Multivariant spatial and geometric interpolation of field maps through model order reduction

Complex three-dimensional field maps can be efficiently denoised and compressed in data volume by high order single value decomposition. Extension of the method to inter/extrapolation and determination of basis function representation is reported here. Inter/extrapolation of denoised and compressed maps is applied to spatial coordinates and especially to the parameters defining the geometry of devices creating such maps. It significantly increases the efficiency of the provision of compact and noise-free maps combined with high resolution. The method is illustrated by creating corresponding maps for rf cells of radio-frequency quadrupoles.

Figure 1

A typical RFQ cell comprises one-half of an oscillation of the sinusoidal vane tip. $a$: minimum distance from the axis, $ma$: maximum distance from the axis, $m$: modulation factor, $l$: rf cell length, $r$: vane tip radius.

Figure 2

Singular vectors ${\lambda }_1^{(4)}$ to ${\lambda }_5^{(4)}$ of $U^{(4)}$ as functions of the $z$ step.

Figure 3

Example for a plot of $U^{(2)}$ (left) and $u^{(2)}$ (right) as functions of the $r$ step. The vertical axis displays the real values of singular vector elements. According to Eq. (3), $U_3^{(2)}\sim U_6^{(2)}$ correspond to singular values that are at least 4 orders of magnitude lower than the largest singular value occurring in the two-mode decomposition.

Figure 4

Interpolation and fitting of preserved singular vectors of $u^{(2)}$. Left: interpolation to $r\mathrm{step}=3.5$ or $r=1.1\mathrm{mm}$. Right: replacement of singular vectors by smooth and best-fitting functions.

Figure 5

Illustration of the executable program producing field maps for typical accelerator elements.

Figure 6

Upper left: simulated potential map $V_s$ for an RFQ cell with parameters $m=2$, $r=1.1$, $l=14$, $z=0.4\mathrm{l}$, $x=[0,2]$, $y=[0,2]$. Upper right: potential map from inter/extrapolation. Lower: relative difference between simulated and inter/extrapolated potential maps.

Figure 7

Potential distribution along three straight paths parallel to the beam axis. In the upper and middle case, the path is partially inside of the metallic vane tip, in which the real potential is constant. However, the extrapolation method fills this volume with a smooth distribution, i.e., it ignores the metallic boundaries and presumes a vacuum instead.

Figure 8

Fitting of $u^{(4)}$ through basis function expressions using four terms of Legendre polynomials. The fittings of other bases $u^{(l)}$ are as good as of $u_1^{(4)}$ and $u_2^{(4)}$. Fitting of $u_3^{(4)}$ is worse but its low singular value makes this loss of accuracy effectively negligible. ${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_3$ are the first three elements of ${\lambda }^{(4)}$ in Eq. (3).

Figure 9

Upper left: simulated potential map $V_s$ for an RFQ cell with parameters $m=2$, $r=1.1$, $l=14$, $z=0.4$, $x=[0,2]$, and $y=[0,2]$. Upper right: basis function potential map $V_a$. Lower: relative difference between basis function potential and the simulated potential map.

Figure 10

Example for one-dimensional interpolation and extrapolation.