Towards stability results for global radial basis function based quadrature formulas

Abstract

Quadrature formulas (QFs) based on radial basis functions (RBFs) have become an essential tool for multivariate numerical integration of scattered data. Although numerous works have been published on RBF-QFs, their stability theory can still be considered as underdeveloped. Here, we strive to pave the way towards a more mature stability theory for global and function-independent RBF-QFs. In particular, we prove stability of these for compactly supported RBFs under certain conditions on the shape parameter and the data points. As an alternative to changing the shape parameter, we demonstrate how the least-squares approach can be used to construct stable RBF-QFs by allowing the number of data points used for numerical integration to be larger than the number of centers used to generate the RBF approximation space. Moreover, it is shown that asymptotic stability of many global RBF-QFs is independent of polynomial terms, which are often included in RBF approximations. While our findings provide some novel conditions for stability of global RBF-QFs, the present work also demonstrates that there are still many gaps to fill in future investigations.

A Moments

Henceforth, we provide the moments for different RBFs. The one-dimensional case is discussed in Sect. A.1, while two-dimensional moments are derived in Sect. A.2.

1.1 A.1 One-dimensional moments

Let us consider the one-dimensional case of $\varOmega = [a,b]$ and distinct data points $x_1,\dots ,x_N \in [a,b]$.

1.1.1 A.1.1 Gaussian RBF

For $\varphi (r) = \exp ( - \varepsilon ^2 r^2 )$, the moment of the translated Gaussian RBF,

$$\begin{aligned} m_n = m(\varepsilon ,x_n,a,b) = \int _a^b \exp ( - \varepsilon ^2 | x - x_n |^2 ) \, \textrm{d}x, \end{aligned}$$

(31)

is given by

$$\begin{aligned} m_n = \frac{\sqrt{\pi }}{2 \varepsilon } \left[ \textrm{erf}( \varepsilon (b-x_n) ) - \textrm{erf}( \varepsilon (a-x_n) ) \right] . \end{aligned}$$

Here, $\textrm{erf}(x) = 2/\sqrt{\pi } \int _0^x \exp ( -t^2 ) \, \textrm{d}t$ denotes the usual error function, [73, Section 7.2].

1.1.2 A.1.2 Polyharmonic splines

For $\varphi (r) = r^k$ with odd $k \in \mathbb {N}$, the moment of the translated PHS,

$$\begin{aligned} m_n = m(x_n,a,b) = \int _a^b \varphi ( x - x_n ) \, \textrm{d}x, \end{aligned}$$

is given by

$$\begin{aligned} m_n = \frac{1}{k+1} \left[ (a-x_n)^{k+1} + (b-x_n)^{k+1} \right] , \quad n=1,2,\dots ,N. \end{aligned}$$

For $\varphi (r) = r^k \log r$ with even $k \in \mathbb {N}$, on the other hand, we have

$$\begin{aligned} m_n = (x_n - a)^{k+1} \left[ \frac{\log ( x_n - a )}{k+1} - \frac{1}{(k+1)^2} \right] + (b - x_n)^{k+1} \left[ \frac{\log ( b - x_n )}{k+1} - \frac{1}{(k+1)^2} \right] . \end{aligned}$$

Note that for $x_n = a$ the first term is zero, while for $x_n = b$ the second term is zero.

1.2 A.2 Two-dimensional moments

Here, we consider the two-dimensional case, where the domain is given by a rectangular of the form $\varOmega = [a,b] \times [c,d]$.

1.2.1 A.2.1 Gaussian RBF

For $\varphi (r) = \exp ( - \varepsilon ^2 r^2 )$, the two-dimensional moments can be written as products of one-dimensional moments. In fact, we have

$$\begin{aligned} \int _a^b \int _c^d \exp ( - \varepsilon ^2 \Vert (x-x_n,y-y_n\Vert _2^2 ) = m(\varepsilon ,x_n,a,b) \cdot m(\varepsilon ,y_n,c,d). \end{aligned}$$

Here, the multiplicands on the right-hand side are the one-dimensional moments from (31).

