Research On The Weighted Compression Sampling Algorithm With Arbitrary Random Input Partial Differential Equations

One of the persistent problems in Uncertainty Quantification(UQ)is partial differ-ential equations with random inputs.The major challenges for PDE with random inputs are the high-dimensionality of the stochastic inputs and the lack of explicit knowledge of their distributions.The main purpose of this thesis is to establish an efficient precondi-tioned l~1-minimization method for PDE with arbitrary random inputs.We first use the statistical moments of the inputs random variables to construct the arbitrary polynomial chaos with the procedure given by[Ahlfeld et al.(2016),[2]].After that we propose a preconditioned?~1-minimization method to recover the coefficients of polynomial chaos expansions and compute the statistical moments of the statistical response.This approach is different from the sparse grid collocation method.We then samples with respect to the equilibrium measure.This sampling strategy is independent of the random inputs and can be used for both bounded and unbounded domains.The stochastic collocation method based on compressive sampling need much less points compared with the cardinality of the polynomial space and the computational efficien-cy is greatly improved.In the numerical experimental part,we first consider using the proposed method to approximate some analytic functions.The numerical results show that sampling with the equilibrium measure is superior to Monte Carlo(MC)sampling of the density of orthogonality for low dimensionality when the polynomial degree is high.For high-dimensional problems,the proposed sampling strategy is better than MC as the number of points increase.We finally consider Kirchhoff plate bending problem with stochastic inputs,where the Young’s Modulus is modeled as a stochastic process.On the collocation points,the deterministic plate problem is solved by the non-conforming Morley element method.The simulation results show the efficiency of our approach.