function space Sh. Typically, Sh is apiecewise polynomial spline space with certain regularity conditions.

In the Galerkin method the approximation is the function gh whose residual isorthogonal to the space Sh. This is equivalent to

T∫

0

φh(t)(

λgh(t)+Kgh(t))

dt =

T∫

0

φh(t) f (t)dt,

for all φh ∈ Sh. A natural choice for Sh is a spline space subject to the partition

Ph = {0 = t0 < τ1 < τ2 < · · · < τN = T} .

A set of basis function is denoted by χ j, 0≤ j ≤ N. The discretized integral operatoris a matrix with coefficients

Kgi, j =

T∫

0

t∫

0

k(t,τ)√t − τ

χi(t)χ j(τ)dτdt.

It is easy to see that in the case of piecewise linear splines this matrix is a lower Hes-senberg matrix, i.e., lower triangular with an additional non-zero super-diagonal. Ifhigher order splines are considered, more non-zero diagonals will appear. Causalityis preserved only with piecewise constant elements.

A different way to seek an approximate gh ∈ Sh is the collocation method. Thisinvolves a set of nodes Xh of cardinality dimSh = N +1 which is selected such thatthe interpolation problem

gh(ti) = gi, 0 ≤ i ≤ N,


has a unique solution gh ∈ Sh for all gi ∈ R. The collocation discretization of (7) is

λgh(ti)+Kgh(ti) = f (ti), ti ∈ Xh.

If gh is represented by the basis functions χ j of Sh, then the coefficients of dis-cretized operator are

Kci, j =

ti∫

0

k(ti,τ)√ti − τ

χ j(τ)dτ

In the case of piecewise linears the node points are ti = τi and the matrix Kc is lowertriangular. However, higher order spaces will again generate non vanishing superdiagonals.

Since Abel integral equations may be regarded as weakly singular Fredholm in-tegral equations with the additional property that the kernel vanishes outside thetriangle ∆T , the standard convergence results apply. More on this can be found inthe introductory texts to integral equations, e.g., [2, 5, 19, 23].

3.2 Product Integration Methods

Historically, product integration methods have received the most interest and havebeen reviewed in, e.g., [25]. They can be regarded as a special case of a Nystrommethod. That is, the integral operator in (7) is replaced by a quadrature rule, and theequation is enforced on the quadrature nodes. Since the integrand is singular at theright endpoint, some care must be taken to select a rule that handles the singularitywith sufficient accuracy.

To derive such a quadrature rule the kernel is replaced by a piecewise polynomialinterpolate and the singularity is integrated exactly. For example, a piecewise linearinterpolant

k(tn,τ)g(τ) ≈

∑j

k(tn, t j)g(t j)χ j(τ), if τ ≤ tn,

0, otherwise,

leads to the approximation

Kg(tn) ≈n∑j=0

k(tn, t j)wn, jg(t j), (8)

where wn, j are quadrature weights

wn, j =

tn∫

0

1√tn − τ

χ j(τ)dτ ,

8 Johannes Tausch

which can be found in closed form by elementary means. Thus the discretization ofthe Abel integral operator is a lower triangular matrix K p with coefficients

K pn, j = k(tn, t j)wn, j.

The numerical analysis of product integration methods involve an estimate of thequadrature error and some stability argument. For integral equations of the secondkind stability can be derived from a Gronwall Lemma [19]. Stability for Abel in-tegral equations of the first kind is considerably more subtle. The case where thekernel is replaced by a piecewise linear has quadratic convergence and been dis-cussed by Eggermont [12]. Unfortunately, the order cannot be increased beyondthree, as the schemes become unstable [25, 26].

3.3 Convolution Quadrature

The limitations of product or desingularized integration methods for first-kind equa-tions can be overcome with the convolution quadrature, an idea introduced in thepaper [26]. There it is shown that the Abel integral operator can be discretized in theform

Kcqg(tn) = hαt

n∑

m=0ωn−mk(tn, tm)g(tm), (9)

where the quadrature weights ωn are derived from a linear multistep method withgenerating polynomials ρ(z) = a0 + za1 + · · ·+ zpap and σ(z) = b0 + zb1 + · · ·+zpbp, with ap,bp 6= 0. The weights are the Maclaurin series coefficients of

∞

∑k=0

ωkzk =

(

σ(1/z)ρ(1/z)

)12.

The stability and consistency of this method is directly related to the properties ofthe underlying multistep method. Unlike product integration methods, it is possibleto obtain stable schemes of arbitrary order.

3.4 Desingularized Quadrature

Another way to handle the singularity in the Abel operator is singularity subtraction.The regularized integral can then be treated with the trapezoidal rule. The derivationis quite simple

Kg(t) =

t∫

0

1√t − τ

(

k(t,τ)g(τ)− k(t, t)g(t))

dτ +2√

tk(t, t)g(t).


The new integrand vanishes at the right endpoint. In fact, if g(t) and k are smooththe intgrand is O(

√t − τ) and the trapezoidal rule converges at rate O(h3/2

t ). Thus

Khg(tn) = htn−1

∑m=0

′ 1√tn − tm

(

k(tn, tm)g(tm)− k(tn, tn)g(tn))

+2√

tnk(tn, tn)g(tn)

= hn−1

∑m=0

′ k(tn, tm)√tn − tm

g(tm)+ wnk(tn, tn)g(tn), (10)

where

wn = htn−1

∑m=0

′ 1√tn − tm

−2√

tn.

Here, the prime at the summation sign indicates that the m = 0 term must be multi-plied by the factor 1/2. The advantage of the desingularized quadrature method overthe product integration and convolution quadrature is that all except for the last termin (10) have the same quadrature weight. In (8) and (9) the weights depend on therow and column index. While this is not very significant for the direct solution of thediscretized equation with forward substitution, it will play an important role whendesigning fast summation methods described below. Higher order methods can beobtained easily by subtracting more terms of the Taylor expansion of the kernel.

The arguments used in the analysis of desingularized quadrature methods is simi-lar to the product quadrature. A form of the Gronwall lemma that is suitable to showstability for second-kind equations is given in [37]. As in the case of the productintegration method, the schemes become unstable for higher orders when they areapplied for first-kind equations . The maximal stable order is O(h5/2

t ), see [40].

4 The Fast Multipole Method in Time Domain

Discretizations of Abel integral operators lead to essentially lower triangular ma-trices with quadratic complexity in the number of unknowns. While for one-dimensional problems this may be still acceptable, the quadratic scaling will be pro-hibitive when parabolic problems in three space dimensions are considered. There-fore it is worthwhile to understand how to accelerate the evaluation of Abel integraloperators, as these techniques can be easily extended to parabolic problems.

