A new algorithm for high-dimensional uncertainty quantification problems based on dimension-adaptive and reduced basis methods

In this work we develop an adaptive and reduced computational framework based on dimension-adaptive hierarchical approximation and reduced basis method for solving high-dimensional uncertainty quantification (UQ) problems. In order to tackle the computational challenge of “curse-ofdimensionality” commonly faced by these problems, we employ a dimension-adaptive tensor-product algorithm [29] and propose a verified version to enable effective removal of the stagnation phenomenon besides automatically detecting the importance and interaction of different dimensions. To reduce the heavy computational cost of UQ problems modelled by partial differential equations (PDEs), we adopt a weighted reduced basis method [18] and develop an adaptive greedy algorithm in combination of the previous verified algorithm for efficient construction of an accurate reduced basis approximation space. The effectivity, efficiency and accuracy of this computational framework are demonstrated and compared to several other existing techniques by a variety of classical numerical examples.