Research On Compressed Sensing Algorithm For Uncertain Quantization Problems Based On Arbitrary Polynomial Approximation

In recent years,uncertainty quantification has attracted much attention.In the field of UQ,one of the fundamental problem is how to quantify the impact of the stochastic random inputs to the stochastic response.Generalized polynomial chaos expansion is an efficient method and has been widely used in the computation of uncertainty quantification.However,in many applications,the random parameters are only observed at discrete values,which implies that a discrete probability measure is more appropriate from the modeling point of view.In this paper,we mainly consider the random parameters subject to a discrete probability measure,and we use the corresponding arbitrary orthogonal polynomials(aPC)with respect to the discrete measure to approximate the stochastic response function to a stochastic differential equation with random inputs.We introduce the methods of generating arbitrary orthonormal polynomial with respect to discrete measure,including Nowak method,Stieltjes method and Lanczos method.Then,just like the most works of generalized polynomial chaos(gPC),based on the arbitrary polynomial chaos with respect to any discrete measure,we combine the aPC with stochastic collocation method to study the model problems which has random inputs subject to discrete measure.To be specific,we presented the stochastic collocation method based on sparse grid and use the arbitrary othonormal polynomials expansion to approximate the partial differential equations with random input.Futhermore,we study the stochastic collocation method via a class of nonconvex compressed sensing algorithms to construct the arbitrary polynomial expansion approximation of a stochastic differential equations with random inputs.In the numerical experimental part,we consider the reconstruction of sparse arbitrary polynomial function.By comparing the success rate of reconstruction,we illustrate the performance of three different kinds of non-convex compressed sensing solutions.Then we consider using the orthogonal polynomial chaos to approximate some analytic functions,by calculating the root mean square error between the approximation and the exact solution,we obtain that via the stochastic collocation method based on non convex compressed sensing can approximate the objective function effectively.This provides the basis for the subsequent section of study the approximation solution of the stochastic response of stochastic differential equations.Then,we consider the solution of the ODE equation with random input.Finally,through the numerical simulation of the fractional diffusion equation with random source,we compare the different results of the stochastic collocation method based on sparse grid and stochastic collocation methods based on nonconvex compressed sensing.We find that in the same precision,the stochastic collocation method based on non convex compressed sensing method need much less points than the sparse grid,and the computational efficiency is greatly improved.All the calculated results show that the random response of a stochastic system can be effectively approximated by the stochastic collocation method based on nonconvex compressed sensing method.And among the three kinds nonconvex optimization algoritms,the smoothed-Log minimization problem performs the best.This provides us an optimal choice when using the stochastic collocation methods to solving complex system with random inputs.