Data-driven Multi-fidelity Sparse Polynomial Approximation And Its Application In Uncertainty Quantification

The field of Uncertainty Quantification（UQ）has attracted wide attention in recent years.The foundamental problem of Uncertainty Quantification is how to quantify the effect of random inputs to the stochastic response of the system.The polynomial chaos method（g PC）is an efficient method widely used in UQ.But the classical g PC method assume that the distribution of random inputs are known.On the other hand,as the number of random inputs increases,the computational complexity of polynomial chaos expansion increases exponentially.It is of great significance to establish a data-driven efficient polynomial chaos method for UQ problems with high-dimensional random inputs.In this thesis,we combine the transformed l₁（TL1）minimization and multi-fidelity methods to develop a data-driven efficient polynomial chaos expansion method.We first review a data-driven basis construction for arbitrary probability measure via Gram-Schmidt orthogonalization process.We then use TL1 minimization to recover the coefficients of polynomial expansion with low-fidelity computations.To obtain the multi-fidelity polynomial chaos expansion,the high-fidelity computations are then employed to correct a subset of recovered coefficients.We define a threshold to select the subsets of coefficients that need to be corrected.We perform two kinds of numerical experiments namely,the Ackley function approximation and the stochastic elliptic equation.The numerical results show that the TL1 minimization multi-fidelity polynomial expansion can get accurate approximation with much lower cost than the classical g PC with least squares method and the TL1 minimization method.The number of samples is greatly reduced for the proposed method.Thus,it is observed that the method developed here provide an efficient approach for UQ.