Sylvester matrix equations are ubiquitous in scientific computing. However, few solution techniques exist for their generalized multiterm version, as they now arise in an increasingly large number of applications. In this work, we consider algebraic parameter-free preconditioning techniques for the iterative solution of generalized multiterm Sylvester equations. They consist in constructing low Kronecker rank approximations of either the operator itself or its inverse. While the former requires solving standard Sylvester equations in each iteration, the latter only requires matrix-matrix multiplications, which are highly optimized on modern computer architectures. Moreover, low Kronecker rank approximate inverses can be easily combined with sparse approximate inverse techniques, thereby enhancing their performance with little or no damage to their effectiveness.

翻译：Sylvester矩阵方程在科学计算中无处不在。然而，随着它们在日益增多的应用中出现，其广义多版本目前仍缺乏有效的求解技术。本文针对广义多Sylvester方程的迭代求解，考虑代数无参数预处理技术。这些技术通过构建算子或其逆的低Kronecker秩近似来实现：前者需要在每次迭代中求解标准Sylvester方程，而后者仅需在现代计算机架构上高度优化的矩阵乘法。此外，低Kronecker秩近似逆可轻松与稀疏近似逆技术相结合，从而在几乎不损害其有效性的前提下提升性能。