A reduced-order model for advection-dominated problems based on Radon Cumulative Distribution Transform

TL;DR

This work tackles the challenge of advection-dominated problems where linear MOR struggles to capture traveling features. It introduces the Radon-Cumulative-Distribution Transform (RCDT), which combines the Radon transform with the Cumulative Distribution Transform to map high-dimensional transport phenomena into a space where linear POD-based compression and interpolation are effective. The authors implement RCDT in Python, analyze intrinsic discretisation errors, and demonstrate improved interpolation and compression on Gaussian pulses, multi-phase waves, and CFD data, while noting artefacts from the discrete transforms. The results suggest that transform-based MOR can significantly reduce required modes and improve predictive capability for transport-dominated flows, with future work aimed at reducing artefacts, extending to 3D, and exploring intrusive extensions.

Abstract

Problems with dominant advection, discontinuities, travelling features, or shape variations are widespread in computational mechanics. However, classical linear model reduction and interpolation methods typically fail to reproduce even relatively small parameter variations, making the reduced models inefficient and inaccurate. This work proposes a model order reduction approach based on the Radon-Cumulative-Distribution transform (RCDT). We demonstrate numerically that this non-linear transformation can overcome some limitations of standard proper orthogonal decomposition (POD) reconstructions and is capable of interpolating accurately some advection-dominated phenomena, although it may introduce artefacts due to the discrete forward and inverse transform. The method is tested on various test cases coming from both manufactured examples and fluid dynamics problems.

Figures (18)

• Figure 1: Snapshots of the manufactured example for $5$ different values on the input parameter in the physical space (on the left) and in the CDT space (on the right).
• Figure 2: Eigenvalue decay (left) and cumulative eigenvalues (right) of the POD computed for the manufactured example. Results are plotted both in the physical and in the CDT space.
• Figure 3: Left: input images of a smoothed circle defined by ones (top) or zeros (bottom) and vice versa for background. Centre: result of the RCDT followed by iRCDT on the input images. Right: difference between the input images and RCDT iRCDT results.
• Figure 4: Left: input images of a circle defined by ones (top) or zeros (bottom) and vice versa for background. Centre: result of the RCDT followed by iRCDT on the input images. Right: difference between stated the input images and RCDT iRCDT results.
• Figure 5: Left: input images of a circular edge ring defined by ones (top) or zeros (bottom) and vice versa for the background. Centre: result of RCDT followed by iRCDT on the input images. Right: difference between stated input images and RCDT iRCDT results.