基于PC-EnKF方法对耦合Burgers方程降阶模型的订正

发布时间：2018-05-30 04:41

本文选题：不确定性分析 + 资料同化　；参考：《南京信息工程大学》2017年硕士论文

【摘要】：在优化设计、优化控制和反问题应用中,人们常采用模型降阶方法来构造低自由度下大规模动力系统的降阶模型,从而满足在保证一定物理精度的同时提高计算效率的要求.特征正交分解法(Proper orthogonal decomposition, POD)由于其恢复数值信息的能力而成为最常用的模型降阶方法之一.本论文首先针对一个典型的耦合Burgers方程,通过POD方法构造其在Galerkin投影下的降阶模型(Reduced-order model, ROM),并引入离散经验插值法(Discrete empirical interpolation method, DEIM)来减少降阶模型中非线性项的计算复杂性.数值求解离散化以及对POD模的部分截断会给降阶模型的数值解精度带来不可避免的损失.在提高数值精度和维持稳定性方面增加POD模数的确会在一定程度上发挥一定作用,但这会增加计算负担.因此本论文发展了一个订正的POD降阶模型,它通过对POD降阶模型分别乘以和加上一组依赖时间的不确定性参数构造而成.这样订正降阶模型就转化为高维随机空间中的参数识别问题.反问题的不适定特征使得相关计算过程具有一定的挑战性.这种情况下,我们采用了基于多项式逼近的集合卡尔曼滤波(Polynomial chaos-based ensemble Kalman Filter, PC-EnKF)方法来处理以上问题,同时引入稀疏化算法来识别出模式输入和模式输出PC展开式中系数近乎为零的基函数,这对集合卡尔曼滤波(EnKF)所需统计矩的快速计算是有帮助的.利用最终更新过的输入参数,大雷诺数Re=10000情况下耦合Burger方程的订正POD降阶模型即被建立起来.数值实验表明PC-EnKF方法在恢复求解精度方面是有效的,在订正动力系统数值解方面是可行性.PC-EnKF方法可作为一个一般的模型订正工具,在当前研究基础上有希望推广应用于更高维模型订正问题中.
[Abstract]:In the optimization design, optimization control and inverse problem application, the model reduction method is often used to construct the reduced order model of the large power system under the low degree of freedom, so as to meet the requirement of improving the computational efficiency while ensuring a certain physical precision. The Proper orthogonal decomposition (POD) is due to its recovery number. The ability of value information is one of the most commonly used model reduction methods. Firstly, this paper constructs a reduced order model (Reduced-order model, ROM) under the Galerkin projection for a typical coupled Burgers equation, and introduces the discrete empirical interpolation (Discrete empirical interpolation method, DEIM) to reduce the reduction of order by POD method. The computational complexity of the nonlinear term in the model. The numerical solution dispersion and the partial truncation of the POD model will bring the inevitable loss to the precision of the numerical solution of the reduced order model. The increase of the POD modulus in the improvement of the numerical accuracy and the maintenance of the stability will certainly play a certain role, but this will increase the calculation burden. This paper develops a revised POD order reduction model, which is constructed by multiplying POD reduced order models respectively and adding a set of uncertain parameters with a set of dependent time. The revised reduced order model is transformed into a parameter identification problem in the high dimensional random space. The discomfort characteristics of the inverse problem make the related computing process a certain challenge. In this case, we use the Polynomial chaos-based ensemble Kalman Filter (PC-EnKF) method based on polynomial approximation to deal with the above problems. At the same time, we introduce a sparsity algorithm to identify the base function with a nearly zero coefficient in the mode input and mode output PC expansion, which is for the collection of Calman filters. The fast calculation of the statistical moments needed for the wave (EnKF) is helpful. Using the final updated input parameters, the revised POD reduced order model of the coupled Burger equation with the large Reynolds number Re=10000 is established. The numerical experiments show that the PC-EnKF method is effective in restoring the accuracy of the solution, and it is feasible in the numerical solution of the revised dynamic system. The sex.PC-EnKF method can be used as a general model correction tool. Based on the current research, it is hoped to be applied to the correction of higher dimensional models.
【学位授予单位】：南京信息工程大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8

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