Stability of radial basis function methods for convection problems

on the circle and sphere∗

Jordan M. Martel† and Rodrigo B. Platte‡

Abstract

This paper investigates the stability of the Radial Basis Function (RBF) collocation methodfor convective problems on the circle and sphere. We prove that the RBF method is Lax-stablefor problems on the circle when the collocation points are equispaced and the transport speed isconstant. We also show that the eigenvalues of discretization matrices are purely imaginary in the caseof variable coefficients and equispaced nodes. By studying the ε-pseudospectra of these matrices weargue that approximations are also Lax-stable in the latter case. Based on these results, we conjecturethat the discretization of transport operators on the sphere present a similar behavior. We providestrong evidence that the method is Lax-stable on the sphere when the collocation points come fromcertain polyhedra. In both geometries, we demonstrate that eigenvalues of the differentiation matrixdeviate from the imaginary axis linearly with perturbations off the set of ideal collocation points.When the ideal set is impractical or unavailable, we propose a least-squares method and presentnumerical evidence suggesting that it can substantially improve stability without any increase tocomputational cost and with only a minor cost to accuracy.

Keywords. Eigenvalue stability; RBF; kernel methods; spectral methods; periodic functions; pseudospectra

AMS subject classification. 65M70 - 65M12 - 65T40

1 Introduction

Radial Basis Function (RBF) methods have generated a great deal of attention in the last decade be-cause of their ease of implementation, spectral convergence (for smooth data and basis functions) andgeometric flexibility. They have been successfully used to solve partial differential equations, includingtime dependent problems in a variety of applications. Of particular interest are transport problems onthe sphere, where RBFs have shown to be very effective [6, 7, 5]. Despite their success, stability analysisremains underdeveloped when compared to more traditional schemes such as finite difference, finite ele-ment, and spectral methods. Although some work has been done in this direction (notably [14]), nonehas attempted showing Lax-stability. Here, we show Lax-stability of RBF methods for convection on thecircle and make some cursory observations about Lax-stability of the method for convection problems onthe sphere. We present numerical evidence suggesting that instabilities arise when collocation points arechosen to be non-equispaced.

In this paper, we write the RBF approximation as

h(x) =

N∑j=1

cjΦ(||x− xj ||) (1)

where x1, . . . , xN ∈ Rn are known as centers, || · || denotes the `2 norm and Φ : [0,∞)→ R is known as theradial function or kernel. While many of the results presented here are true of any positive definite kernel,

∗This work is supported in part by NSF-DMS-CSUMS 0703587 and AFOSR FA9550-12-1-0393.†University of Colorado, Leeds School of Business, Boulder CO 80309 (jordan.martel@colorado.edu)‡Arizona State University, Department of Mathematics and Statistics, Tempe, AZ 85287 (rbp@asu.edu).

1

Figure 1: A deformational velocity field on the unit sphere.

we choose the so-called inverse multiquadric (IMQ) radial function for our computational experiments.The IMQ is given by

Φ(r) =1√

1 + (εr)2. (2)

Collocating the IMQ basis functions {Φ(||x−xj ||)}Nj=1 over their centers results in an interpolation matrixthat is guaranteed to be positive-definite, even when the centers are scattered in Rn. Other popularradial functions include multiquadics (

√1 + (εr)2) and Gaussians (e−(εr)2). We point out, however, that

multiquadrics are not positive definite. The parameter ε is known as the shape parameter and governsthe steepness (or flatness) of the radial function and we assume it is constant for all basis functions. Fora thorough treatment of RBF approximation, see [2, 4, 17].

It was shown in [14] that the spectrum of the RBF discretization of the transport operator (u · ∇)on the sphere is purely imaginary when the velocity field u is non-deformational [9]. This result holdsindependently of node location, although these matrices can be far from normal if discretization pointsare poorly placed. For deformational flows such as the one presented in Figure 1, on the other hand, thisis no longer the case. Moreover, as shown in [6] the eigenvalues of such operators might have positivereal part, a clear indication of instabilities.

