Highly localized RBF Lagrange functions for finite difference methods on spheres

Abstract

The aim of this paper is to show how rapidly decaying RBF Lagrange functions on the sphere can be used to create a numerically feasible, stable finite difference method based on radial basis functions (an RBF-FD-like method). For certain classes of PDEs this approach leads to rigorous convergence estimates for stencils which grow moderately with increasing discretization fineness.

Appendices

A Kernels on the sphere

In this section, we consider SBFs of the form $\varPhi (x,y) = \phi (x\cdot y)$. Many SBFs are the restriction to ${\mathbb {S}}^2\times {\mathbb {S}}^2$ from radial, translation invariant kernels (RBFs) on ${\mathbb {R}}^3$: i.e., restrictions of kernels of the form $\varPsi (x,y) = \psi (|x-y|)$. By writing $|x-y| = \sqrt{2-2t}$, we may write the restricted RBF in terms of a single real variable $t = x\cdot y$, so

$$\begin{aligned} \phi (t) = \psi (\sqrt{2-2t}). \end{aligned}$$

(A.1)

The restricted polyharmonic splines are a well-known class of conditionally positive SBFs given by

$$\begin{aligned} \phi (t)= \left\{ \begin{array}{ll }(1-t)^{s} \log (1- t), &{} s \in {\mathbb {N}}, \\ (1-t)^{s}, &{} s \in {\mathbb {N}} - \frac{1}{2}, \end{array} \right. \end{aligned}$$

for all $t\in [-1,1]$. In terms of $R(t):= \sqrt{1-t}$ (which is invertible on $[-1,1]$), we can express $\phi (t)$ as

$$\begin{aligned} \phi (t)= \left\{ \begin{array}{ll }R(t)^{2s} \log R(t), &{} s \in {\mathbb {N}}, \\ R(t)^{2s}, &{} s \in {\mathbb {N}} - \frac{1}{2}. \end{array} \right. \end{aligned}$$

In this appendix, we focus on strictly positive definite SBFs. In contrast to the restricted polyharmonic splines, which are discussed throughout the body of the article, such kernels require tuning of a shape parameter.

1.1 A.1 Strictly positive definite kernels and shape parameters

By rescaling the RBF by $\sigma >0$ as $\varPsi _{\sigma }(x,y) = \psi (\frac{|x-y|}{\sigma })$, we obtain a new positive definite radial basis function.¹ This means that the RBF has a parameter (roughly the reciprocal of the “shape parameter” $\epsilon$ defined below) which must me be set. By changing this parameter, one alters the scaling of the RBF; this is desirable (indeed, necessary) for many practical problems: e.g., for a finite point set $X_N\subset {\mathbb {R}}^d$, the collocation matrix

$$\begin{aligned} {{\varvec{\Psi }}}_{X_N} =\bigl (\varPsi _{\sigma }(x_j,x_k)\bigr )_{j,k} \end{aligned}$$

will have a large condition number if $\sigma$ is large relative to the nearest neighbor distance $q= \min _{j\ne k} |x_j-x_k|$. Thus, in order to treat (e.g., by interpolation, quadrature, etc.) densely sampled data, one may wish to choose a small value of $\sigma$ to stabilize the problem. This is also the case when using the SBF obtained by restricting the RBF $\varPsi _{\sigma }$.

Choosing small $\sigma$ may come at a cost, however: the shape parameter affects the native space, both by modifying the inner product, and in some cases by altering the underlying set. The known convergence results for RBF approximation and interpolation are most often assume a fixed shape parameter. Indeed, for most RBFs (which have a positive, continuous Fourier transform and therefore lack a Strang-Fix condition [3]), choosing $\sigma \propto h$, will lead to non-convergence [27]. To date there are few direct approximation results which deal with multi-scale RBF approximation [4, 13, 17, 23].

In what follows, we’ll use the notation $R(t):= \sqrt{1-t}$ (which is invertible on $[-1,1]$) and the modified “shape parameter” $\epsilon := \frac{\sqrt{2}}{\sigma }$, which yields the relation, via (A.1), $\phi (t) = \psi (\epsilon R(t))$.

