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Transformed Snapshot Interpolation with High Resolution Transforms

G. Welper

TL;DR

This work addresses the challenge of jump-induced non-smoothness in parametric PDEs by developing high-resolution, ODE-built transforms $X(\eta;\mu,x)$ within Transformed Snapshot Interpolation (TSI). It establishes a discretization framework for the transforms, proves existence/uniqueness under mild conditions, and analyzes stability under transform perturbations, introducing a transport-field optimization that can be regularized via Laplace smoothing or multilevel frames. The authors demonstrate through 1D and 2D experiments that high-resolution transforms significantly improve jump alignment and reconstruction near topology changes, with smoothing strategies enabling practical optimization and stability. The results suggest that high-resolution, optimally learned transforms extend the applicability and efficiency of TSI for parametric PDEs with shocks and complex jump structures, offering a scalable path to accurate reduced-order representations in challenging regimes.

Abstract

In the last few years, several methods have been developed to deal with jump singularities in parametric or stochastic hyperbolic PDEs. They typically use some alignment of the jump-sets in physical space before performing well established reduced order modelling techniques such as reduced basis methods, POD or simply interpolation. In the current literature, the transforms are typically of low resolution in space, mostly low order polynomials, Fourier modes or constant shifts. In this paper, we discuss higher resolution transforms in one of the recent methods, the transformed snapshot interpolation (TSI). We introduce a new discretization of the transforms with an appropriate behaviour near singularities and consider their numerical computation via an optimization procedure.

Transformed Snapshot Interpolation with High Resolution Transforms

TL;DR

This work addresses the challenge of jump-induced non-smoothness in parametric PDEs by developing high-resolution, ODE-built transforms within Transformed Snapshot Interpolation (TSI). It establishes a discretization framework for the transforms, proves existence/uniqueness under mild conditions, and analyzes stability under transform perturbations, introducing a transport-field optimization that can be regularized via Laplace smoothing or multilevel frames. The authors demonstrate through 1D and 2D experiments that high-resolution transforms significantly improve jump alignment and reconstruction near topology changes, with smoothing strategies enabling practical optimization and stability. The results suggest that high-resolution, optimally learned transforms extend the applicability and efficiency of TSI for parametric PDEs with shocks and complex jump structures, offering a scalable path to accurate reduced-order representations in challenging regimes.

Abstract

In the last few years, several methods have been developed to deal with jump singularities in parametric or stochastic hyperbolic PDEs. They typically use some alignment of the jump-sets in physical space before performing well established reduced order modelling techniques such as reduced basis methods, POD or simply interpolation. In the current literature, the transforms are typically of low resolution in space, mostly low order polynomials, Fourier modes or constant shifts. In this paper, we discuss higher resolution transforms in one of the recent methods, the transformed snapshot interpolation (TSI). We introduce a new discretization of the transforms with an appropriate behaviour near singularities and consider their numerical computation via an optimization procedure.

Paper Structure

This paper contains 14 sections, 3 theorems, 56 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Transformed Snapshot Interpolation
  3. An Example Problem
  4. Transforms as solutions of ODEs
  5. A note about stability
  6. Optimizing Singular Transforms
  7. Difficulties with the optimization of the transform
  8. Optimization in Banach Spaces
  9. Numerical Experiments
  10. Testing the Optimizers
  11. Testing Transform Resolutions
  12. Testing Transform Orders
  13. $2d$ Example
  14. Proof of Proposition \ref{['prop:stab']}

Key Result

Proposition 4.1

Let $I$ be an interval that contains the interpolation points $\mathcal{P}_n$ and the initial point $\mu$.

Figures (6)

  • Figure 1: Blue: Jump set of the example function \ref{['eq:example-func-1']}, \ref{['eq:example-func-2']} in the $x-\mu$-plane. Dotted lines: Curves $\eta \to X(\eta; \mu, x)$ defined in \ref{['eq:example-transform']}. Black: Inappropriate transform $\eta \to X(\eta; \mu, x)$ crossing a jump.
  • Figure 2: Results for the $1d$ Example in Section \ref{['sec:optimizers-1d']}. Left: Snapshots (gray), exact solution (solid red) and reconstructions at $\mu = 0$: TSI with Laplace smoothing (cyan, dashed), multilevel smoothing (violet, dotted) and a TSI with low resolution transform \ref{['eq:transform-interpol']} (black, dashed). Right: corresponding gradient descent errors with the same color coding. In addition the figure shows the errors for high resolution transforms without smoothing (olive dotted).
  • Figure 3: Results for the Example in Section \ref{['sec:collision-1d']}. Left: Snapshots (gray), exact function at $\mathcal{P}_T$ (red) and TSI with low resolution (black dashed) and high resolution with Laplace smoothing (cyan, dashed) and multilevel smoothing (violet, dotted). Right: The corresponding transform with the same color coding. The red lines are the locations of the jumps of the training snapshot (vertically) and the target snapshots (horizontally).
  • Figure 4: Gradient descent errors for the examples in Sections \ref{['sec:collision-1d']} and \ref{['sec:numerical-2d-example']}. The left picture uses the color coding of Figure \ref{['fig:1d-solutions']}.
  • Figure 5: Results for the Example in Section \ref{['sec:curve-1d']}. Left: Snapshots (gray), exact function at $\mathcal{P}_T$ (red) and TSI reconstruction with linear in $\eta$ (black dashed) and quadratic (cyan dashed) transforms. Right: The corresponding gradient descent errors.
  • ...and 1 more figures

Theorems & Definitions (6)

  • Proposition 4.1
  • proof
  • Remark 4.2
  • Corollary 4.3
  • proof
  • Proposition 5.1