论文标题
多项矩阵方程的交替能量最小化方法
Alternating Energy Minimization Methods for Multi-term Matrix Equations
论文作者
论文摘要
我们开发用于近似低等级的线性多项矩阵方程的解的计算方法。我们遵循一个交替的最小化框架，其中解决方案表示为两个矩阵的乘积，并且通过反复解决某些最小化问题来寻求每个矩阵的近似值。我们提出的解决方案方法基于交替的能量最小化方法的等级自适应变体，该变体通过在每个步骤中连续计算排名一的解决方案组件来构建近似迭代。我们还开发了有效的程序，以提高使用这些连续的排名良好更新技术计算出的低级别近似解决方案的准确性。我们探索了与线性多项矩阵方程相关的方法，这些方程是由随机的盖尔金有限元元素离散化的参数椭圆形PDE，并通过数值研究证明了它们的有效性。
We develop computational methods for approximating the solution of a linear multi-term matrix equation in low rank. We follow an alternating minimization framework, where the solution is represented as a product of two matrices, and approximations to each matrix are sought by solving certain minimization problems repeatedly. The solution methods we present are based on a rank-adaptive variant of alternating energy minimization methods that builds an approximation iteratively by successively computing a rank-one solution component at each step. We also develop efficient procedures to improve the accuracy of the low-rank approximate solutions computed using these successive rank-one update techniques. We explore the use of the methods with linear multi-term matrix equations that arise from stochastic Galerkin finite element discretizations of parameterized linear elliptic PDEs, and demonstrate their effectiveness with numerical studies.