When the kernel is a convolution, several of the discussed discretization methodspreserve this structure which suggests to use fast Fourier transforms for efficientnumerics. However, a straight forward application of Fourier methods enables thefast application of the operator to a vector, but does not facilitate solving a lowertriangular system by forward elimination. The latter can be accomplished by a hier-archic splitting of the matrix into smaller blocks, where local FFTs are employed.An O(Nt log2 Nt) scheme based on such ideas has been introduced in [20] to solve

10 Johannes Tausch

nonlinear Volterra equations. Here and in the following Nt denotes the number oftime steps in the interval [0,T ].

Unfortunately, many interesting problems are not convolutional. For instance,this happens in parabolic problems when surfaces are time dependent. Thereforewe focus on extending the fast multipole method for elliptic problems to parabolicproblems.

Although the multipole expansion is specific to the Laplace kernel, it has becomecustomary to refer to a fast algorithm that is based on a hierarchic decompositionof the problem domain and some form of kernel expansions as a fast multipolemethod, see, e.g., [42]. We will follow this slight misuse of terminology. A one-dimensional FMM that can be applied to very general operators was discussed in theafore mentioned paper. Instead of approximating the kernel in a truncated series, wewill consider an interpolation approximation of the kernel. This has the advantagethat only the kernel, and not its derivatives must be computed. In the context ofelliptic problems, this approach is used in [13, 29].

In the following we will summarize the basic ideas in the fast multipole algo-rithm for fast evaluations of discretized Abel integral operators. A straight forwardapplication of the FMM is not causal and hence not very useful in the context ofVolterra or Abel equations. Therefore we introduce a causal version of the FMM insection 4.5 that enables fast forward substitutions.

4.1 Separation of Variables

The first idea is the separation of variables in the kernel of the integral operator.Since the kernel is singular on the diagonal τ = t this will succeed only locally, in arectangular region (t,τ)∈ I×J where the intervals I and J are sufficiently separated.If this is the case, an approximation of the form

k(t,τ)√t − τ

≈ ∑i, j

ki, jLi(t)L j(τ) (11)

can be achieved with few terms in the summation on the right hand side. An exampleis the truncated Taylor expansion of the kernel. Here the Li’s are monomials and theki, j’s are partial derivatives of the kernel at the centroid of I×J. This may not be theoptimal choice as the Taylor polynomials may converge slowly and derivatives areoften difficult to compute. A more straight forward approximation is a two-variatepolynomial interpolation. To this end, let I and J be intervals of same length 2δ withcenters t and τ , respectively. We introduce local variables with the transform

t = t +δ t ′,τ = τ +δτ ′, −1 ≤ t ′,τ ′ ≤ 1.

and select interpolation nodes {ω p0 , . . . ,ω p

p} ⊂ [−1,1]. Then


k(t,τ)√t − τ

≈p

∑i, j=0

k(ti,τ j)√ti − τ jLi(t ′)L j(τ ′) (12)

whereti = t +δω p

i , τi = τ +δω pi ,

and Li is a Lagrange polynomial

Li(t ′) = ∏k 6=i

t ′−ω pk

ω pi −ω p

k. (13)

Because of their well-known suitability for interpolation, we let ω pk be the roots of

the (p+1)-st Chebyshev polynomial

ω pk = cos

(

π2

2k +1p+1

)

. (14)

The coefficients of the expansion (11) follow from the interpolation property

ki, j =k(ti, t j)√ti − t j

, 0 ≤ i, j ≤ p.

If the kernel is sufficiently smooth it is possible to show that the interpolation errorcan be bounded independently of δ provided that |t − τ | > rδ for some fixed r > 2.This will motivate the subdivision scheme in section 4.2.

4.2 Hierarchy of Intervals

The second idea of the Fast Multipole Method is to evaluate potentials using a hi-erarchical subdivision of the time interval [0,T ]. The coarsest level is the intervalitself, which is subdivided into two finer intervals of equal length. These intervalsare divided into two equal subinterval each and the process is repeated until thefinest intervals contain a small pre-determined number of time steps. The result isa tree of intervals. The coarsest level is level L, the finest level zero. The l-th levelconsists of the intervals

Ilk =

T2L−l [k,k +1), k ∈ {0, . . . ,Kl}.

where Kl = 2L−l − 1. Each interval Ilk above level zero has two children which are

given byK

lk = {Il−1

2k , Il−12k+1}.

The binary representation of an integer is

k = (σR . . .σ1σ0)2 = σR2R + · · ·+2σ1 +σ0

12 Johannes Tausch

where σi ∈ {0,1}, 0 ≤ i < R and σR = 1. The level-l parent of interval I0k is the

interval Ilkl

where the binary representation of kl is

kl = (σR . . .σl+1σl)2 = σR2R−l + · · ·+2σl+1 +σl

The neighbors of Ilk are the intervals in the same level that are near to the given

interval including the interval itself. Since the kernel vanishes for τ > t it suffices toconsider only intervals with a smaller or equal k-index. Here we define the intervalto be a neighbor if it shares at least one endpoint, thus the set of neighbors is

N (Ilk) =

{

{Il0}, if k = 0,

{Ilk−1, Il

k} , else.

Related to the neighbors is the concept of the interaction list of an interval. It consistsof intervals whose parents are neighbors but which are not neighbors themselves,i.e.,

I (Ilk) =

/0, if k ∈ {0,1},{Il

k−2}, if k ≥ 2, even,{Il

k−3, Ilk−2}, if k > 2, odd.

Because of the singularity at t = τ approximation (11) of The kernel in the Abelintegral operator is valid only if the source and evaluation intervals are sufficientlyseparated and holds for larger intervals if their separation is larger. The interactionlists provide a systematic way to split a time interval into well separated intervals ofincreasing size

k−2⋃

j=0I0

j =R−2⋃

l=0

⋃

J∈I (Ilkl

)

J.

The right hand side consists of the union of the interaction lists of all parents of I0k .

For the evaluation point tn ∈ I0k , the sum in the desingularized quadrature rule

(10) is written as

(Khg)(tn) = wnk(tn, tn)g(tn)+ht ∑t≤tm<tn

k(tn, tm)√tn − tm

g(tm)+φsm(tn) (15)

where t is the left endpoint of interval I lk−1 and φsm is the smooth potential given by

φsm(tn) = htR−2

∑l=0

∑J∈I (Il

kl)

∑tm∈J


g(tm). (16)

The first terms in (15) make up the local part. Since the finest level intervals containa fixed number of quadrature points the cost for the evaluation of the local part isindependent of the total number of time steps. If evaluated directly, the cost for thesmooth potential dominates if n is large. The fast multipole method accelerates the


computation of φsm by replacing the kernels in the smooth potential by expansion(11). The details are described in the following.

