發(fā)布時(shí)間：2018-11-22 16:11

【摘要】：近些年來(lái),不確定量化問(wèn)題的計(jì)算受到廣泛關(guān)注.如何量化系統(tǒng)的隨機(jī)輸入對(duì)系統(tǒng)輸出的影響是不確定性量化的核心問(wèn)題.隨機(jī)參數(shù)空間的廣義正交多項(xiàng)式(generalized Polynomial Chaos)逼近是一種有效的方法,被成功地應(yīng)用到不確定量化問(wèn)題的計(jì)算中.然而,在很多情況下,我們獲取的隨機(jī)參數(shù)是一些離散的值,從建模角度來(lái)看,這也就意味著采用離散測(cè)度更適合解決我們的問(wèn)題.因此,在本文中,我們著重處理當(dāng)隨機(jī)參數(shù)服從離散概率測(cè)度時(shí),利用基于離散測(cè)度下的任意正交多項(xiàng)式(a PC)對(duì)隨機(jī)模型的輸出進(jìn)行逼近,期望為該問(wèn)題提供快速有效的計(jì)算方法.我們首先介紹了關(guān)于離散測(cè)度下正交的任意多項(xiàng)式的生成方法,包括Nowak方法、Stieltjes方法和Lanczos方法.接著我們以關(guān)于離散測(cè)度正交的任意多項(xiàng)式為基函數(shù),利用基于非凸壓縮感知的隨機(jī)配置方法來(lái)處理一些常見(jiàn)的帶有離散型隨機(jī)輸入的不確定性量化問(wèn)題.具體地說(shuō),我們給出了具有隨機(jī)輸入的偏微分方程的基于任意正交多項(xiàng)式逼近的稀疏網(wǎng)格隨機(jī)配置方法;研究了利用任意多項(xiàng)式展開(kāi)求解具有隨機(jī)輸入偏微分方程的幾類非凸壓縮感知算法的隨機(jī)配置方法;給出了smoothed-log優(yōu)化算法、smoothed-l_q優(yōu)化算法和l_1-l_2算法求解多項(xiàng)式展開(kāi)的稀疏逼近的隨機(jī)配置方法的基本框架.在數(shù)值實(shí)驗(yàn)部分,我們首先考察重構(gòu)稀疏多項(xiàng)式函數(shù),通過(guò)計(jì)算重構(gòu)成功率比較了基于三種不同的非凸壓縮感知求解器的表現(xiàn)效果;然后我們考慮函數(shù)的稀疏多項(xiàng)式逼近,通過(guò)計(jì)算其均方根誤差來(lái)說(shuō)明以a PC為基函數(shù),基于非凸的壓縮感知隨機(jī)配置方法可以有效地逼近目標(biāo)函數(shù),這為后續(xù)求解隨機(jī)微分方程的隨機(jī)響應(yīng)的逼近提供了基礎(chǔ).接著,我們考慮帶有隨機(jī)輸入的ODE的求解.最后,通過(guò)具有隨機(jī)源項(xiàng)的分?jǐn)?shù)階擴(kuò)散方程的數(shù)值模擬,比較了基于稀疏網(wǎng)格的隨機(jī)配置方法和基于非凸壓縮感知的隨機(jī)配置方法.基于非凸壓縮感知的隨機(jī)配置方法在同樣的精度要求下,比稀疏網(wǎng)格用的樣本點(diǎn)個(gè)數(shù)少很多,大大提高了計(jì)算效率.所有的計(jì)算結(jié)果表明,給定服從任意離散測(cè)度的隨機(jī)變量,通過(guò)基于非凸壓縮感知的隨機(jī)配置方法,可以有效地求解系統(tǒng)的隨機(jī)響應(yīng)在任意正交多項(xiàng)式下的逼近.同時(shí),在我們所選定的三種非凸的壓縮感知求解器中(smoothed-log優(yōu)化算法、smoothed-l_q優(yōu)化算法和l_1-l_2算法),smoothed-Log在所有的數(shù)值例子中呈現(xiàn)較大的優(yōu)勢(shì),這對(duì)于我們利用基于非凸壓縮感知的隨機(jī)配置方法求解大規(guī)模隨機(jī)問(wèn)題時(shí)關(guān)于求解器的選取具有參考作用.
[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.
【學(xué)位授予單位】：上海師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O241

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