  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.