As we describe in this paper, a similar shift in spectral properties of the discretized transport operatoralso takes place on the unit circle. That is, if the initial condition is transported around the circle withoutbeing deformed, operators are guaranteed to have purely imaginary eigenvalues (as proved in [14]). Whenthe transport takes place with spatially variable speeds, on the other hand, we show that eigenvalues willdeviated from the imaginary axis unless nodes are place perfectly equispaced.

This paper is structured as follows: in Section 2 we consider the stability of the RBF collocationmethod on the circle; in Section 3 we address the stability on the sphere; in Section 4 we propose aleast-squares method for ameliorating instabilities arising from non-equispaced centers. Our concludingremarks are presented in Section 5.

2 Stability on the Circle

Notice that the Euclidian distance between two points on the unit circle is given by

‖xi − xj‖22 = 2− 2 cos(θi − θj),

where θj is the angular component of the point xj in polar coordinates. For simplicity, and without lossof generality, we can write (1) as

h(x) =

N∑j=1

cjφ(1− cos(θ − θj)), (3)

2

with the understanding that φ(1− cos(θ − θj)) = Φ(‖x− xj‖2).Now consider the problem: ht + u(θ)hθ = 0 0 < θ < 2π, t > 0

h(θ, 0) = h0(θ) 0 ≤ θ ≤ 2πh(0, t) = h(2π, t) t ≥ 0

(4)

where u is continuous and satisfies u(0) = u(2π) and u > 0. Before studying the spectral properties ofthe RBF differential operator, it is worthwhile to state the spectral properties of the exact differentialoperator L = −u(θ)∂θ. Consider the eigenvalue problem

Lh = µh (5)

where h is continuous and satisfies h(0) = h(2π). A solution to (5) is

h(θ) = e−µ∫ θ0

dϕu(ϕ) (6)

and enforcing the periodicity condition, we have that

e0 = h(0) = h(2π) = e−µ∫ 2π0

dϕu(ϕ) (7)

which implies that <{−µ∫ 2π

0dϕu(ϕ)

}= 0 and =

{−µ∫ 2π

0dϕu(ϕ)

}= nπ for n ∈ Z so that

µn = − 2nπi∫ 2π

0dϕu(ϕ)

. (8)

Hence, the spectrum of the exact spatial operator is purely imaginary.Now let Φ be a radial function. We seek a collocation solution to (4) of the form

h(θ, t) =

N∑k=1

ck(t)φ(1− cos(θ − θk)). (9)

Note that (9) automatically satisfies the periodicity condition. Substituting (9) into (4), we obtain

N∑j=1

c′j(t)φ(1− cos(θ − θj)) +

N∑j=1

cj(t)u(θ) sin(θ − θj)φ′(1− cos(θ − θj)) = 0. (10)

Requiring that (10) be exact at each center, we obtain

N∑j=1

c′j(t)φ(1− cos(θi − θj)) +

N∑j=1

cj(t)u(θi) sin(θi − θj)φ′(1− cos(θi − θj)) = 0 (11)

for 1 ≤ i ≤ N , which can be written in matrix×vector form as

Ac′(t) + UBc(t) = 0 (12)

where

A = [aij ] ≡ [φ(1− cos(θi − θj))] (13)

B = [bij ] ≡ [sin(θi − θj)φ′(1− cos(θi − θj))] (14)

U = [uij ] ≡ [u(θi)δij ] (15)

c(t) ≡ (c1(t), . . . , cN (t))ᵀ. (16)

3

If we writeh(t) ≡ (h(θ1, t), . . . , h(θN , t))

ᵀ, (17)

then c and h are related by h(t) = Ac(t) so that (12) can be rewritten as the semi-discrete problem

h′(t) = Dh(t) (18)

where D ≡ −UBA−1 is the differentiation matrix (DM). We have tacitly assumed that φ is such thatA is non-singular. For a time-step ∆t > 0, we numerically integrate (18) by a time-integration formulaF∆t. For ease of exposition, we will assume that F∆t is an explicit Runge-Kutta formula. Since theproblems we are interested in are hyperbolic, this assumption is quite reasonable. Equation (18) can nowbe rewritten as the fully-discrete problem

hn+1 = F∆t(D)hn. (19)

We say that (19) is Lax-stable if||F∆t(D)n|| ≤ C (20)

for some constant C > 0 and for all n ≥ 0. If D is normal, it suffices to show that the eigenvalues of Dlie in the stability domain of F∆t.

