Nonlinear Model Reduction via an Adaptive Weighting of Snapshots
For computational scientists solving parameterized PDEs, this method offers improved accuracy over global POD with the same subspace dimension, though it is an incremental improvement combining existing ideas.
The paper proposes an adaptive reduced basis method for nonlinear parameterized PDEs that uses multiple localized subspaces from weighted snapshots, achieving higher accuracy than classical POD for a fixed subspace dimension and demonstrating large speedups with good accuracy on elliptic and parabolic examples.
In this paper, we propose a new approach to model reduction of parameterized partial differential equations (PDEs) based on the concept of adaptive reduced bases. The presented approach is particularly suited for large-scale nonlinear systems characterized by parameter variations. Instead of using a global basis to construct a global reduced model, the proposed method approximates the original system by multiple lower-dimensional subspaces. Each localized reduced basis is generated by the SVD of a weighted snapshot ensemble; here, each weighting coefficient is a function of the input parameter. Compared with a global model reduction method, such as the classical POD, the adaptive model reduction method could yield a more accurate solution with a fixed subspace dimension. Moreover, we combine the adaptive reduced model with the chord iteration to solve elliptic PDEs in a computationally efficient fashion. The potential of the method for achieving large speedups, while maintaining good accuracy, is demonstrated for both elliptic and parabolic PDEs in a few numerical examples.