4.3 Translation Operators

4.3.1 Moment-to-Local Operator

We now consider the contribution of a certain interval J in the smooth potentialevaluated in interval I, which is a parent of I0

k in the same level as J. If I and Jare sufficiently separated then the kernel can be replaced by its interpolation (11),which results in a low-rank approximation of the operator. This follows from thecalculation

∑tm∈J


g(tm) ≈p

∑i=0

ki, j

(

p

∑j=0

∑tm∈J

′L j(tm)g(tm)

)

Li(tn), tn ∈ I.

Thus the potential due to interval J is approximated by a combination of the Li’s ininterval I

ht ∑tm∈J

k(tn, tm)√tn − tm

g(tm) ≈p

∑i=0

λi(I)Li(tn), tn ∈ I,

whereλi(I) =

p

∑j=0

ki, jµ j(J) (17)

andµ j(J) = ht ∑

tm∈J

′L j(tm)g(tm). (18)

The λi’s are the local expansion coefficients, which depend on the moments µ j de-fined in (18) and the moment-to-local (MtL) transformation (17). If the intervals Iand J contain sufficiently many points, then computing the potential via this processcan be much more efficient than computing the potential directly. The fundamen-tal principle behind the Fast Multipole Method is to compute the potentials of allinteractions in the smooth potential in this manner.

In (16) J is in the interaction list of I and thus both intervals are in the same leveland are separated by either one or two intervals. That is,

I = Ilk and J = Il

k−d, d ∈ {2,3}.

In matrix notation, we write (17) in the form

λλλ lk = MtL µµµk−d , d ∈ {2,3}, (19)

14 Johannes Tausch

where µµµ lk−r is a vector of moments in the interval I l

k−r and λλλ lk is the vector of

expansion coefficients for the interval I lk and MtL is a matrix with coefficients ki, j.

Of course, the matrix depends on the direction d, but our notations suppress thisdependence, because it is implied by the input and output vectors.

4.3.2 Source-to-Moment and Local-to-Potential Operators

In matrix notation, the source-to-moment operation (18) is written as

µµµ lk = QtMgggl

k,

where µµµ lk is the vector of moments of interval I l

k and ggglk is the vector of nodal values

of the function g in Ilk.

Once the expansion coefficients for every parent of I0k have been determined, the

smooth potential is evaluated by the series expansion (17). We call this the local-to-potential translation and write it as

ΦΦΦ lk = LtPλλλ l

k,

where ΦΦΦ lk is the vector of nodal values of the function φsm in Il

k. The matrices QtMand LtP have coefficients L j(tm) and are transposes of each other.

At first glance, one might consider computing all moments in all levels usingQtM transformations. Since every time step contributes to a moment in any level,the complexity of computing all moments is order pNL.

Likewise, after all expansion coefficients in all levels have been computed, thetotal potential at a point can be obtained by adding the LtP transformations of the in-terval and all its parents where the point is located. This type of potential evaluationis also order pNL.

A great deal of the computations can be saved if the moments and expansioncoefficients are computed in a systematic manner, exploiting the hierarchy of theintervals. Such a hierarchic scheme can reduce the complexity of the moment andpotential evaluations to order pN. This is based on the observation that the momentsin an interval only depend on the moments of the interval’s children. Likewise, theexpansion coefficients depend on the coefficients of the parent. This will be de-scribed below in more detail.

4.3.3 Moment-to-Moment Operator

The moments in the coarser levels are computed in an upward pass, where the mo-ments of a given interval are computed from the moments of the children. Likewise,the expansion coefficients a computed in a downward pass, where the coefficientsof a given interval are computed from the parent’s coefficients.


The translation formulas are derived from a linear relationship between the localcoordinates of the parent and its children. It is given by

τ ′ =12τ ′′± 1

2 .

Here the plus sign is for the right and the minus sign for the left child of I lk. Since

the local variables are affinely related, a polynomial in the parent’s coordinates re-mains a polynomial of the same degree in the child’s coordinates. Thus the Lagrangepolynomials satisfy the addition theorem

Li

(

12τ ′′± 1

2

)

=p

∑j=0

q±i, jL j(τ ′′), (20)

where the coefficients follow from the interpolation property

q±i, j = Li

(

12ω p

k ± 12

)

.

We begin with the translation of the moments. Consider a moment in interval I lk in

some level l > 0

µ i(Ilk) = ht ∑

tm∈Il−12k

Li(t ′m)g(tm)+ht ∑tm∈Il−1

2k+1

Li(t ′m)g(tm)

= ht ∑tm∈Il−1

2k

Li

(

12 t ′′m − 1

2

)

g(tm)+ht ∑tm∈Il−1

2k+1

Li

(

12 t ′′m +

12

)

g(tm)

where Il−12k and Il−1

2k+1 are the two children and t ′m and t ′′m are local coordinates of tm.From the addition theorem it follows that

µ i(Ilk) =

p

∑j=0

q−i, jµj(Il−1

2k )+p

∑j=0

q+i, jµ

j(Il−12k+1).

We also write the last relation in matrix-vector notation as

µµµ lk = MtMµµµ l−1

2k +MtMµµµ l−12k+1.

where MtM is the matrix with coefficients q±i, j. Our notations omit the dependenceon the direction because it is implied by the input and output vectors.

4.3.4 Local-to-Local Operator

We turn to the translation of the local expansion coefficients. Suppose the polyno-mial fp(t) is given in terms of the Lagrange polynomials of I l

k. That is,

16 Johannes Tausch

fp(t) =p

∑i=0

λ i(Ilk)Li(t ′),

where t ′ is the local coordinate of Ilk and λ i(Il

k) is the function value of fp at the i-thinterpolation node of Il

k. The same polynomial can also be expressed in terms of theLagrange polynomials of I l

k’s children. For instance, the addition theorem gives forthe left child

fp(t) =p

∑i=0

λ i(Ilk)Li

(

12 t ′′− 1

2

)

=p

∑i=0

λ i(Ilk)

p

∑j=0

q−i, jL j(t ′′)

=p

∑j=0

λ j(Il−12k )L j(t ′′)

The new expansion coefficients are given by the local-to-local (LtL) translation

λ j(Il−12k ) =

p

∑i=0

q−i, jλi(Il

k).