2.1 Equispaced Centers

If we take θj ≡ 2πj/N for 1 ≤ j ≤ N , then (13), (14) and (15) become

A = [aij ] = [φ(1− cos(2π(i− j)/N))] (21)

B = [bij ] = [sin(2π(i− j)/N)φ′(1− cos(2π(i− j)/N))] (22)

U = [uij ] = [u(2π(i− j)/N)] (23)

By assuming that the centers are equispaced, a number of nice results follow:

Lemma 1. A and B commute.

The proof amounts to computing AB−BA. A proof is included in the appendix for the interested reader.

Lemma 2. BA−1 is skew-adjoint.

Proof. From (21), A is self-adjoint and from (22), B is skew-adjoint. Furthermore, since A and Bcommute by Lemma 1, so too do A−1 and B. Hence,

(BA−1)∗ = −A−1B = −BA−1, (24)

where (∗) denotes the complex conjugate transpose.

2.1.1 Non-Deformational Flow

Assume u is constant (without loss of generality u ≡ 1). Then D = −BA−1.

Theorem 1. The discretization in (19) is Lax-stable.

Proof. By Lemma 2, D is skew-adjoint and hence normal. Since D is skew-adjoint, its eigenvalues arepurely imaginary. Since D is normal, a sufficient condition for Lax-stability of (19) is that the eigenvaluesof D lie within the stability region of F∆t. If we choose F∆t to be the classical 4th-order Runge-Kuttamethod (RK4), a ∆t can easily be chosen so that this condition is satisfied, since the stability domain ofRK4 covers a neighborhood of the origin along the imaginary axis.

4

Exact (see equation (8)) RBF (128 centers) Fourier-PS (129 modes)0 0.0000 0.00002i/3 0.6640i 0.6667i4i/3 1.3264i 1.3333i6i/3 1.7416i 2.0000i8i/3 1.9875i 2.6667i

Table 1: First Few Eigenvalues for Deformation Flow (u(θ) = 1/(sin2(θ) + 1))

2.1.2 Deformational Flow

Assume u is non-constant. Then D = −UBA−1.

Lemma 3. The spectrum of UBA−1 is purely imaginary.

Proof. Let λ ∈ C and v ∈ Cn be such that

UBA−1v = λv. (25)

u > 0 implies that U > 0. So U is invertible and U−1 > 0. We now consider the generalized eigenvalueproblem

BA−1v = λU−1v. (26)

Applying 〈 · , v〉 to (26), we obtain ⟨BA−1v, v

⟩= λ

⟨U−1v, v

⟩. (27)

Using the fact that BA−1 is skew-adjoint and U is diagonal,

−⟨BA−1v, v

⟩=⟨v,BA−1v

⟩(28)

= 〈BA−1v, v〉

= λ 〈U−1v, v〉= λ

⟨v, U−1v

⟩= λ

⟨U−1v, v

⟩.

Adding (27) and (28) yields0 = (λ+ λ)

⟨U−1v, v

⟩. (29)

But since U−1 > 0, we have that⟨U−1v, v

⟩> 0. Hence, λ+ λ = 0. So the spectrum of UBA−1 is purely

imaginary.

Although the eigenvalues of the RBF DM are purely imaginary, they do not line up with the eigenvaluesof the exact spatial operator as the eigenvalues of a Fourier-Pseudospectral (Fourier-PS) DM do. Table 1shows the numerical values of the first five eigenvalues of the RBF DM. Unfortunately, UBA−1 is generallynon-normal and so we require additional machinery to prove the Lax-stability of (18). So we look to theε-pseudospectra of the DM for further insight:

Definition 1. The ε-pseudospectra of a matrix D is the set

Λε(D) = {λ ∈ C | λ ∈ Λ(D + E) for some ||E|| < ε} (30)

where Λ(D) denotes the spectrum of D.