1.2.2 A.2.2 Polyharmonic splines and other RBFs

If it is not possible to trace the two-dimensional moments back to the one-dimensional ones, we are in need of another approach. This is, for instance, the case for PHS. We start by noting that for a data points $(x_n,y_n) \in [a,b] \times [c,d]$ the corresponding moment can be rewritten as follows:

$$\begin{aligned} m(x_n,y_n) = \int _{a}^b \int _{c}^d \varphi ( \Vert (x-x_n,y-y_n)^T \Vert _2 ) \, \textrm{d}y \, \textrm{d}x = \int _{\tilde{a}}^{\tilde{b}} \int _{\tilde{c}}^{\tilde{d}} \varphi ( \Vert (x,y)^T \Vert _2 ) \, \textrm{d}y \, \textrm{d}x \end{aligned}$$

with translated boundaries $\tilde{a} = a - x_n$, $\tilde{b} = b - x_n$, $\tilde{c} = c - y_n$, and $\tilde{d} = d - y_n$. We are not aware of an explicit formula for such integrals for most popular RBFs readily available from the literature. That said, such formulas were derived in [78,79,80] (also see [95, Chapter 2.3]) for the integral of $\varphi$ over a right triangle with vertices $(0,0)^T$, $(\alpha ,0)^T$, and $(\alpha ,\beta )^T$. Assuming $\tilde{a}< 0 < \tilde{b}$ and $\tilde{c}< 0 < \tilde{d}$, we therefore partition the shifted domain ${\tilde{\varOmega } = [\tilde{a},\tilde{b}] \times [\tilde{c},\tilde{d}]}$ into eight right triangles. Denoting the corresponding integrals by $I_1, \dots , I_8$, the moment $m(x_n,y_n)$ correspond to the sum of these integrals. The procedure is illustrated in Fig. 11.

Fig. 11

Illustration of how the moments can be computed on a rectangle in two dimensions

Table 3 The reference integral $I_{\text {ref}}(\alpha ,\beta )$—see (32)—for some PHS

The special cases where one (or two) of the edges of the rectangle align with one of the axes can be treated similarly. However, in this case, a smaller subset of the triangles is considered. We leave the details to the reader, and note the following formula for the weights:

$$\begin{aligned} \begin{aligned} m(x_n,y_n)&= \left[ 1 - \delta _0\left( \tilde{b} \tilde{d}\right) \right] \left( I_1 + I_2 \right) + \left[ 1 - \delta _0\left( \tilde{a} \tilde{d}\right) \right] \left( I_3 + I_4 \right) \\&\quad + \left[ 1 - \delta _0\left( \tilde{a} \tilde{c}\right) \right] \left( I_5 + I_6 \right) + \left[ 1 - \delta _0\left( \tilde{b} \tilde{c}\right) \right] \left( I_7 + I_8 \right) \end{aligned} \end{aligned}$$

Here, $\delta _0$ denotes the usual Kronecker delta defined as $\delta _0(x) = 1$ if $x = 0$ and $\delta _0(x) = 0$ if $x \ne 0$. The above formula holds for general $\tilde{a}$, $\tilde{b}$, $\tilde{c}$, and $\tilde{d}$. Note that all the right triangles can be rotated or mirrored in a way that yields a corresponding integral of the form

$$\begin{aligned} I_{\text {ref}}(\alpha ,\beta ) = \int _0^{\alpha } \int _0^{\frac{\beta }{\alpha }x} \varphi ( \Vert (x,y)^T \Vert _2 ) \, \textrm{d}y \, \textrm{d}x. \end{aligned}$$

(32)

More precisely, we have

$$\begin{aligned} \begin{aligned} I_1&= I_{\text {ref}}(\tilde{b},\tilde{d}), \quad I_2 = I_{\text {ref}}(\tilde{d},\tilde{b}), \quad I_3 = I_{\text {ref}}(\tilde{d},-\tilde{a}), \quad I_4 = I_{\text {ref}}(-\tilde{a},\tilde{d}), \\ I_5&= I_{\text {ref}}(-\tilde{a},-\tilde{c}), \quad I_6 = I_{\text {ref}}(-\tilde{c},-\tilde{a}), \quad I_7 = I_{\text {ref}}(-\tilde{c},\tilde{b}), \quad I_8 = I_{\text {ref}}(\tilde{b},-\tilde{c}). \end{aligned} \end{aligned}$$

Finally, explicit formulas of the reference integral $I_{\text {ref}}(\alpha ,\beta )$ over the right triangle with vertices $(0,0)^T$, $(\alpha ,0)^T$, and $(\alpha ,\beta )^T$ for some PHS can be found in Table 3. Similar formulas are also available, for instance, for Gaussian, multiquadric and inverse multiquadric RBFs.

We note that the approach presented above is similar to the one in [85], where the domain $\varOmega = [-1,1]^2$ was considered. Later, the same authors extended their findings to simple polygons [84] using the Gauss–Grenn theorem. Also see the recent work [86], addressing polygonal regions that may be nonconvex or even multiply connected, and references therein. It would be of interest to see if these approaches also carry over to computing products of RBFs corresponding to different centers or products of RBFs and their partial derivatives, again corresponding to different centers. Such integrals occur as elements of mass and stiffness matrices in numerical PDEs. In particular, they are desired to construct linearly energy stable (global) RBF methods for hyperbolic conservation laws [35, 39, 40].