含参数偏微分方程的Greedy-KPOD模型降阶  

Greedy-KPOD Model Order Reduction for Parametrized Partial Differential Equations

作　　者：邢秩源 王丽 蒋耀林[1,2] XING Zhi-yuan;WANG Li;JIANG Yao-lin(College of Mathematics and System Sciences,Xinjiang University,Urumqi Xinjiang 830046,China;School of Mathematics and Statistics,Xi'an Jiaotong University,Xi'an Shaanxi 710049,China)

机构地区：[1]新疆大学数学与系统科学学院,新疆乌鲁木齐830046 [2]西安交通大学数学与统计学院,陕西西安710049

出　　处：《计算机仿真》2022年第11期376-381,共6页Computer Simulation

基　　金：国家自然科学基金(11871393);陕西省重点研发计划国际合作项目(2019KWZ-08)。

摘　　要：许多工程领域中的问题都需要对带有参数的偏微分方程来进行模拟。在方程离散规模较大和参数空间较复杂的情形下,求解这类问题需要大量时间成本。为了提高含参数偏微分方程的求解效率,提出了新的含参数偏微分方程的模型降阶方法,即单边及双边Greedy-KPOD模型降阶方法。首先,根据Galerkin变分理论对含参数偏微分方程进行有限元离散,得到含参系数矩阵的微分方程组。其次,利用Greedy算法,通过迭代选出最优参数,进一步构造基于块Arnoldi过程的单边及双边Krylov子空间,给出了系统间的矩匹配的性质,并生成两种Greedy-KPOD变换矩阵。基于参数分离的系数矩阵,对该系统降阶,得到降阶参数系统,使得降阶系统保持原始系统的参数结构。最后,数值算例比较了两种Greedy-KPOD降阶解、Greedy-POD降阶解与有限元解的相对误差以及生成降阶矩阵所用时间,验证了所提方法对含参数偏微分方程的求解优势。Many engineering problems require the simulation of partial differential equations with parameters.It takes a lot of time to solve this kind of problem when the discrete scale of the equation is larger and the parameter space is more complex.In order to improve the solving efficiency of partial differential equations with parameters,new model order reduction methods for partial differential equations with parameters are proposed.i.e.,the one-sided and two-sided Greedy-KPOD model order reduction methods are proposed.First,the finite element discretization of partial differential equations with parameters was carried out according to Galerkin variational theory,the differential equations with parameter coefficient matrices were obtained.Next,the optimal parameters were selected iteratively using the Greedy algorithm,further the one-sided and two-sided Krylov subspaces based on the block Arnoldi process were constructed,and the properties of moment matching between systems were given.Then the two Greedy-KPOD transformation matrices were generated.Based on the coefficient matrices by parameter separation,the order of the system was reduced and the reduced parameter system was obtained,which could maintain the parameter structure of the original system.Finally,a numerical example was used to compare the relative errors of the solution by the finite element method and the reduced solutions which are from the two Greedy-KPOD methods and the Greedy-POD method.The time to generate the transformation matrices of these methods was also provided.The advantages of the proposed methods for solving parametrized partial differential equations are verified.

关 键 词：含参数偏微分方程 模型降阶方法 有限元离散 贪婪算法 矩匹配 

分 类 号：O29[理学—应用数学]