Sylvester matrix equations are ubiquitous in scientific computing. However, few solution techniques exist for their generalized multiterm version, as they recently arose in stochastic Galerkin finite element discretizations and isogeometric analysis. In this work, we consider preconditioning techniques for the iterative solution of generalized Sylvester equations. They consist in constructing low Kronecker rank approximations of either the operator itself or its inverse. In the first case, applying the preconditioning operator requires solving standard Sylvester equations, for which very efficient solution methods have already been proposed. In the second case, applying the preconditioning operator only requires computing matrix-matrix multiplications, which are also highly optimized on modern computer architectures. Moreover, low Kronecker rank approximate inverses can be easily combined with sparse approximate inverse techniques, thereby further speeding up their application with little or no damage to their preconditioning capability.

翻译：Sylvester矩阵方程在科学计算中广泛存在。然而，针对近期随机伽辽金有限元离散和等几何分析中出现的广义多变量形式，目前仅有少量求解方法。本文研究了广义Sylvester方程迭代求解的预处理技术，主要包括构造算子本身或其逆的低Kronecker秩近似。对于前者，应用预处理算子需要求解标准Sylvester方程，而此类方程已有高效解法；对于后者，应用预处理算子仅需计算矩阵乘法，这在现代计算机架构上已高度优化。此外，低Kronecker秩近似逆矩阵可便捷地与稀疏近似逆技术结合，从而在几乎不影响预处理能力的前提下进一步加速其应用。