  This work proposes a systematic model reduction approach based on rank
adaptive tensor recovery for partial differential equation (PDE) models with
high-dimensional random parameters. Since the standard outputs of interest of
these models are discrete solutions on given physical grids which are
high-dimensional, we use kernel principal component analysis to construct
stochastic collocation approximations in reduced dimensional spaces of the
outputs. To address the issue of high-dimensional random inputs, we develop a
new efficient rank adaptive tensor recovery approach to compute the collocation
coefficients. Novel efficient initialization strategies for non-convex
optimization problems involved in tensor recovery are also developed in this
work. We present a general mathematical framework of our overall model
reduction approach, analyze its stability, and demonstrate its efficiency with
numerical experiments.