Let us now give a few examples of zonal kernels, along with their native spaces.

B Differential equations on the sphere: assembling the FD matrix

In this section, we consider how to assemble the kernel FD matrix. Generally, this requires some understanding of the expression of ${\mathscr {L}}$ in spherical coordinates (a different method to construct ${\textbf{M}}_{X_N}$ in Cartesian coordinates has been given in [5]) and how to use it to obtain an expression for ${\mathscr {L}}^{(1)} \varPhi$.

The basic challenge is to assemble the Kansa type matrix ${\textbf{K}}_{X_N} = \bigl ({\mathscr {L}}^{(1)} \varPhi (x_j,x_k)\bigr )_{j,k}$ for a few first and second order linear differential operators ${\mathscr {L}}$.

1.1 B.1 Working in coordinates

In what follows, we use spherical coordinates $(\theta ,\varphi )\in [0,2\pi )\times [0,\pi ]$ to describe points on the sphere. This involves the convention $x =\cos \theta \sin \varphi$, $y= \sin \theta \sin \varphi$ and $z=\cos \varphi$.

The surface gradient is $\nabla f(\theta , \varphi ) = \frac{\partial f}{\partial \varphi } \varvec{\varphi } + \frac{1}{\sin ^2 \varphi }\frac{\partial f}{\partial \theta } \varvec{\theta }$. Here $\varvec{\varphi }$ and $\varvec{\theta }$ are the basic tangent vectors for the spherical coordinate system:

$$\begin{aligned} \varvec{\varphi } = \begin{pmatrix}\cos \theta \cos \varphi \\ \sin \theta \cos \varphi \\ - \sin \varphi \end{pmatrix} \quad \text { and } \quad \varvec{\theta } = \begin{pmatrix}-\sin \theta \sin \varphi \\ \cos \theta \sin \varphi \\ 0\end{pmatrix}. \end{aligned}$$

The spherical divergence operator applied to a vector field $F = F_{\varphi } \varvec{\varphi }+F_{\theta } \varvec{\theta }$ gives

$$\begin{aligned} \textrm{div} F = \frac{1}{\sin \varphi }\bigl (\frac{d}{d\theta } (\sin \varphi F_{\theta }) +\frac{d}{d\varphi } (\sin \varphi F_{\varphi })\bigr ) = \frac{dF_{\theta }}{d\theta } + \cot \varphi F_{\varphi } + \frac{d F_{\varphi }}{d \varphi }. \end{aligned}$$

(B.1)

An example of divergence-free vector field (tangent to the sphere) is, for an angle $\alpha$, is

$$\begin{aligned} \varvec{u} =u_{\varphi }\varvec{\varphi }+u_{\theta }\varvec{\theta } = - \sin \alpha \cos \theta \varvec{\varphi } + (\cos \alpha +\sin \alpha \cot \varphi \sin \theta )\varvec{\theta }. \end{aligned}$$

The fact that this is divergence-free is evident from (B.1). Writing $x= \cos \theta \sin \varphi$, $y= \sin \theta \sin \varphi$ and $z=\cos \varphi$, we have

$$\begin{aligned} \varvec{u}(x,y,z) = \begin{pmatrix} -\sin \varphi \sin \theta \cos \alpha - \cos \varphi \sin \alpha \\ \sin \varphi \cos \theta \cos \alpha \\ \sin \varphi \cos \theta \sin \alpha \end{pmatrix} = \begin{pmatrix} -y \cos \alpha - z\sin \alpha \\ x\cos \alpha \\ x \sin \alpha \end{pmatrix}. \end{aligned}$$

An example of a transport term We consider a first order operator of the form ${\mathscr {L}}= \varvec{u}\cdot \nabla$, where $\nabla$ is the “surface” gradient and $\varvec{u} = u_\phi \varvec{\phi } + u_{\theta } \varvec{\theta }$ is a (tangent) vector field. Carrying out the (Cartesian) inner product simply produces

$$\begin{aligned} {\mathscr {L}}f = u_{\varphi }\frac{\partial f}{\partial \varphi }\langle \varvec{\varphi },\varvec{\varphi }\rangle + u_{\theta }\frac{1}{\sin ^2\varphi } \frac{\partial f}{\partial \theta } \langle \varvec{\theta },\varvec{\theta }\rangle = u_{\varphi }\frac{\partial f}{\partial \varphi } +u_{\theta } \frac{\partial f}{\partial \theta }. \end{aligned}$$