The translation to the right child is completely analogous. In matrix vector notationswe write

λλλ l−1k = LtLλλλ l

k, k ∈ {2k,2k +1},where the LtL operators are the transposes of the MtM operators.

4.4 The Standard FMM

The standard FMM is an efficient scheme to compute the smooth potential using thesplitting (16). It proceeds by computing all moments in all levels, then all expansioncoefficients and finally the potentials by evaluating the expansion coefficients. Thedetails are given in algorithm 1.

To compute the discrete integral operator in (15) the local potential must be addedin each time step. This version of the FMM can be regarded as the fast applicationof a vector g to a lower triangular matrix. In particular, the vector g must be knownin all time steps before the potential can be computed at any time step. Thus thisalgorithm is useful only if the linear system is solved by an iterative procedure. Ofcourse it is more natural to solve lower triangular systems by forward substitution,taking advantage of the causality of the Abel integral operator. This can be accom-plished by rearranging the order in which moments and expansion coefficients arecomputed. We call the resulting method the causal FMM and describe it in the fol-lowing section.


Algorithm 1 The Standard Fast Multipole Algorithm

% Moment Calculation.for k = 0 to K0 do

µµµ0k = QtM g0

kend for% Upward Pass.for l = 1 to L−2 do

for k = 0 to Kl doµµµ l

k = MtMµµµ l−12k +MtMµµµl−1

2k+1end for

end for% Interaction Phase.for l = 0 to L−2 do

for k = 2 to Kl doif k even then

λλλ lk = MtL µµµ l

k−2else

λλλ lk = MtL µµµ l

k−2 +MtL µµµ lk−3

end ifend for

end for% Downward Pass.for l = L−2 down to 1 do

for k = 0 to Kl doλλλ l−1

2k = λλλ l−12k +LtLλλλ l

kλλλ l−1

2k+1 = λλλ l−12k+1 +LtLλλλ l

kend for

end for% Potential evaluation.for k = 0 to K0 do

Φ0k = LtPλλλ 0

kend for

4.5 The Causal FMM

To enable the fast computation of gn from the gm, m < n, the order in which mo-ments, expansion coefficients and potentials are computed must be changed. In par-ticular, the outermost loop is over all time steps, and moments and expansion coef-ficients are computed as soon as they become available as time progresses.

To describe how this process works we introduce further notations

R: Index of the highest digit in the binary representation of k.S: Index of the lowest non-zero digit in the binary representation of k.

For example, the binary representation of the number 20 is (10100)2 which impliesthat R = 4 and S = 2.

18 Johannes Tausch

t

µ λµ µ

µ

µ λ

λ

µ

µ

λµµ

µ

0 1 2 20

Fig. 1 Operations for the 20th interval. Quantities on white ground are computed in this time step.Quantities on dark ground have been computed in earlier time steps.

Suppose now that the forward substitution process has moved to interval I0k .

Hence g0k , where k = k − 1, has been computed and the moment µµµ0

k can be de-termined by a QtM transform. Furthermore, there may be moments in the parentsof I0

k that also become available at this time step. These are exactly the levels wherethe binary representation changes bits as k is increased to k. This happens exactly inlevels 1 to S. Consider, for instance, the case k = 20, which is illustrated in Figure 1.The moment µµµ0

19 is new and can be computed by a QtM. Since k = 19 = (10011)2and k = 20 = (10100)2 the bits in levels 0, 1 and 2 have change. Therefore the mo-ments in the coarser levels µµµ1

9 and µµµ24 are are also new and can be computed by

MtMs. Note that the moments that are necessary to do that have been computed inearlier time steps. The parent moments in levels above S will be computed in a latertime step.

The expansion coefficients are always computed for all parents of the currentinterval. Thus the coefficients above level S are known as the time stepping movesinto interval I0

k , and we only need to compute the coefficients λλλ 0k to λλλ S

kS by MtLtransformations. Since MtLs always go back in time, the moments that are necessaryto do that are already known. Finally, the contributions of the parents to the childrenis added in levels S down to 0. In the previous example, we compute λλλ 2

5,λλλ110 and

λλλ 020. The coefficient λλλ 3

2 was already computed when k was 16. The coefficients inthe higher levels are all zero because the interaction lists are empty.

Since only the order is changed in which quantities are evaluated, the causalFMM has the same complexity as the standard FMM. Unlike the standard FMM, itis not necessary to store all moments and expansion coefficients in all levels. In fact,it is easy to see that in algorithm 2 we only need to keep three moment vectors and


Algorithm 2 The Causal Fast Multipole Algorithm

for k = 0 to K0 doCompute S and R in the binary representation of k.Set k = k−1.

% Moment Calculation.µµµ0

k = QtM g0k

% Upward Pass.for l = 1 to S do

µµµ lkl

= MtMµµµl−12kl

+MtMµµµl−12kl+1

end for% Interaction Phase.λλλ S

kS = MtL µµµSkS−2 +MtL µµµS

kS−3for l = S−1 down to 0 do

λλλ lkl = MtL µµµ l

kl−2end for% Downward Pass.for l = S down to 0 do

λλλ lkl = λλλ l

kl +LtLλλλ l+1kl+1

end for% Potential evaluation.Φ0

k = LtPλλλ 0k

% Compute g0k using the smooth potential Φ0

k and the local part.end for

one coefficient vector per level. It is possible to rearrange computations such thatonly two moments per level must be stored. This was described in [38].

5 The Parabolic FMM

5.1 Discretization of thermal layer potentials

For the time discretization we take the view that thermal layer potentials are general-ized Abel integral operators with operator-valued kernels. Hence the desingularizedquadrature rule (10) applied to the thermal single layer operator leads to

Vht g(tn) := ht√4π

n−1

∑m=0

′ 1√tn − tm

V (tn − tm)g(tm) + wng(tn). (21)

The surface integral operators in the above quadrature rule have smooth kernels,thus the Nystrom method can also be used for the spatial discretization. Quadra-ture rules for smooth surface integrals are usually constructed in a triangulation of

20 Johannes Tausch

the parameter space and lifted on the surface. Details of the derivation with errorestimates can be found in Atkinson [2] and Chien [6]. We write the rule as

∫

S

f (x)ds(x) ≈NS

∑l=1

f (xl)wl . (22)

where xl ∈ S are the nodes and wl are the quadrature weights.All kernels of the surface integral operators in (21) are smooth, but for values of

the summation index m near n they become very peaked. To resolve this behaviorthe mesh width of the surface triangulation must be decreased as the temporal timestep is decreased. The relationship between ht and hs should be

hs√ht

→ 0 as ht → 0, (23)

to preserve the asymptotic convergence rate of the time discretization scheme,see [39].

