Nonlinear model reduction via a locally weighted POD method

In this article, we propose a new approach for model reduction of parameterized partial differential equations (PDEs) by a locally weighted proper orthogonal decomposition (LWPOD) method. 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, LWPOD approximates the original system by multiple local reduced bases. Each local reduced basis is generated by the singular value decomposition of a weighted snapshot matrix. Compared with global model reduction methods, such as the classical proper orthogonal decomposition, LWPOD can yield more accurate solutions with a fixed subspace dimension. As another contribution, we combine LWPOD 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. Copyright © 2016 John Wiley & Sons, Ltd.