Data-dependent approximation through RBF

Abstract

In this article we present a modification of classical Radial Basis Function (RBF) interpolation techniques aimed at reducing oscillations near discontinuities in one and two dimensions. Our approach introduces an adaptive mechanism by varying the shape parameter of the RBFs and making it data-dependent, forcing it to tend to infinity in the vicinity of discontinuities. This modification results in kernel functions that locally resemble %Kronecker delta functions, effectively minimizing spurious oscillations. To detect discontinuities, we employ smoothness indicators: for grid-based data, these are computed as undivided second-order differences squared. For scattered data, we use least squares approximations of the Laplacian multiplied by the square of the mean local separation of the stencil points, and then squared. These indicators guide the adaptive adjustment of the shape parameter. We prove the invertibility of the resulting interpolation matrix and propose a solution strategy that maintains the condition number comparable to that of a system where points near discontinuities are excluded. Numerical experiments in one and two dimensions demonstrate that the proposed method significantly reduces oscillations near discontinuities across various kernel types, whether locally or globally supported. At the same time, the interpolation accuracy and matrix conditioning in smooth regions remain essentially unchanged, as measured by the infinity norm of the error and the condition number.

Figures (14)

• Figure 1: Approximation to the function $g$ (red solid dots), Eq. \ref{['funciong']}, using a uniform grid of $n=32$ points and generating ten evaluation points between each of them. In each plot the classical and data-dependent RBF algorithms have been used. From left to right and top to bottom these algorithms are: RBF$_{\text{G}}$ and RBF$_{\text{G}}$, RBF$_{\text{IMQ}}$ and DD-RBF$_{\text{IMQ}}$, RBF$_{\text{W2}}$ and DD-RBF$_{\text{W2}}$, RBF$_{\text{W4}}$ and DD-RBF$_{\text{W4}}$, RBF$_{\text{M2}}$ and DD-RBF$_{\text{M2}}$, RBF$_{\text{M4}}$ and DD-RBF$_{\text{M4}}$.
• Figure 2: Approximation to the function $g$ (red solid dots), Eq. \ref{['funciong']}, using $n=32$ Halton points and generating ten Halton evaluation points between each of them. In each plot the classical and data-dependent RBF algorithms have been used. From left to right and top to bottom these algorithms are: RBF$_{\text{G}}$ and RBF$_{\text{G}}$, RBF$_{\text{IMQ}}$ and DD-RBF$_{\text{IMQ}}$, RBF$_{\text{W2}}$ and DD-RBF$_{\text{W2}}$, RBF$_{\text{W4}}$ and DD-RBF$_{\text{W4}}$, RBF$_{\text{M2}}$ and DD-RBF$_{\text{M2}}$, RBF$_{\text{M4}}$ and DD-RBF$_{\text{M4}}$.
• Figure 3: Approximation of the function $f_1$, defined in Eq. \ref{['frankesdisc']}, using $n = 50 \times 50$ initial gridded data points. The left column shows the results obtained with the classical RBF$_{\text{G}}$ algorithm, while the right column corresponds to the data-dependent DD-RBF$_{\text{G}}$ algorithm. The first row displays the final approximations, and the second row presents the error distributions across the domain.
• Figure 4: Approximation of the function $f_1$, defined in Eq. \ref{['frankesdisc']}, using $n = 50 \times 50$ initial gridded data points. The left column shows the results obtained with the classical RBF$_{\text{IMQ}}$ algorithm, while the right column corresponds to the data-dependent DD-RBF$_{\text{IMQ}}$ algorithm. The first row displays the final approximations, and the second row presents the error distributions across the domain.
• Figure 5: Approximation of the function $f_1$, defined in Eq. \ref{['frankesdisc']}, using $n = 50 \times 50$ initial gridded data points. The left column shows the results obtained with the classical RBF$_{\text{W2}}$ algorithm, while the right column corresponds to the data-dependent DD-RBF$_{\text{W2}}$ algorithm. The first row displays the final approximations, and the second row presents the error distributions across the domain.

Theorems & Definitions (9)

• Proposition 2.1
• Proposition 2.2
• Lemma 3.1
• Proof 3.1
• Proposition 3.2
• Proof 3.2
• Lemma 4.1
• Proof 4.1
• Remark 1