Computationally, one uses the equivalent definition

Λε(D) = {λ ∈ C | σmin(λ−D) ≤ ε} (31)

where σmin denotes the smallest singular value. We can now state the following theorem relating to theLax-stability of (18):

5

Theorem 2. (Reddy and Trefethen, 1992) Let S denote the stability region of the time-integration formulaF∆t. If D satisfies

supλε∈Λε(∆tD)

{sups∈S|λε − s|} ≤ C1ε (32)

for some constant C1, then||F∆t(D)n|| ≤ C2 min{N,n} (33)

for sufficiently small ∆t, for some constant C2 and for all n ≥ 0.

Further details can be found in [15]. A thorough treatment of non-normal operators and pseudospectracan be found in [16]. Figure 2 shows contours of the ε-pseudospectra of the DM using the deformationalvelocity u(θ) = sin2(θ) + 1. To satisfy the hypothesis of Theorem 2, the ε-pseudospectra contours mustdecay in distance to the imaginary axis like O(ε). The lower plot shows this to be the case, with C1 beingapproximately unity.

2.2 Condition Number of the Eigenvector Matrix

Another way to arrive at the hypothesis of Theorem 2 is by following result found in [16]:

Theorem 3. If D is diagonalizable (i.e. D = V EV −1), then Λε(D) ⊂ ∆κ(V )ε.

Here, κ(V ) denotes the condition number of V and

∆r =⋃

λ∈Λ(D)

Br(λ), (34)

i.e., ∆r is the union of all open balls of radius r centered at an eigenvalue λ of D. Hence, if κ(V ) is’well-behaved,’ the method will be Lax-stable according to Theorem 2. In Figure 3, we depart from ourtypical choice of ε and instead choose for each N an ε such that κ(A) = 100. The plot changes little fora different choice of the wave speed c. This shows that although κ(V ) does not favor large or small N ,an exact estimate on κ(V ) may be difficult to obtain.

2.3 Non-Equispaced Centers

We have studied the RBF method on equispaced centers largely out of theoretical interest. In practice,the Fourier-Pseudospectral (Fourier-PS) method is used since the coefficients can be rapidly obtainedvia the FFT. The draw of the RBF method is that it promises convergence on non-equispaced centers.Unfortunately, it is rarely Lax-stable in this non-equispaced regime. It is worth noting that when the ve-locity is constant and the radial function is positive definite, the spectrum of the DM is purely imaginaryeven on non-equispaced centers [14]. The result breaks down for non-constant velocities, however.

In order to explain why RBF solutions become unstable on non-equispaced centers, we consider theperturbed centers

θk = 2πk/N + αXk (35)

where α > 0 is a parameter controlling the magnitude of the perturbation and Xk ∼ U(−1, 1) is auniformly distributed random variable. Figure 4 shows how the magnitude of the largest real-part andimaginary-part of the spectrum of the DM vary with the magnitude of α. While the largest imaginary-part is unaffected by perturbations in the centers, the largest real-part grows almost linearly with α.We can express the perturbed regime in terms of entry-by-entry Taylor expansions. Make the following

6

Real

Ima

gin

ary

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

ε = 10−1

ε = 100

Real

Ima

gin

ary

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

ε = 10−10

ε = 10−9

ε = 10−8

ε = 10−7

ε = 10−6

ε = 10−5

ε = 10−4

ε = 10−3

ε = 10−2

Figure 2: Contours of ε-pseudospectra for deformational flow DM on the circle. The eigenvalues of theDM are located in each of the small circular contours in the top plot. The contours of ε-pseudospectra oforder 10−2 and less are circles surrounding each eigenvalue. For computational and visualization reasons,we plot only the contours of this order around the zero eigenvalue and in the first quadrant.

7

0 50 100 150 200 250 3000

5

10

15

20

25

30

35

40

45

N

Conditio

n N

um

ber

of V

Figure 3: κ(V ) for each N with c(θ) = sin2(θ) + 1.

10−15

10−10

10−5

10−20

10−15

10−10

10−5

100

105

Magnitude of Perturbation (α)

Largest Real−Part of Λ(D)

Largest Imaginary−Part of Λ(D)

slope =1.0057

Figure 4: Magnitude of largest real/imaginary-parts of the spectrum of the deformational flow DM onthe circle plotted against magnitude of perturbation off of equispaced grid. 10,000 samples shown.

