[Paper Review] $h$ and $hp$-adaptive Interpolation by Transformed Snapshots for Parametric and Stochastic Hyperbolic PDEs

Gerrit Welper|arXiv (Cornell University)|Oct 31, 2017

Probabilistic and Robust Engineering Design35 references15 citations

TL;DR

This paper proposes $h$ and $hp$-adaptive interpolation using transformed snapshots to address parametric and stochastic hyperbolic PDEs with shock topology changes. By locally refining parameter space and leveraging coordinate-aligned transforms, the method achieves exponential convergence rates despite discontinuities, outperforming standard reduced basis and POD methods in problems with evolving shock structures.

ABSTRACT

The numerical approximation of solutions of parametric or stochastic hyperbolic PDEs is still a serious challenge. Because of shock singularities, most methods from the elliptic and parabolic regime, such as reduced basis methods, POD or polynomial chaos expansions, show a poor performance. Recently, Welper [Interpolation of functions with parameter dependent jumps by transformed snapshots. SIAM Journal on Scientific Computing, 39(4):A1225-A1250, 2017] introduced a new approximation method, based on the alignment of the jump sets of the snapshots. If the structure of the jump sets changes with parameter, this assumption is too restrictive. However, these changes are typically local in parameter space, so that in this paper, we explore $h$ and $hp$-adaptive methods to resolve them. Since local refinements do not scale to high dimensions, we introduce an alternative "tensorized" adaption method.

Motivation & Objective

• Address the poor performance of traditional reduced basis, POD, and polynomial chaos methods on parametric and stochastic hyperbolic PDEs due to shock discontinuities.
• Overcome the limitation of transformed snapshot interpolation (TSI), which assumes fixed jump set topology across parameters.
• Develop $h$ and $hp$-adaptive refinement strategies in parameter space to resolve local topology changes in shock structures.
• Introduce a tensorized adaptation framework to mitigate scalability issues in high-dimensional parameter spaces.
• Demonstrate exponential convergence rates for model problems and validate with numerical experiments on benchmark PDEs.

Proposed method

• Apply transformed snapshot interpolation (TSI) with parameter-dependent domain transforms $\phi(\mu,\eta)(x)$ to align jump discontinuities across parameter values.
• Use $h$- and $hp$-adaptive refinement in parameter space to resolve local regions where shock topology changes occur, such as shock collisions or transitions from shocks to rarefaction waves.
• Construct a tensorized adaptation strategy by aligning local refinements with coordinate axes, enabling scalability in high-dimensional parameter spaces.
• Employ Lagrange interpolation in parameter space using transformed snapshots: $ u(x,\mu) \approx \sum_{\eta \in \mathcal{P}_n} \ell_\eta(\mu) \, u(\phi(\mu,\eta)(x), \eta) $.
• Use finite element-based transforms $\phi(\mu,\eta)(x) = x + \varphi(\mu,\eta)(x)$ with degree-two elements and boundary conditions $\varphi \cdot \nu = 0$ to ensure diffeomorphism and domain preservation.
• Apply coarse quadrature and polynomial approximations of degree two for spatial resolution in transforms and solution evaluation.

Experimental results

Research questions

• RQ1 Can $h$ and $hp$-adaptive refinement in parameter space effectively resolve local topology changes in shock structures of parametric hyperbolic PDEs?
• RQ2 Does a tensorized adaptation strategy based on coordinate-aligned transforms improve scalability in high-dimensional parameter spaces compared to standard local refinement?
• RQ3 What convergence rates can be achieved by the proposed adaptive TSI method for problems with evolving shock topologies?
• RQ4 How does the method compare to standard reduced basis and POD methods in terms of error and robustness for problems with discontinuous solutions?
• RQ5 Can the method capture critical transitions such as shock collisions and shock-to-rarefaction wave transitions with minimal refinement?

Key findings

• The method achieves exponential convergence rates for model problems with evolving shock topologies, demonstrating superior performance over standard reduced basis and POD methods.
• Numerical experiments on the inviscid Burgers' equation with initial conditions containing two jumps show that local refinements capture shock collisions and rarefaction wave transitions effectively.
• For $\mu = -0.5$, refinement is concentrated at the time of shock collision; for $\mu = 0.5$, refinement occurs at $t=0$ where the initial jump transitions to a rarefaction wave.
• The overall error in the test case is already close to the error level of snapshots with spatial resolution $h = 0.01$, indicating high accuracy with minimal refinement.
• In the forward-facing step flow example, the method captures the formation of a new shock at higher Mach numbers, with refinement localized near the transition point.
• The method avoids stair-casing effects in solutions and maintains errors approximately twice the spatial mesh size ($h = 0.025$), indicating robustness and accuracy.