基于任意多项式逼近的不确定量化问题的压缩感知算法的研究
发布时间：2018-11-22 16:11
【摘要】：近些年来,不确定量化问题的计算受到广泛关注.如何量化系统的随机输入对系统输出的影响是不确定性量化的核心问题.随机参数空间的广义正交多项式(generalized Polynomial Chaos)逼近是一种有效的方法,被成功地应用到不确定量化问题的计算中.然而,在很多情况下,我们获取的随机参数是一些离散的值,从建模角度来看,这也就意味着采用离散测度更适合解决我们的问题.因此,在本文中,我们着重处理当随机参数服从离散概率测度时,利用基于离散测度下的任意正交多项式(a PC)对随机模型的输出进行逼近,期望为该问题提供快速有效的计算方法.我们首先介绍了关于离散测度下正交的任意多项式的生成方法,包括Nowak方法、Stieltjes方法和Lanczos方法.接着我们以关于离散测度正交的任意多项式为基函数,利用基于非凸压缩感知的随机配置方法来处理一些常见的带有离散型随机输入的不确定性量化问题.具体地说,我们给出了具有随机输入的偏微分方程的基于任意正交多项式逼近的稀疏网格随机配置方法;研究了利用任意多项式展开求解具有随机输入偏微分方程的几类非凸压缩感知算法的随机配置方法;给出了smoothed-log优化算法、smoothed-l_q优化算法和l_1-l_2算法求解多项式展开的稀疏逼近的随机配置方法的基本框架.在数值实验部分,我们首先考察重构稀疏多项式函数,通过计算重构成功率比较了基于三种不同的非凸压缩感知求解器的表现效果;然后我们考虑函数的稀疏多项式逼近,通过计算其均方根误差来说明以a PC为基函数,基于非凸的压缩感知随机配置方法可以有效地逼近目标函数,这为后续求解随机微分方程的随机响应的逼近提供了基础.接着,我们考虑带有随机输入的ODE的求解.最后,通过具有随机源项的分数阶扩散方程的数值模拟,比较了基于稀疏网格的随机配置方法和基于非凸压缩感知的随机配置方法.基于非凸压缩感知的随机配置方法在同样的精度要求下,比稀疏网格用的样本点个数少很多,大大提高了计算效率.所有的计算结果表明,给定服从任意离散测度的随机变量,通过基于非凸压缩感知的随机配置方法,可以有效地求解系统的随机响应在任意正交多项式下的逼近.同时,在我们所选定的三种非凸的压缩感知求解器中(smoothed-log优化算法、smoothed-l_q优化算法和l_1-l_2算法),smoothed-Log在所有的数值例子中呈现较大的优势,这对于我们利用基于非凸压缩感知的随机配置方法求解大规模随机问题时关于求解器的选取具有参考作用.
[Abstract]:In recent years, the calculation of uncertain quantization problems has been paid more and more attention. How to quantify the effect of random input on system output is the core problem of uncertainty quantization. Generalized orthogonal polynomial (generalized Polynomial Chaos) approximation in random parameter spaces is an effective method, which has been successfully applied to the computation of uncertain quantization problems. However, in many cases, the random parameters we get are discrete values, which means that the discrete measure is more suitable to solve our problem from the point of view of modeling. Therefore, in this paper, we focus on the approximation of the output of the stochastic model by any orthogonal polynomial (a PC) based on the discrete measure when the random parameter is subjected to the discrete probability measure. It is expected to provide a fast and effective calculation method for this problem. We first introduce the methods of generating orthogonal arbitrary polynomials under discrete measure, including Nowak method, Stieltjes method and Lanczos method. Then we use the random collocation method based on the nonconvex contractive perception to deal with some common uncertain quantization problems with discrete random input by taking any polynomial of orthogonal discrete measure as the basis function. Specifically, we give a sparse grid random collocation method based on arbitrary orthogonal polynomial approximation for partial differential equations with random input. A random collocation method for solving some kinds of nonconvex contractive perception algorithms with stochastic input partial differential equations by arbitrary polynomial expansion is studied. The basic framework of smoothed-log optimization algorithm, smoothed-l_q optimization algorithm and l_1-l_2 algorithm for solving the sparse approximation of polynomial expansion is presented. In the part of numerical experiment, we first investigate the sparse polynomial function, and compare the performance of three kinds of non-convex compression perceptual solver based on three kinds of non-convex compression perceptual solver by calculating the success rate of reconstruction. Then we consider the sparse polynomial approximation of the function. By calculating the root mean square error of the function, we show that a PC is used as the basis function, and the nonconvex contractive perceptual random collocation method can effectively approximate the objective function. This provides a basis for the approximation of stochastic responses to the subsequent solutions of stochastic differential equations. Then, we consider the solution of ODE with random input. Finally, through the numerical simulation of fractional diffusion equation with random source term, we compare the random collocation method based on sparse mesh with the random collocation method based on non-convex compression perception. The random collocation method based on non-convex compression sensing is much less than the number of sample points used in sparse mesh under the same precision requirement, which greatly improves the computational efficiency. All the results show that for a given random variable with arbitrary discrete measure, the approximation of the random response of the system under any orthogonal polynomial can be effectively solved by a random collocation method based on nonconvex contractive perception. At the same time, in the three non-convex compression-aware solvers we selected (smoothed-log optimization algorithm, smoothed-l_q optimization algorithm and l_1-l_2 algorithm), smoothed-Log shows great advantages in all numerical examples. This provides a reference for the selection of solvers for solving large-scale stochastic problems using the stochastic collocation method based on nonconvex compression perception.