8

definitions

Aα = [aαij ] = [φ(1− cos(2π(i− j)/N + α (Xi −Xj)))] ≡ [ξij(α)] (36)

A′ = [a′ij ] =[(Xi −Xj) ξ

′ij(α)

](37)

Bα = [bαij ] = [sin(2π(i− j)/N + α (Xi −Xj))φ′(1− cos(2π(i− j)/N + α (Xi −Xj)))] ≡ [ηij(α)] (38)

B′ = [b′ij ] =[(Xi −Xj) η

′ij(α)

](39)

Uα = [uαij ] = [u(2π(i− j)/N + α (Xi −Xj))] ≡ [ζij(z)] (40)

U ′ = [u′ij ] =[(Xi −Xj) ζ

′ij(α)

](41)

Now consider the DM in the perturbed regime and simplify by neglecting O(α2) terms:

Dα ≡ −UαBαA−1α (42)

∼ −(U + αU ′)(B + αB′)(A+ αA′)−1

= −(U + αU ′)(B + αB′)[A(I − (−αA−1A′))]−1

= −(U + αU ′)(B + αB′)(I − (−αA−1A′))−1A−1

∼ −(U + αU ′)(B + αB′)(I − αA−1A′)A−1

∼ D − α(U ′B + CB′ −DA′)A−1

where we assume that α is chosen small enough so that ||A−1A′|| < 1/α. A direct application of theBauer-Fike Theorem yields:

Theorem 4. If µ ∈ Λ(Dα) and D can be diagonalized as D = V EV −1, then

minλ∈Λ(D)

|λ− µ| ≤ ακ(A)κ(V )

(||U ||||B′||+ ||U ′||||B||+ ||D||||A′||

||A||

)(43)

Theorem 4 provides a liberal estimate on how far the eigenvalues of the DM are perturbed when thecenters are perturbed. While the values of ||A′|| and ||B′|| depend strongly on the radial function, it isapparent that the condition numbers κ(A) and κ(V ) are critical quantities.

3 Stability on the Sphere

The original impetus for studying the stability of the RBF method on the circle was the discovery ofinstabilities of the RBF method on the sphere, where large-scale geoscience calculations are of interest.Figure 5 shows the eigenvalues of the DM for a convection problem (to be described later in this section).It clearly shows that eigenvalues with positive real-part are present in the DM, which makes stabletime-integration impossible. Our hypothesis is that irregularities in node location cause these spuriouseigenvalues. For the discussion that follows, we will use the latitude-longitude coordinates given by

x = cos(λ) cos(θ) (44)

y = sin(λ) cos(θ) (45)

z = sin(θ). (46)

Given a velocity field u, consider the 2D convection problem

ht + (u · ∇)h = 0 (47)

on the sphere with suitable periodicity and initial conditions. Setting the latitudinal component of u tozero and eliminating the longitudinal dependence of the longitudinal component of u to zero results inthe simplified problem

ht + sec(θ)u(θ)hλ = 0 −π < λ < π, −π/2 < θ < π/2, t > 0h(λ, θ, 0) = h0(λ, θ) −π < λ < π, −π/2 < θ < π/2h(−π, θ, t) = h(π, θ, t) t ≥ 0h(λ,−π/2, t) = h(λ, π/2, t) t ≥ 0

. (48)

9

−0.1 −0.05 0 0.05 0.1−10

−5

0

5

10nodes=1849, centers=1849

ℜ (z)

ℑ(z

)

−0.1 −0.05 0 0.05 0.1−10

−5

0

5

10nodes=2601, centers=1296

ℜ (z)

ℑ(z

)

max real−part:1.3e−3

max real−part:6.4e−2

Figure 5: Spectrum of DM for deformational flow on the sphere.

Figure 6: From left to right, the numerical solution of the polar vortex problem at t = 0, 8, 16, 24 reps.as seen from the north pole.