Combining (21) and (22) leads to the quadrature rules

Vhs,ht g(xk, tn) := htn−1

∑m=0

′Ns

∑l=1

G(xk − xl, tn − tm)wlg(xl , tm) + wng(xk, tn),

Khs,ht g(xk, tn) := htn−1

∑m=0

′Ns

∑l=1

∂∂nl

G(xk − xl, tn − tm)wlg(xl , tm) + wnH(xk)g(xk, tn).

The Nystrom discretization of the Green’s formula replaces the thermal layer po-tentials by their discrete counterparts. This results in an explicit scheme for the un-known boundary data in every time step.

For the Neumann problem on a smooth surface Green’s formula is an integralequation of the second kind with compact operator. In this setting it is possible toprove stability and convergence at same rate as the temporal discretization providedthat assumption (23) holds [39]. Of course, it is hard to extend these results to first-kind equations and non smooth geometries.

Because of its simplicity, the Nystrom discretization is an effective method tosolve the heat equation. However, the refinement of the mesh that can be used isseverely limited by the O(N2

t N2s ) complexity of the direct evaluation of Vhs,ht and

Khs,ht . Fortunately, the causal FMM can be easily modified to rapidly evaluate thediscrete thermal potentials. We call this extension parabolic FMM and describe thedetails in the following section.

5.2 Approximation Theory for the Heat Kernel

We have seen that integral operators can be evaluated efficiently if the kernel is ap-proximated by a truncated series expansion to achieve separation of variables. By


definition, Abel integral operators are singular for t = τ . Therefore the approxima-tion can be accomplished only locally, for rectangular regions in the (τ , t) plane thatare sufficiently separated from the diagonal.

In addition to the two temporal variables, the heat kernel depends on six spatialvariables. When t > τ , the heat kernel is smooth, which somewhat simplifies theapproximation scheme in space. However, when τ is near t the heat kernel is peakednear x = y and decays rapidly as x and y separate. In this range space must be sub-divided into small clusters, but only nearby cluster interactions must be considered.When τ and t are farther separated, the heat kernel is more smooth, but decays moreslowly. In this range space is subdivided into larger clusters. Since the clusters arebigger, it suffices again to consider only neighboring cluster interactions.

Before we discuss the subdivision scheme of space and time in more detail, weconsider different possibilities to approximate the heat kernel. To that end, let ν bea cube in R

3 with center x side length 2hx and ν be a cube with same side lengthand center x, furthermore let I and I be non overlapping intervals of R with length2ht and centers t and t, respectively. For x ∈ ν , y ∈ ν , t ∈ I and τ ∈ I we introducelocal variables with the transformation

x = x+ x′hx, y = x+ y′hx, −1 ≤ x′σ ,y′σ ≤ 1, σ ∈ {1,2,3}t = t + t ′ht , τ = t + τ ′ht , −1 ≤ t ′,τ ′ ≤ 1.

In the local variables, the heat kernel is

Gloc(x′,y′, t ′,τ ′) =1

[4π(t − τ)]32

exp(

− |x− y|24(t − τ)

)

=1

[(4πht)(d′ + t ′− τ ′)]32

exp(

−ρ|r′ + x′− y′|2

d + t ′− τ ′

)

. (24)

The constants d′ > 0, r′ ∈ R3 and ρ > 0 are given by

d′ = (t − τ)/ht , (25)r′ = (x− y)/hx, (26)

ρ =h2

x4ht

. (27)

The fast evaluation of potentials is based on separation of variables of the heatkernel in (24). Unfortunately, the multivariate interpolation approximation results ina very large number of terms: For order p polynomials in time and order q polyno-mials in space the multivariate version of (11) has (p+1)2(q+1)6 terms, which isprohibitively large even for moderate values of p and q. We will show below that amore efficient approximation can be obtained if the spatial variables are expandedin Chebyshev instead of Lagrange polynomials.

For the temporal variables it is still more efficient to use the Lagrange form.Although this leads to more terms, the benefit is that the tensor product form of the

22 Johannes Tausch

heat kernel is preserved. Thus we begin by approximating (24) in time as follows

Gloc(x′,y′, t,τ ′)

≈ 1(4πht)

32

p

∑i, j=0

1(d′ +ω p

i −ω pj )

32

exp(

−ρ|r′ + x′− y′|2d′ +ω p

i −ω pj

)

Li(t ′)L j(τ ′). (28)

Since the heat kernel is analytic in the local variables, the convergence rate of theapproximation is exponential in p. Each term in the sum is a Gaussian. Before wediscuss how the multivariate Gaussian can be approximated we will first focus onthe one dimensional case in the following section.

5.3 Chebyshev Expansion of the Gauss Kernel

It is well known that Chebyshev polynomials have distinct advantages for the ap-proximation of functions. In the interval [−1,1] they are defined by

Tn(x) = cos(narccos(x)).

The functions Tn are polynomials of degree n which are L2w[−1,1]-orthogonal with

weight function w(x) = (1− x2)−12 . That is,

1∫

−1

Tn(x)Tm(x)w(x)dx =πγn

δn,m,

where γ0 = 1 and γn = 2 for n ≥ 1. The roots of Tp+1 are given by (14). By orthog-onality, the expansion of the one-dimensional Gauss Kernel is

exp(

− (r + x− y)2

δ

)

=∞

∑k,l=0

Ek,l(r,δ )Tk(x)Tl(y), x,y ∈ [−1,1], (29)

where

Ek,l(r,δ ) =γkγlπ2

1∫

−1

1∫

−1

exp(

− (r + x− y)2

δ

)

Tk(x)Tl(y)w(x)w(y)dxdy. (30)

Separation of variables is accomplished by truncating the expansion (29) Since themagnitude of the Chebyshev polynomials bounded by unity, the approximation er-ror depends on how rapidly the coefficients En(r,δ ) approach zero. The followingbound can be derived.

Lemma 1. For every a > 0 and n = k + l the bound


∣

∣Ek,l(r,δ )∣

∣≤ γkγlan exp

(

4δ

(

a− 1a

)2)

holds.

Note that a > 0 is a free parameter, which means that the geometric progression canbe made arbitrarily fast by increasing the value of a. The trade off is that a larger aimplies that the constant factor is also larger. For given k and l the optimal a can bedetermined by simple calculus. This leads to an estimate of the form

∣

∣Ek,l(r,δ )∣

∣≤ γkγl exp(

−κ(δn)n)

where κ(t) ∼ ln(4t)/4. The super exponential behavior has to do with the fact thatthe Gaussian is not only analytic but also entire. This becomes evident from thefollowing proof, which follows a similar argument as in [41].

