Nonlinear Model Reduction via an Adaptive Weighting of Snapshots

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.