A numerical simulation of this problem is shown in Figure 6. A quick calcuation shows that the exactspatial operator L ≡ − sec θu(θ)∂λ is skew-adjoint:

〈[Lf ], g〉 =

∫ π/2

−π/2

∫ π

−π[Lf ](λ, θ)g(λ, θ) cos(θ)dλdθ (49)

=

∫ π/2

−π/2

∫ π

−π(sec(θ)[ufλ](λ, θ))g(λ, θ) cos(θ)dλdθ

=

∫ π/2

−π/2

∫ π

−π[ufλg](λ, θ)dλdθ

=

∫ π/2

−π/2

([[ufg](λ, θ)]λ= π

λ=−π −∫ π

−π[uλfg](λ, θ)dλ−

∫ π

−π[ufgλ](λ, θ)dλ

)dθ

= −∫ π/2

−π/2

∫ π

−π[ufgλ](λ, θ)dλdθ

= −∫ π/2

−π/2

∫ π

−πf(λ, θ)(sec(θ)[ugλ](λ, θ)) cos(θ)dλdθ

= −〈f, [Lg]〉

10

Figure 7: Dodecahedral points. The latitude-longitude mesh is shown for comparison.

where we have assumed that f and g are suitably continuous and periodic on the domain [−π,−π] ×[−π/2, π/2]. Hence, the spectrum of L is purely imaginary. In our computations, we take u to be

u(θ) =

√3

2sech2(3 cos(θ)) tanh(3 cos(θ)). (50)

This velocity profile has been extensively used in the RBF literature for convection problems on thesphere (see [6], [7]).

3.1 Equispaced Centers

The notion of ’equispacing’ is more ambiguous on the sphere than it is on the circle. One popular testfor equispacing on the sphere is that the quantity

`j ≡ mink 6=j{||xj − xk||} (51)

be independent of j. The vertex sets of a number of polyhedra satisfy this criterion, with the largestset belonging to the dodecahedron. Prior to normalization, the dodecahedral vertices have cartesiancoordinates

{(±1,±1,±1), (0,±1/ϕ,±ϕ), (±1/ϕ,±ϕ, 0), (±ϕ, 0,±1/ϕ)} (52)

where ϕ is the golden ratio. These points are plotted in Figure 7. Using the Variable Precision Arithmatic(VPA) function in MATLAB’s Symbolic Math Toolbox, we computed the magnitude of the largestimaginary-part of the spectrum of the RBF DM to be 1.1×10−1 and the largest real-part to be 1.7×10−35.For comparison, we consider the same discretization on the so-called minimum engery (ME) points, whichare minimizers of the energy function

E(x1, . . . , xN ) =

N∑i=1

i∑j=1

1

||xi − xj ||2. (53)

ME points belong to a class of pseudo-equispaced points that approximately satisfy (52). An eigenvaluecomputation using 25 of these points resulted in the largest imaginary-part being 1.1 × 10−2 and thelargest real-part being 9.9 × 10−4, which suggests that the dodecahedral points lead to a more stablemethod. As on the circle, we look to the ε-pseudospectra of the DM for further insight. Figure 8 showsthat the ε-pseudospectra contours decay in distance to the imaginary axis like O(ε) with C1 ≈ 1/10,suggesting that the RBF method may in fact be Lax-stable on dodecahedral centers.

11

Real

Ima

gin

ary

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

ε = 10−2

ε = 10−1

ε = 100

Real

Ima

gin

ary

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

ε = 10−10

ε = 10−9

ε = 10−8

ε = 10−7

ε = 10−6

ε = 10−5

ε = 10−4

ε = 10−3

Figure 8: Contours of ε-pseudospectra for deformational flow DM on the sphere. The bottom log-log plotshows contours in the first quadrant about the zero eigenvalue.

12

10−15

10−10

10−5

10−20

10−15

10−10

10−5

100

105

Magnitude of Perturbation (α)

Largest Real−Part of Λ(D)

Largest Imaginary−Part of Λ(D)

slope =0.99888

Figure 9: Magnitude of largest real/imaginary-parts of the spectrum of the deformational flow DM onthe sphere plotted against magnitude of perturbation off of dodecahedral points. 10,000 samples shown.

3.2 Non-Equispaced Centers

As on the circle, the RBF method is rarely stable in the non-equispaced regime. It is again worth notingthat when the velocity is given by u(θ) = cos(θ) and the radial function is positive definite, the spectrumof the DM is purely imaginary even on non-equispaced centers [14].