Proof. After changing variables x = cosθ , y = cosφ and using the symmetry prop-erties integral (30) becomes

Ek,l(r,δ ) =γkγlπ2

π∫

0

π∫

0

exp(

− (r + cosθ − cosφ)2

δ

)

cos(kθ )cos(lφ)dθdφ

=γkγl

(2π)2

2π∫

0

2π∫

0

exp(

− (r + cosθ − cosφ)2

δ

)

exp(−ikθ )exp(−ilφ)dθdφ .

The integrand is analytic and periodic in θ and φ . By Cauchy’s theorem, we canshift both intervals into the complex plane without changing the integral. For anya > 0 we obtain

Ek,l(r,δ )

=γkγl

(2π)21an

2π∫

0

2π∫

0

exp(

− (r + cos(θ + ia)− cos(φ + ia))2

δ

)

exp(−i(kθ + lφ))dθdφ ,

where a = exp(a). The magnitude of the integral is determined by the real part ofthe argument to the exponential function. We find that

Re[

(r + cos(θ + ia)− cos(φ + ia))2]

=[

r +(cos(θ )− cos(φ))cosh a]2

−[

(sin(θ )− sin(φ))sinh a]2

≥ −4sinh2 a = −4(

a− 1a

)2.

From this the assertion follows easily.

24 Johannes Tausch

The estimate of the coefficient Ek,l(r,δ ) depends only on the sum of the coefficients.This suggests to retain only terms in expansion (29) that are indexed by the triangu-lar region

S2q = {(k, l) : 0 ≤ k + l ≤ q},

where q is the expansion order for the space variable. The truncation error

Rq(x,y) = exp(

− (r + x− y)2

δ

)

− ∑(k,l)∈S2q

En(r,δ )Tk(x)Tl(y)

can be estimated by

∣

∣Rq(x,y)∣

∣≤∞

∑n=q+1

∑k+l=n

∣

∣Ek,l(r,δ )∣

∣≤ (2+q)

(

1a−1

)q+1exp(

4δ

(

a− 1a

)2)

.

Since a is arbitrary the convergence is super exponential. The approximation hasabout half as many terms as the corresponding interpolation approximation (12).

The effect is more pronounced in the three dimensional case. From the propertiesof the exponential function we have that

exp(

− 1δ∣

∣r′ + x′− y′∣

∣

2)

= ∑α,β

Eα,β (δ ,r)Tα (x′)Tβ (y′). (31)

Here α and β are multi-indices, α = (α1,α2,α3), |α | = α1 +α2 +α3 and Tα(x′) =Tα1(x′1)Tα2(x′2)Tα3(x′3). Because of the basic properties of the exponential functionthe coefficients of the three dimensional Gauss kernel are products of the one-dimensional coefficients

Eα,β (δ ,r) = Eα1,β1(δ ,r1)Eα2,β2(δ ,r2)Eα3,β3(δ ,r3).

Thus it suffices to retain the terms in (31) whose indices add up to the given expan-sion order q. These indices are in the set

S6q = {(α ,β ) : 0 ≤ |α +β | ≤ q}.

with cardinality

#S6q =

(

q+6q

)

∼ q6

6! as q → ∞.

Even for small expansion orders this number is much less than the (q+1)6 terms inthe interpolation approximation.

The truncation error can be bounded by

∣

∣Rp(x,y)∣

∣≤ exp(

12δ

(

a− 1a

)2)

∞

∑n=q+1

(

n+5n

)

1an ≤ q6C(a,δ )

(

1a−1

)q+1


where C(a,δ ) is independent of q. This shows that the error is super exponential.Replacing the Gaussians in (28) by the truncation of (31) leads to

Gloc(x′,y′, t,τ ′) ≈1

(4πht)32

p

∑i, j=0

∑|α+β |≤q

Eiα, jβ Tα(x′)Tβ (y′)Li(t ′)L j(τ ′), (32)

where

Eiα, jβ =1

(d′ +ω pi −ω p

j )32

Eα1,β1

(

ρd′ +ω p

i −ω pj,r1

)

· · · · ·Eα3,β3

(

ρd′ +ω p

i −ω pj,r3

)

.

The integrals in (30) are not available in closed form and must be computedby quadrature. The most natural way to do this is to apply the Gauss-Chebyshevquadrature rule

Ek,l(δ ,r′) ≈(

2q+1

)2

∑0≤n≤q0≤m≤q

Tk(ωqn )Tl(ωq

m)exp(

− (r′ +ωqn −ωq

m)2

δ

)

.

5.4 Space-Time Subdivision

The parabolic FMM relies on a hierarchic subdivision of time and space. The tem-poral subdivision into intervals was already described in section 4.2, we now turnour attention to the spatial subdivision into cubes. Its construction is familiar fromthe elliptic FMM: The coarsest cube of the spatial tree contains the entire boundarysurface. This cube is refined into eight cubes of half the side length. The process isrepeated until the finest cubes contain at most a predetermined number of surfacequadrature points. Since the boundary is a surface of dimension two, most cubes inthe finer levels will be empty and do not have to be considered in a fast algorithm.

With a parabolic operator every temporal interaction involves calculating a sur-face potential with a smooth kernel. In the parabolic FMM these interactions areagglomerated in the cubes of the spatial tree. The level where the agglomerationtakes place is selected such that the truncation error of an MtL translation is inde-pendent of the level of the temporal tree.

Lemma 1 and the form of the local kernel in (28) show how this can be done.It follows that the spatial truncation error depends on the variance δ of the Gausskernel, which are given by

δ =d′ +ω p

i −ω pj

ρ. (33)

A smaller value of δ implies a larger truncation error. Since the Gauss kernel issmooth, the approximation (28) is valid even when the source and destination cubesare close or identical.

26 Johannes Tausch

The interaction list of a temporal interval I lk in any level consists of the interval

Ilk−2 and, if k is odd, also of Il

k−3. Thus the value of d′ is either four or six and hencethe numerator in (33) is bounded below by two. To ensure uniform error bounds,the factor ρ in (27) must be bounded independently of the level of the temporal tree.This suggests to compute MtL translations in the spatial level

ls = min(

trunc(l/2),Ls)

, (34)

where l is the temporal level. To see this, let h(l)t be the half-length of a temporal

interval in the l-th level and h(ls)s be the half-length of a cube in the ls-th spatial level.

