[Paper Review] Real-time solution of computational problems using databases of parametric linear reduced-order models with arbitrary underlying meshes

David Amsallem, Radek Tezaur+1 more|arXiv (Cornell University)|Jun 23, 2015

Model Reduction and Neural Networks26 references3 citations

TL;DR

This paper proposes a real-time computational framework using a database of parametric linear reduced-order models (ROMs) with arbitrary underlying meshes. By enforcing consistency of reduced operators via congruent transformations and interpolating them on matrix manifolds, the method enables fast, accurate online solutions for parametric problems, demonstrated in real-time on a mobile device for aeroelastic flutter and inverse acoustic scattering.

ABSTRACT

A comprehensive approach for real-time computations using a database of parameterized linear reduced-order models (ROMs) is proposed. The method proceeds by sampling offline ROMs for specific values of the parameters and interpolating online the associated reduced operators. In the offline phase, a pre-processing step transforms the reduced operators into consistent sets of generalized coordinates prior to their interpolation. The present paper also introduces a consistency enforcement approach for models defined on arbitrary underlying meshes. In the online phase, the operators are interpolated on matrix manifolds. The proposed framework is illustrated on two realistic multi-physics problems: an inverse acoustic scattering problem around a submarine and flutter predictions for a wing-tank system. The second application is implemented on a mobile device, illustrating the capability of the proposed framework to operate in real-time.

Motivation & Objective

• Address the challenge of real-time solution of parametric linear PDEs with arbitrary, non-matching meshes in the offline and online phases.
• Overcome the inconsistency of reduced operators across different meshes by enforcing generalized coordinate consistency through congruent transformations.
• Enable robust and efficient online interpolation of reduced operators by mapping them to matrix manifolds to preserve structural properties.
• Demonstrate the framework’s capability in real-time applications, including mobile deployment for complex aeroelastic and acoustic problems.
• Provide a scalable, database-driven approach for high-fidelity simulations under parameter variation, suitable for design optimization and inverse problems.

Proposed method

• Offline: Pre-compute reduced-order models (ROMs) for selected parameter values using proper orthogonal decomposition (POD) and Galerkin projection.
• Enforce operator consistency across different meshes by transforming reduced operators into a common generalized coordinate system via congruent transformations.
• Store the consistent ROM operators in a database indexed by parameters for fast retrieval.
• Online: Interpolate the reduced operators on matrix manifolds (e.g., symmetric positive definite matrices) using tangent space methods to preserve structural properties like symmetry and definiteness.
• Use a Riemannian optimization framework to ensure interpolated operators remain on the manifold, maintaining stability and accuracy.
• Solve the online problem by projecting the parametric system onto the interpolated reduced space, enabling real-time predictions.

Experimental results

Research questions

• RQ1 How can reduced-order models with arbitrary underlying meshes be made consistent for interpolation in parametric model order reduction?
• RQ2 What is an effective interpolation strategy for reduced operators that preserves their intrinsic matrix structure (e.g., symmetry, definiteness) across different parameter values?
• RQ3 Can a database of pre-computed ROMs enable real-time solution of complex, multi-physics problems on lightweight devices like mobile phones?
• RQ4 How does the proposed method compare to standard ROMs in terms of accuracy and computational speed for inverse problems and aeroelasticity?
• RQ5 What is the role of matrix manifold interpolation in ensuring numerical stability and physical consistency of online predictions?

Key findings

• The proposed method enables real-time solution of parametric linear PDEs using a database of ROMs, even when the underlying high-fidelity models are defined on arbitrary, non-matching meshes.
• The consistency enforcement via congruent transformations successfully aligns reduced operators across different meshes, enabling stable interpolation.
• Interpolation on matrix manifolds preserves key structural properties (e.g., symmetry, positive definiteness) of the reduced operators, ensuring numerical stability and accuracy.
• The framework achieves real-time performance on a mobile device for flutter prediction in a wing-tank system across subsonic to supersonic regimes.
• The method successfully solves an inverse acoustic scattering problem around a submarine, demonstrating robustness and accuracy in parameter estimation.
• The online interpolation cost is negligible compared to full-order model solves, enabling thousands of predictions per second on standard mobile hardware.