We will proceed as in the previous section by considering the perturbed centers

θk = 2π(k/N + αXk)− π (54)

λk = π(k/N + αYk)− π/2 (55)

where α > 0 is a parameter controlling the magnitude of the perturbation and Xk, Yk ∼ U(−1, 1)are uniformly distributed random variables and Xk and Yk are independent. Figure 9 shows how themagnitude of the largest real-part and imaginary-part of the spectrum of the DM vary with the magnitudeof α. Just as on the circle, the largest imaginary-part is unaffected by perturbations in the centers whilethe largest real-part grows almost linearly with α.

4 RBF Least-Squares

Consider again the problem in (48) with L ≡ − sec θu(θ)∂λ, which was shown to be skew-adjoint inSection 3. We now seek a Galerkin solution to (48) of the form

h(λ, θ, t) =

N∑j=1

cj(t)φj(λ, θ) (56)

13

where we defineφj(λ, θ) ≡ φ(1− cos(θ) cos(θj) cos(λ− λj)− sin(θ) sin(θj)). (57)

Note that (57) automatically satisfies the periodicity condition. Substituting (56) into (48), we obtain

N∑j=1

c′j(t)φj(λ, θ) +

N∑j=1

c′j(t) [Lφj ] (λ, θ) = 0. (58)

Applying the functional 〈φi(λ, θ), · 〉 to (58), we obtain

N∑j=1

c′j(t) 〈φi(λ, θ), φj(λ, θ)〉+

N∑j=1

c′j(t) 〈φi(λ, θ), [Lφj ] (λ, θ)〉 = 0 (59)

for 1 ≤ i ≤ N , which can be written in matrix×vector form as

Ac′(t) + Bc(t) = 0 (60)

where

A = [aij ] ≡ [〈φi(λ, θ), φj(λ, θ)〉] (61)

B = [bij ] ≡ [〈φi(λ, θ), [Lφj ] (λ, θ)〉] (62)

c(t) ≡ (c1(t), . . . , cN (t))ᵀ. (63)

Two observations can be made:

1. A is a Gramian matrix and hence A ≥ 0.

2. Since L is skew-adjoint, so too is B (i.e. 〈φi,Lφj〉 = −〈φj ,Lφi〉).

When A > 0, an argument similar to that used in the proof of Lemma 3 shows that the spectrum ofD is purely imaginary. In practice, the inner-products in (62) and (63) must be approximated by somequadrature rule, say by

〈φi(λ, θ), φj(λ, θ)〉 ≈ κMM∑k=1

φi(λk, θk)φj(λk, θk) (64)

〈φi(λ, θ), [Lφj ] (λ, θ)〉 ≈ κMM∑k=1

φi(λk, θk) [Lφj ] (λk, θk) (65)

where M > N , κM is a constant and (λ1, θ1), . . . , (λM , θM ) ∈ [−π, π] × [−π/2, π/2] are the quadraturenodes. The idea is that as M → ∞, the quadratures become exact and the spectrum of D becomespurely imaginary. (60) can be rewritten as

A∗Ac′(t) +A∗Bc(t) = 0 (66)

where

A = [aij ] ≡ φj(λi, θj) (67)

B = [bij ] ≡ [Lφj ] (λi, θi). (68)

It is worth noting that (67) and (68) are equivalent to the matrices obtained from collocating over a setof quadrature nodes instead of centers. A further discussion of the connection between collocation andGalerkin methods can be found in [1]. If we again write

h(t) ≡ (h(λ1, θ1, t), . . . , h(λ1, θN , t))ᵀ, (69)

14

then c and h are related by h(t) = Ac(t) so that (12) can be rewritten as the semi-discrete problem

h′(t) = Dh(t). (70)

where D ≡ −B(A∗A)−1A∗. In practice, (A∗A)−1A∗ is computed using a QR-decomposition to ensure nu-merical stability. Furthermore, D is only partially assembled so that the complexity in multipling D andh(t) is only #ops ≡ 4M2N2−M−N (instead of #ops ≡ 2(2M2−M) which may be substantially larger).

