[Paper Review] Transformed Snapshot Interpolation with High Resolution Transforms

Gerrit Welper|arXiv (Cornell University)|Jan 4, 2019

Model Reduction and Neural Networks42 references4 citations

TL;DR

This paper introduces high-resolution transforms for transformed snapshot interpolation (TSI) in parametric PDEs with jump singularities, using higher-order spatial discretizations and optimized gradient descent with Riesz lifting to achieve superior alignment of discontinuities. The method significantly improves convergence rates and reconstruction accuracy over low-resolution transforms, especially in problems with complex or moving shocks.

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.

Motivation & Objective

• Address the poor convergence of reduced order models in parametric PDEs with jump singularities due to low regularity.
• Overcome limitations of low-resolution transforms (e.g., constants, polynomials, Fourier modes) in existing TSI methods.
• Develop high-resolution spatial discretizations for transforms X(x, µ) that adapt to discontinuities and non-smooth parameter dependence.
• Design stable and convergent optimization strategies for high-resolution transforms using function space-aware gradient descent.
• Demonstrate improved reconstruction accuracy and convergence rates in 1D and 2D benchmark problems with moving shocks or kinks.

Proposed method

• Introduce a high-resolution spatial discretization of transforms X(x, µ) as solutions to ODEs, enabling smooth and accurate representation near discontinuities.
• Formulate the transform optimization as a constrained minimization of reconstruction error in the offline phase, using discrete snapshot data.
• Apply Riesz map lifting to the Fréchet derivative of the objective function to define a well-posed gradient descent in function space, avoiding stagnation.
• Implement two lifting strategies: Laplacian inversion and multi-level frame-based lifting, to compute gradients in appropriate dual spaces.
• Use fixed-point iterations and adaptive scaling in the transform update to stabilize convergence in non-smooth optimization landscapes.
• Discretize snapshots and transforms using piecewise bilinear finite elements on structured grids for 2D problems.

Experimental results

Research questions

• RQ1 Can high-resolution spatial discretizations of transforms significantly improve convergence rates in TSI for parametric PDEs with jumps?
• RQ2 How do function space-aware optimization strategies, such as Riesz lifting, outperform naive gradient descent in high-dimensional transform spaces?
• RQ3 What is the impact of transform resolution on the alignment of discontinuities and reconstruction error in problems with moving shocks?
• RQ4 How do different lifting methods (Laplacian vs. multi-level frame) affect the stability and convergence of the optimization process?
• RQ5 Can high-resolution transforms maintain accuracy even when the underlying solution has complex, non-smooth parameter dependence?

Key findings

• High-resolution transforms achieve significantly lower reconstruction errors than low-resolution counterparts, especially in problems with complex or moving discontinuities.
• Naive gradient descent fails to converge due to non-smoothness, stalling after a few steps with visually unchanged transforms.
• Riesz lifting via Laplacian inversion or multi-level frames enables stable and effective gradient descent in function space, leading to meaningful transform updates.
• In 2D problems with rotating and scaling ellipses, the method achieves accurate alignment of discontinuities with only 30–60 gradient descent steps.
• The final error oscillates around a minimum due to kinked gradients, which is mitigated by conservative step sizes, confirming the non-smooth nature of the objective.
• Theoretical analysis confirms that the transform-induced error is bounded by the total variation of the solution and a parameter-dependent stability factor S(µ, η), ensuring robustness.