Then

ρ =(h(ls)

s )2

4h(l)t

=22ls(h(0)

s )2

4 ·2lh(0)t

≤ (h(0)s )2

4h(0)t

.

Hence ρ is indeed bounded.The variance also determines the decay rate of the Gauss kernel in space. Inter-

actions of cubes can be neglected when they are sufficiently well separated in spaceand only a certain number N of neighboring cubes in a linear direction must be in-cluded in the spatial interaction lists. Condition (34) implies that N is always thesame in every level.

Since the spatial level is coarsened only in every second temporal level therewill be translation operators that will either go in space and time or only in time.To simplify their description we introduce the following notations. The nonemptycubes that must be considered in a temporal level are

C (l) ={

nonempty cubes in level ls}

.

For a cube ν ∈ C (l) the neighbors are defined as

N (ν) ={

ν ∈ C (l) : ‖ν − ν‖∞ ≤ N}

.

MtM and LtL translations always go from children to parents of time intervals butmay stay within a spatial cube. Thus we extend the concept of the child of a cubeν ∈ C (l) as

K (ν) =

{{

nonempty spatial children of ν}

if ls 6= (l −1)s,{

ν}

if ls = (l −1)s.

Likewise, the definition of a parent of ν ∈ C (l −1) is extended as

π(ν) =

{{

spatial parent of ν}

if ls 6= (l −1)s,{

ν}

if ls = (l −1)s.


5.5 Space-Time Translation operators

We proceed by describing the translation operators of the parabolic FMM. Thisis analogous to the temporal FMM except for the fact that the spatial dependencesignificantly adds to the complexity.

5.5.1 Moment-to-Local Translations

Similar to what we had done in section 4.3.1 we consider the influence of one timeinterval on another in the interaction list. We set

Φ(x, t) = ∑tm∈J,xP∈S

′G(x− xP, t − tm)g(xP, tm)wPht , x ∈ S, t ∈ I. (35)

For J ∈ I (I) the heat kernel is smooth, but may be very peaked if the intervals areclose to each other, which happens if they are in a low level of the temporal tree.Thus we cannot expect that the heat kernel can be approximated with low degreepolynomials. Towards that goal we break up the surface into pieces that intersectwith the cubes of the spatial tree and choose the level according to formula (34),because this will lead to uniform error bounds in any level. We write

S =⋃

ν∈C (ls)Sν .

where Sν is the piece of the surface that intersects with cube ν . Since the heat kerneldecays exponentially only the neighboring cubes contribute in a significant way. Forx ∈ Sν the potential in (35) is approximated by

Φ(x, t) ≈ ∑ν∈N (ν)

∑tm∈J,xP∈Sν

′G(x− xP, t − tm)g(xP, tm)wPht , t ∈ I,x ∈ Sν . (36)

We proceed by replacing the heat kernel in each term of (36) by expansion (32).Because of the scaling we have r′ = 2(ν − ν) and obtain for the potential generatedby Sν

Φν (x, t) ≈p

∑i=0

∑|α|≤q

λ iα(I×Sν)Li(t ′)Tα(x′), t ∈ I,x ∈ Sν ,

whereλ iα(I ×Sν) =

p

∑j=0

∑|α+β |≤q

Eiα, jβ µ jβ (J×Sν) (37)

and µ jβ (J×Sν) are the moments, given by

µ jβ (J ×Sν) = ∑tm∈J,xP∈Sν

′L j(t ′m)Tβ (x′P)g(xP, tm)wPht .

28 Johannes Tausch

The MtL translation of (37) does not have tensor product structure. To understandthat, re-write on term in (37) more explicitly as

λ iαj =

q−|α|∑

β3=0E(3)

α3,β3

q−|α|−β3

∑β2=0

E(2)α2,β2

q−|α|−β2−β3

∑β1=0

E(1)α1,β1

µ iβ .

The inner summations depend on the order of index α and thus the tensor form islost. However, it can be easily restored by including more terms of the expansion

λ iαj =

q−α1−α3

∑β3=0

E(3)α3,β3

q−α2−β3

∑β2=0

E(2)α2,β2

q−β2−β3

∑β1=0

E(1)α1,β1

µ iβ .

This sum can be evaluated in three stages, working from the inside to the outside

λ (1)α1,β2,β3

=q−β2−β3

∑β1=0

E(1)α1,β1

µi,β1,β2,β3 ,

λ (2)α1,α2,β3

=q−α1−β3

∑β2=0

Eα2,β2λ (1)α1,β2,β3

,

λ (3)α1,α2,α3 =

q−α1−α2

∑β3=0

Eα3,β3λ (2)α1,α2,β3

.

At the end of this computation λ (3)α = λ iα

j holds. Inspection of the upper bound inthe summation shows that in every step we only need to compute coefficients thatare indexed by the set S3

q. This algorithm entails 3 ·#S3q multiplications and additions

which is much less than the cost to evaluate (37).

5.5.2 Moment-to-Moment Translations

From (34) it follows that the spatial level is coarsened only in every other tempo-ral level. Thus there are two types of MtM translations, depending on whether thespatial level of the parent interval is the same or different from the child’s level. Toderive the actual form of the MtM matrix recall that the temporal translation followsfrom the addition theorem for Lagrange polynomials (20). For space translations weneed the analogous formula for Chebyshev polynomials. It is given by

Tk

(

12x′± 1

2

)

=k∑l=0

ak,lTl(x′), (38)

whereak,l =

2p+1

p

∑k=0

Tl(ωpk )Tk

(

12ω p

k ± 12

)

,


and p is any integer greater than k. This formula follows from the discrete orthogo-nality of Chebyshev polynomials, see, e.g. [38].

Equipped with the addition theorems, the MtM translation operator can be de-rived in a similar manner as in section 4.3.3. If the translation involves space andtime then

µ jβ (I ×Sν) = ∑I∈K (I)ν∈K (ν)

∑0≤i≤p|α|≤β

q j,iaβ ,α µ jα(I ×Sν).

Here, the coefficients depend on direction of the translation from the child to theparent. This computation can be tensorized in a similar manner as the MtL transla-tions.

If there is only a temporal translation we have

µ jβ (I×Sν) = ∑I∈K (I)

∑0≤i≤p|β |≤α

q j,iµ iβ (I×Sν).

Using matrix notations, and the extended definition of a child, both cases can bewritten as

µµµ lkl ,ν = ∑

ν∈K (ν)

MtM µµµ l−12kl ,ν +MtM µµµ l−1

2kl+1,ν .

where µµµ lk,ν is the moment vector of cube ν and time interval I l

k.