Our numerical experiments show that for a fixed #ops, the least-squares method results in more de-sirable spectral properties than the collocation method. Figure 10 shows the magnitude of the largestreal and imaginary parts of the spectrum of D for two least-squares methods and the collocation method.Since we fix #ops, an increase in the number of quadrature nodes must come with a decrease in the num-ber of centers. Figure 11 shows that for a given shape parameter, the least-squares method results in lessdesirable accuracy than the collocation method. This loss of accuracy should be expected, as the least-squares method uses fewer basis functions. Figure 11 also shows, however, that by properly adjustingthe shape parameter, the least-squares method can obtain comparable accuracy to the collocation method.

It has been well documented that the problem of inverting (or pseudo-inverting) A becomes ill-conditioned as ε → 0 [13]. This small-ε regime is often desirable, however, with RBFs reproducingpolynomials in certain cases [3]. A number of algorithms have been proposed to bypass the conditioningproblem, notably RBF-QR ([10], [11], [8]) and Contour-Pade ([12]). For our numerical experiments, weuse RBF-QR since it was originally developed for use on the sphere.

5 Conclusion

In this paper, we presented the first known Lax-stability analysis of the RBF collocation method forconvection problems on the circle and sphere. Our analysis suggests that stability can be achievedon equispaced collocation points, with instabilities developing almost linearly with perturbations offthese points. Liberal estimates suggest that the condition numbers of the interpolation matrix and theeigenvector matrix of the DM are important quantities in controlling instabilities. Since the RBF methodis most attractive when collocation points are not equispaced, we proposed a least-squares method tominimize the effect of spurious eigenvalues in the DM. By varying the shape parameter, we showed thatthe least-squares method can achieve a competitive accuracy without any increase in computational cost.

Acknowledgements

The authors would like to thank Anne Gelb, Toby Driscoll, Bengt Fornberg and Natasha Flyer for theirinsight and suggestions. The first author is particularly grateful for the support of Bengt Fornberg andNatasha Flyer at CU Boulder and NCAR, respectively, who gave invaluable commentary on the presentmanuscript.

Appendix

Proof that A and B commute

Proof. Fix 1 ≤ i, j ≤ N and let ψij : {1, . . . , N} → {1, . . . , N} be given by

ψij(k) = i+ j − k +N(1i+j−k < 1 − 1N < i+j−k) (71)

15

10−1

100

101

10−4

10−3

10−2

10−1

100

101

ε

Larg

est R

eal−

Part

of

Λ(D

)

M ≈ 1 × N (3136,3136)

M ≈ 2 × N (4489,2209)

M ≈ 3 × N (5329,1849)

10−1

100

101

8

10

12

14

16

18

20

ε

Larg

est Im

agin

ary

−P

art

of

Λ(D

)

M ≈ 1 × N (3136,3136)

M ≈ 2 × N (4489,2209)

M ≈ 3 × N (5329,1849)

Figure 10: Magnitude of largest real and imaginary parts of the spectrum of the deformational flow DMon the sphere plotted against the shape parameter.

16

10−1

100

101

10−2.23

10−2.22

10−2.21

10−2.2

10−2.19

ε

Err

or

at t =

3

M ≈ 1 × N (3136,3136)

M ≈ 2 × N (4489,2209)

M ≈ 3 × N (5329,1849)

Figure 11: `2 error at t = 3 for the deformational flow problem on the sphere plotted against the shapeparameter.

where 1 denotes the indicator function. It can be shown that ψij is an involution (i.e. ψij=ψ−1ij ) and

hence a bijection. Next, fix 1 ≤ k ≤ N and observe that

cos(2π(ψij(k)− j)/N) = cos(2π(i− k)/N − 2π(1i+j−k < 1 − 1N < i+j−k)) (72)

= cos(2π(i− k)/N),

cos(2π(i− ψij(k))/N) = cos(2π(k − j)/N − 2π(1i+j−k < 1 − 1N < i+j−k)) (73)

= cos(2π(k − j)/N)

and similarly for sine. Comparing (72)-(73) to (21)-(22), we see that

aikbkj = biψij(k)aψij(k)j . (74)

Since ψij is a bijection,

AB ≡ X = [xij ] =

[N∑k=1

aikbkj

]=

[N∑k=1

biψij(k)aψij(k)j

]= [yij ] = Y ≡ BA (75)

and so A and B commute.

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