Multivariant spatial and geometric interpolation of field maps through model order reduction.
Complex three-dimensional field maps can be efficiently de-noised and compressed by High-Order Singular Value Decomposition (HOSVD). Building on the original application of the method to the electric field map of a drift-tube linac (X. Du & L. Groening, Phys. Rev. Accel. Beams 21, 084601, 2018), this repository accompanies the follow-up paper (X. Du & L. Groening, Phys. Rev. Accel. Beams 25, 124601, 2022), which extends it to inter-/extrapolation and to the determination of a basis-function representation. Inter-/extrapolation is applied both to spatial coordinates and to the parameters defining the geometry of the devices that create the maps. This yields compact, noise-free maps at high resolution. The method is illustrated on the RF cells of a radio-frequency quadrupole (RFQ).
- 🗜️ Compression — store a multi-dimensional field map in a tiny core tensor plus a few factor matrices (hundreds- to thousands-fold smaller).
- 🧹 De-noising — discarding the small singular values removes simulation/measurement noise.
- 🔍 Inter-/extrapolation — resample the compressed model onto an arbitrary grid and onto arbitrary device-geometry parameters.
- 📐 Closed-form fitting — fit the singular vectors with Legendre polynomials for a continuous, analytic model.
No external data required — the example builds its own synthetic field map.
git clone https://github.com/duxngsi/Field_maps_HOSVD-compress-fitting.git
cd Field_maps_HOSVD-compress-fitting
pip install -r requirements.txt
python quickstart.pyExpected output (abridged):
synthetic field map: shape=(6, 6, 6, 40, 21, 21), 3,810,240 elements
injected noise (rel. L2): 4.998%
...
compressed to ranks (4, 4, 4, 4, 4, 4)
stored elements : 4,496 (core (4, 4, 4, 4, 4, 4))
compression : 847x
error vs clean : 0.168% <- de-noised reconstruction
error vs noisy : 4.989%
-> reconstruction is closer to ground truth than the noisy input (de-noising works).
resampled mode 3 (z) from 40 to 80 nodes -> recovered shape (6, 6, 6, 80, 21, 21)
A 6-D field map of 3.8 M values is compressed 847× while the reconstruction sits within 0.2 % of the ground truth — well below the 5 % injected noise:
import numpy as np
from HOSVD import hosvd
data = np.load("my_field_map.npy") # an N-dimensional array
model = hosvd(data) # decompose every mode
model.trim((2, 2, 2, 3, 3, 3)) # keep the dominant modes -> compress & de-noise
model.save("./saved_HOSVD/") # persist core tensor + factors
approx = model.recover() # reconstruct the compressed map
# Inter-/extrapolate: resample chosen modes onto new (normalized 0..1) coordinates.
model.resample([[], [], [], np.linspace(0, 1, 400), [], []])
fine = model.recover() # higher z-resolution than the original| Method / attribute | Purpose |
|---|---|
hosvd(data) |
Decompose an N-D array (covariance-based HOSVD). |
hosvd(load_from=path) |
Reload a previously saved decomposition. |
.trim(new_shape) |
Keep the leading singular vectors per mode (compression / de-noising). |
.recover() |
Reconstruct the (compressed) tensor. |
.resample(new_v) |
Resample factor matrices onto new coordinates (inter-/extrapolation). |
.save(path) / .load(path) |
Persist / reload CoreT, v, s. |
.v, .s, .CoreT |
Factor matrices, singular values, core tensor. |
basis.py provides Legendre and Polynomial sampled bases used to fit the factor matrices.
| Path | Description |
|---|---|
HOSVD.py |
Core library: the hosvd class and CST field-map readers. |
basis.py |
Legendre / polynomial basis functions for fitting the factors. |
make_synthetic_data.py |
Generate a self-contained synthetic field-map tensor. |
quickstart.py |
End-to-end demo on synthetic data (no external files needed). |
demo.py |
Reproduce the paper figures from the original RFQ data set. |
RFQ_cell_map_producer.py |
Tkinter GUI to generate RFQ cell maps from a compressed model. |
tests/ |
pytest unit tests. |
demo.py operates on the original RFQ data set (a 6-D tensor of CST-simulated RF-cell
maps). The raw exports are large and are shared separately:
Data: https://drive.google.com/drive/folders/1cYFu_IA5WmKiDpTkB89WaOOAJSO79YnZ
Download the saved_HOSVD/ folder into the repository root, then call the relevant figure function
(comp_para_expo, show_v, base_fitting_show, V_trim_demo, …) from demo.py.
benchmarks/compare_eightterm.py compares the polynomial-fit
HOSVD model against the classic RFQ eight-term potential on the original check cell — for field
evaluations used in particle tracking. Inside the aperture:
| metric (vs. CST reference) | HOSVD polynomial fit | eight-term potential (best least-squares) |
|---|---|---|
| relative L2 error | 1.9 % | 5.4 % |
| evaluation throughput | 2.7 Mpoint/s | 1.0 Mpoint/s |
The HOSVD representation is both more accurate (a data-adaptive basis captures the real, non-ideal field) and faster to evaluate (pure polynomial arithmetic instead of modified-Bessel and trigonometric functions). The eight-term coefficients and its longitudinal/transverse wavenumbers are fitted to give it its best case; the throughput is NumPy-vectorized, so a compiled tracking code would widen the gap further. (Requires the data set above; numbers are from one run.)
pip install -r requirements.txt pytest
pytestIf you use this code, please cite the paper it accompanies:
@article{Du2022Multivariant,
title = {Multivariant spatial and geometric interpolation of field maps through model order reduction},
author = {Du, Xiaonan and Groening, Lars},
journal = {Physical Review Accelerators and Beams},
volume = {25},
issue = {12},
pages = {124601},
year = {2022},
doi = {10.1103/PhysRevAccelBeams.25.124601}
}This work builds on the foundational method introduced in:
@article{Du2018FieldMaps,
title = {Compression and noise reduction of field maps},
author = {Du, Xiaonan and Groening, Lars},
journal = {Physical Review Accelerators and Beams},
volume = {21},
issue = {8},
pages = {084601},
year = {2018},
doi = {10.1103/PhysRevAccelBeams.21.084601}
}(See CITATION.cff.)
Released under the MIT License.