5.5.3 Local-to-Local Translations

As in the temporal FMM, the LtL translations in the parabolic case are the trans-poses of the MtM translations. Their derivation is completely analogous and is omit-ted here to avoid repetition. We only state the resulting translation formulas. In thecase of a spatio-temporal translation we have

λ iα(I ×Sν) = ∑0≤ j≤p|β |≤α

q j,iaβ ,αλ jβ (I ×Sν),

where I ∈ K (I) and ν is a child of ν .In the case of a time-only translation the formula simplifies to

λ iα(I×Sν) = ∑0≤ j≤p

q j,iλ jα(I ×Sν).

In matrix notation, both cases can be written as

λλλ l−1k,π(ν)

= LtLλλλ lk,ν , k ∈ {2k,2k +1}.

30 Johannes Tausch

5.6 The Parabolic FMM

The general structure of the parabolic FMM is the same as the causal FMM, withthe exception that translations are in time and space. The details are given in algo-rithm 3.

Algorithm 3 The Parabolic Fast Multipole Algorithm

for k = 0 to K0 doCompute S and R in the binary representation of k.Set k = k−1.

% Moment Calculation.for ν ∈ C (0) do

µµµ0k,ν = QtM g0

k,νend for% Upward Pass.for l = 1 to S do

for ν ∈ C (l) doµµµ l

kl= ∑

ν∈K (ν)

MtMµµµl−12kl ,ν

+MtMµµµl−12kl+1,ν

end forend for% Interaction Phase.for ν ∈ C (S) do

λλλ SkS ,ν = ∑

ν∈N (ν)

MtL µµµSkS−2,ν +MtL µµµS

kS−3,ν

end forfor l = S−1 down to 0 do

for ν ∈ C (l) doλλλ l

kl ,ν = ∑ν∈N (ν)

MtL µµµ lkl−2,ν

end forend for% Downward Pass.for l = S down to 0 do

for ν ∈ C (l) doλλλ l

kl ,ν = λλλ lkl ,ν +LtLλλλ l+1

kl+1 ,π(ν)

end forend for% Potential evaluation.for ν ∈ C (0) do

Φ0k,ν = LtPλλλ 0

k,νend for% Compute g0

k,ν ’s using the smooth potential Φ0k,ν and the local part.

end for


6 A Numerical Example

To illustrate the convergence properties of the method we solve the Dirichlet andthe Neumann problem of the heat equation in the exterior of the unit sphere for thetime interval [0,1]. The boundary conditions are chosen such that the solution isu(x, t) = G(x−x0, t) where x0 = (0.5,0,0). Hence the initial condition vanishes andthe numerical results can be compared with the given analytical solution.

Green’s formula (1) is an integral equation of the first kind for the Dirich-let and of the second kind for the Neumann problem. The sphere is triangulatedand the weights in the spatial quadrature rule are selected to obtain degree ofprecision two. The resulting rule has nodes on the midpoints of the edges, seeequation (5.1.44) in [2]. The coarsest refinement has 288 quadrature nodes, andis four times uniformly refined. From relation (23) it follows that hs ∼ h1/2

t isnot sufficient to maintain the asymptotic convergence of the time discretizationmethod. We therefore adjust the space and time mesh according to hs ∼ hλ

t , withλ = ln(10)/ ln(4) ≈ 0.6021. Table 1 shows the discretization parameters and theexpansion orders for the parabolic FMM.

Mesh 1 2 3 4Nt 25 80 250 800Ns 288 1152 4608 18432

NsNt 7,200 92,160 1,252,000 14,745,600p/q 3/12 4/22 5/16 6/24

Table 1 Mesh parameters and expansion orders for the different refinements.

The errors as a function of time are shown in figure 2 and the overall errorsin figure 4. It is apparent that the results obtained with both integral formulationsreproduce the theoretical O(h3/2

t ) estimate well. The integral equation of the firstkind shows some minor oscillations in the finest meshes which are caused by thetruncation error of the fast method.

The CPU time per time step and the total times are shown in figures 3 and 4, re-spectively. Because of the fast method, the CPU time in a time step mainly dependson how many translation operators must be evaluated, but not so much on how farthe computation has progressed in time.

The overall computational complexity is strongly dependent on the choice ofthe expansion orders of the fast method. The values of p and q in Table 1 havebeen determined experimentation such that the error of the fast method is negli-gible compared with the discretization error. The total complexity of the schemeis O(p2q4Nt Ns), see [38]. Since the expansion orders also depend on the discretiza-tion, the timings are, except for logarithmic terms, linear in the number of quadraturepoints. Our numerical experiments are in good agreement with this observation.

32 Johannes Tausch

0 0.2 0.4 0.6 0.8 1

10−4

10−2

100

time

L 2(S) n

orm

0 0.2 0.4 0.6 0.8 1

10−4

10−2

100

time

L 2(S) n

orm

0 0.2 0.4 0.6 0.8 1

10−4

10−2

100

time

L 2(S) n

orm

0 0.2 0.4 0.6 0.8 1

10−4

10−2

100

time

L 2(S) n

orm

0 0.2 0.4 0.6 0.8 1

10−4

10−2

100

time

L 2(S) n

orm

0 0.2 0.4 0.6 0.8 1

10−4

10−2

100

time

L 2(S) n

orm

Fig. 2 Computational results as a function of time for the integral equation of the first kind (left)and second kind (right). The top curve is the L2(S) norm of the solution, the curves below are theL2(S)-errors for meshes 1 to 4 of table 1.

0 0.2 0.4 0.6 0.8 110−2

10−1

100

101

102

103

time

sec

M 1M 2M 3M 4

Fig. 3 CPU time per time step for the four meshes of table 1.

Acknowledgements The work of Johannes Tausch is in part funded by the National Science Foun-dation under grant number 0915222.


10−3 10−2 10−1

10−4

10−3

10−2

ht

L 2(S×[

0,1]

) nor

m

1st2ndO(ht

3/2)

10−3 10−2 10−1

10−4

10−3

10−2

ht

L 2(S×[

0,1]

) nor

m

1st2ndO(ht

3/2)

10−3 10−2 10−1

10−4

10−3

10−2

ht

L 2(S×[

0,1]

) nor

m

1st2ndO(ht

3/2)

104 106

100

102

104

106

Nt*Ns

sec

cpu timelinarquadratic

Fig. 4 Left: L2(S× [0,1]) errors versus temporal step length. Right: Total CPU time versus numberof quadrature nodes.

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