Model reduction of traveling-wave problems via Radon cumulative distribution transform

Traveling-wave problems, due to their sizable Kolmogorov $n$-width, have brought critical challenges to conventional model reduction techniques. This study aims to provide insights into this problem by exploiting the Radon cumulative distribution transform (R-CDT) [Kolouri, Park, and Rohde, IEEE Trans. Image Process. 25, 920 (2016)], which has emerged in the sector of computer vision science. The core lies in the unique property of the nonlinear invertible R-CDT that renders both traveling and scaling components into amplitude modulations. In contrast to the physical space, a substantial model reduction is achieved in the R-CDT space while sustaining high accuracy. The method is parameter-free and data-driven and lends itself to problems regardless of the dimensions or boundary conditions. Examples start with a one-dimensional Burgers' equation subject to nonperiodic boundary conditions, where both traveling and diffusion dominate the physics. In higher-dimensional problems, we show the model reduction of traveling Gaussian solitons. In addition to foreseeable motions, the proposed method is capable of handling random traveling with a nondifferentiable trajectory.

Figure 1

Traveling Gaussian soliton $f(x,t)$ (a) and its CDT transform $\widehat{f}(x,t)$ (b). Solid lines are colored by time traveled, while the dashed line indicates the template. The pseudocolor plot of the correlation matrix ${\mathit{R}}_{ij}=\langle f(x,t_i),f(x,t_j)\rangle$ in physical (c) and CDT (d) space.

Figure 2

Flow chart of the proposed method.

Figure 3

Model reduction of the viscous Burgers' equation. The evolution of the wave profiles is visualized in panel (a), where the lines are colored by the time traveled. We show the original signal and its two-mode ROM in the $x\text{−}t$ frame in panels (b) and (c). With the proposed method, the physical signal is transformed into the CDT space (d), where the two-mode ROM is built (e), and this is followed by an iCDT to recover the data in physical space (f).

Figure 4

Comparison of the singular value decay (a) and error of reduced order models (b) in the physical and CDT space.

Figure 5

The procedure to build a reduced-order model via R-CDT. The original signal (a) is transformed into the R-CDT space by sequentially applying the Radon transform (b) and CDT (c). The four-mode ROM is built in the R-CDT space (d); this is followed by iCDT (e) and iRadon to recover the physical space (f). For comparison, the 25-mode and 50-mode ROMs built in the physical space are shown in panels (g) and (h). This figure shows a snapshot corresponding to $t=10$; a video covering the whole time horizon is available in the Supplemental Material [20].

Figure 6

Comparison of the singular value decay (a) and error of reduced order models (b) in the physical and R-CDT space.

Figure 7

The 3D traveling Gaussian soliton visualized with an isosurface of $f(x,y,z,t)=0.05$ (a). Comparison of the singular value drop in the physical and R-CDT space (b). The reconstructed data from four-mode ROM built in the physical (c) and R-CDT (d) spaces, respectively. Shadows in panels (a), (c), and (d) indicate the projection of isosurfaces onto three coordinate planes. A movie showing the temporal evolution is available in the Supplemental Material [20].