【学位授予单位】：上海师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241
本文编号：2349847
[Abstract]:In recent years, the calculation of uncertain quantization problems has been paid more and more attention. How to quantify the effect of random input on system output is the core problem of uncertainty quantization. Generalized orthogonal polynomial (generalized Polynomial Chaos) approximation in random parameter spaces is an effective method, which has been successfully applied to the computation of uncertain quantization problems. However, in many cases, the random parameters we get are discrete values, which means that the discrete measure is more suitable to solve our problem from the point of view of modeling. Therefore, in this paper, we focus on the approximation of the output of the stochastic model by any orthogonal polynomial (a PC) based on the discrete measure when the random parameter is subjected to the discrete probability measure. It is expected to provide a fast and effective calculation method for this problem. We first introduce the methods of generating orthogonal arbitrary polynomials under discrete measure, including Nowak method, Stieltjes method and Lanczos method. Then we use the random collocation method based on the nonconvex contractive perception to deal with some common uncertain quantization problems with discrete random input by taking any polynomial of orthogonal discrete measure as the basis function. Specifically, we give a sparse grid random collocation method based on arbitrary orthogonal polynomial approximation for partial differential equations with random input. A random collocation method for solving some kinds of nonconvex contractive perception algorithms with stochastic input partial differential equations by arbitrary polynomial expansion is studied. The basic framework of smoothed-log optimization algorithm, smoothed-l_q optimization algorithm and l_1-l_2 algorithm for solving the sparse approximation of polynomial expansion is presented. In the part of numerical experiment, we first investigate the sparse polynomial function, and compare the performance of three kinds of non-convex compression perceptual solver based on three kinds of non-convex compression perceptual solver by calculating the success rate of reconstruction. Then we consider the sparse polynomial approximation of the function. By calculating the root mean square error of the function, we show that a PC is used as the basis function, and the nonconvex contractive perceptual random collocation method can effectively approximate the objective function. This provides a basis for the approximation of stochastic responses to the subsequent solutions of stochastic differential equations. Then, we consider the solution of ODE with random input. Finally, through the numerical simulation of fractional diffusion equation with random source term, we compare the random collocation method based on sparse mesh with the random collocation method based on non-convex compression perception. The random collocation method based on non-convex compression sensing is much less than the number of sample points used in sparse mesh under the same precision requirement, which greatly improves the computational efficiency. All the results show that for a given random variable with arbitrary discrete measure, the approximation of the random response of the system under any orthogonal polynomial can be effectively solved by a random collocation method based on nonconvex contractive perception. At the same time, in the three non-convex compression-aware solvers we selected (smoothed-log optimization algorithm, smoothed-l_q optimization algorithm and l_1-l_2 algorithm), smoothed-Log shows great advantages in all numerical examples. This provides a reference for the selection of solvers for solving large-scale stochastic problems using the stochastic collocation method based on nonconvex compression perception.
【学位授予单位】：上海师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241
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