1.2 B.2 First order operators

We apply this to a zonal kernel $\varPhi (x,y) = \phi (x\cdot y)$ in the first argument. So we can write the dot product as

$$\begin{aligned} x\cdot y= & {} \sin \varphi _1 \sin \varphi _2 \bigl (\cos \theta _1 \cos \theta _2 + \sin \theta _1 \sin \theta _2\bigr ) + \cos \varphi _1 \cos \varphi _2\\= & {} \sin \varphi _1 \sin \varphi _2 \bigl (\cos (\theta _1 - \theta _2)\bigr ) + \cos \varphi _1 \cos \varphi _2 \end{aligned}$$

For first order differential problems, we need ${\mathscr {L}}^{(1)}\phi (x\cdot y)$, which requires an expression for $\phi '(t)$. Indeed, note that

$$\begin{aligned} \frac{\partial }{\partial \varphi }^{(1)} \phi (x\cdot y)= & {} \left[ \cos \varphi _1 \sin \varphi _2 \bigl ( \cos (\theta _1 - \theta _2)\bigr ) - \sin \varphi _1 \cos \varphi _2\right] \phi '(x\cdot y) = (\varvec{\varphi }\cdot y) \phi '(x\cdot y) \nonumber \\ \end{aligned}$$

(B.2)

$$\begin{aligned} \frac{\partial }{\partial \theta }^{(1)} \phi (x\cdot y)= & {} -\sin \varphi _1 \sin \varphi _2 \bigl (\sin (\theta _1 - \theta _2)\bigr ) \phi '(x\cdot y) = (\varvec{\theta }\cdot y) \phi '(x\cdot y) \end{aligned}$$

(B.3)

From this, it follows that the surface gradient is

$$\begin{aligned} \nabla ^{(1)} \phi (x\cdot y)= & {} (\varvec{\varphi } \cdot y) \phi '(x\cdot y) \varvec{\varphi } + \frac{1}{\sin ^2 \varphi }(\varvec{\theta } \cdot y) \phi '(x\cdot y) \varvec{\theta }. \end{aligned}$$

(B.4)

For the transport term we get in this way the formula

$$\begin{aligned} {\mathscr {L}}^{(1)} \phi (x \cdot y) = u_{\varphi } (\varvec{\varphi } \cdot y) \phi '(x\cdot y) + u_{\theta } (\varvec{\theta } \cdot y) \phi '(x\cdot y). \end{aligned}$$

1.3 B.3 s order operators

In principle, the kernel derivative formulas (B.2), (B.3) and (B.4) along with (B.1) are sufficient to calculate second order operators in divergence form ${\mathscr {L}}= \textrm{div} (\varvec{a} \nabla f)$ for a sufficiently smooth tensor field $\varvec{a}$, although this may be too cumbersome to carry out by hand.

Laplace-Beltrami For ${\mathscr {L}}= \Delta$ (so $\varvec{a} = \textrm{Id}$), it is much easier to use the rotation invariance of $\Delta$. In that case, we can write $x\cdot y = \cos (\vartheta )$, with $\vartheta$ the solid angle between x and y (equivalently, we can perform a rotation mapping y to the north pole). In this case, rotation invariance gives $\Delta ^{(1)} \phi (x\cdot y) = \Delta \phi (\cos \vartheta )$ and

$$\begin{aligned} \Delta \phi (\cos \vartheta ) =\frac{1}{\sin \vartheta } \frac{\partial }{\partial \vartheta } \bigl (\sin \vartheta \frac{\partial }{\partial \vartheta } \phi (\cos \vartheta )\bigr ) = -2\cos \vartheta \phi '(\cos \vartheta ) +(1-\cos ^2\vartheta )\phi ''(\cos \vartheta ). \end{aligned}$$

which simplifies to

$$\begin{aligned} \Delta ^{(1)} \phi (x\cdot y) = (1-(x\cdot y)^2)\phi ''(x\cdot y) -2(x\cdot y)\phi '(x\cdot y). \end{aligned}$$

