A Dirac-Frenkel-Onsager principle: Instantaneous residual minimization with gauge momentum for nonlinear parametrizations of PDE solutions
Matteo Raviola, Benjamin Peherstorfer
TLDR
This paper introduces a Dirac-Frenkel-Onsager principle that uses gauge momentum to resolve ill-conditioning in PDE solution parametrizations.
Key contributions
- Resolves ill-conditioning and non-unique parameter dynamics in Dirac-Frenkel methods for PDE solutions.
- Interprets non-uniqueness as gauge freedom, using nullspace directions to select optimal parameter velocities.
- Introduces a "gauge momentum" history variable, based on Onsager's principle, injected only along nullspace.
- The new Dirac-Frenkel-Onsager dynamics enhance robustness and temporal smoothness without introducing bias.
Why it matters
This paper offers a novel approach to stabilize numerical methods for PDEs, particularly those with nonlinear parametrizations. By leveraging gauge freedom and a momentum-like variable, it enhances robustness in challenging regimes without compromising accuracy, which is crucial for reliable simulations.
Original Abstract
Dirac-Frenkel instantaneous residual minimization evolves nonlinear parametrizations of PDE solutions in time, but ill-conditioning can render the parameter dynamics non-unique. We interpret this non-uniqueness as a gauge freedom: nullspace directions that leave the time derivative unchanged can be used to select better-conditioned parameter velocities. Building on Onsager's minimum-dissipation principle, we introduce a history variable -- interpretable as momentum -- and inject it only along the nullspace directions. The resulting Dirac-Frenkel-Onsager dynamics preserve instantaneous residual minimization, in contrast to standard regularization that can introduce bias, while promoting temporally smooth parameter evolutions. Examples demonstrate that the approach leads to increased robustness in singular and near-singular regimes.
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