A second example: In the basis $\{ \varvec{\varphi }, \varvec{\theta } \}$, we consider the tensor $\varvec{a}$ given by

$$\begin{aligned} \varvec{a}= a(\theta ,\varphi ) \begin{pmatrix} 1 &{} 0 \\ 0 &{} \sin ^2\varphi \end{pmatrix}, \end{aligned}$$

i.e. $\varvec{a}$ lies parallel to the metric tensor of the unit sphere. In this case, we can simplify the second order differential operator ${\mathscr {L}}$. Because $\nabla f = \frac{\partial f}{\partial \theta } \varvec{\varphi } + \frac{1}{\sin ^2\varphi } \frac{\partial f}{\partial \varphi } \varvec{\theta }$, we can write

$$\begin{aligned} {\mathscr {L}}f= & {} \textrm{div}(\varvec{a} \nabla f) \\= & {} \frac{1}{\sin \varphi } \frac{\partial }{\partial \varphi } \sin \varphi \, a(\theta ,\varphi ) \frac{\partial f}{\partial \varphi } + \frac{\partial }{\partial \theta } a(\theta ,\varphi ) \frac{\partial f}{\partial \theta }\\= & {} a(\theta ,\varphi ) \left( \Delta f - \cot ^2 \varphi \frac{\partial ^2 f}{\partial \theta ^2}\right) + \frac{\partial a}{\partial \varphi } \frac{\partial f}{\partial \varphi }+ \frac{\partial a}{\partial \theta }\frac{\partial f}{\partial \theta } \end{aligned}$$

For the kernel function $\phi (x\cdot y)$ centered at a fixed y we therefore get, similarly as for the Laplace-Beltrami operator and the surface gradient the formula

$$\begin{aligned} {\mathscr {L}}^{(1)} \phi (x \cdot y)= & {} a(\theta ,\varphi ) \left( \Delta ^{(1)} \phi (x \cdot y) - \cot ^2 \varphi \Big ( \phi ''(x \cdot y) (\varvec{\theta } \cdot y)^2 + \phi '(x \cdot y) ( \frac{\partial \varvec{\theta }}{\partial \theta } \cdot y) \Big ) \right) \\{} & {} + \left( \frac{\partial a}{\partial \varphi } \varvec{\varphi } \cdot y + \frac{\partial a}{\partial \theta }\varvec{\theta } \cdot y \right) \phi '(x \cdot y). \end{aligned}$$

Table 1 Some SBFs and their native spaces

Considering this with $a(\varphi ,\theta )= (1-\frac{1}{2} \cos \varphi )$ gives an elliptic PDE considered in [22]. This simplifies to

$$\begin{aligned} {\mathscr {L}}^{(1)} \phi (x \cdot y)= & {} (1-\frac{1}{2} \cos \varphi ) \left( \Delta ^{(1)} \phi (x \cdot y) - \cot ^2 \varphi \Big ( \phi ''(x \cdot y) (\varvec{\theta } \cdot y)^2 + \phi '(x \cdot y) ( \frac{\partial \varvec{\theta }}{\partial \theta } \cdot y) \Big ) \right) \\{} & {} + \frac{1}{2} \sin {\varphi } \, (\varvec{\varphi } \cdot y) \phi '(x \cdot y), \end{aligned}$$

where $(\varphi ,\theta )$ are the spherical coordinates corresponding to x (Tables 1, 2).

Table 2 Derivatives of SBFs appearing in Table 1

1.4 B.4 Derivatives of well known SBFs

Given an SBF of the form $\phi (t) = \psi (\epsilon R(t))$ (as described in A), the first and second derivatives can be written as:

$$\begin{aligned} \phi ' = - \frac{\epsilon }{2R} \psi '(\epsilon R) =: \epsilon ^2 w(\epsilon R)\qquad \text {and} \qquad \phi '' = -\frac{\epsilon ^3}{2R}w'(\epsilon R). \end{aligned}$$

(B.5)