求解涉及矩阵Kronecker和的线性系统的高效张量方法
Efficient tensor-based approach to solving linear systems involving Kronecker sum of matrices
AI总结：
本文提出一种基于张量的新公式求解涉及Kronecker和的线性系统，证明Sylvester张量方程在谱和不为零时具有唯一解，并通过高效求解二维、三维Poisson方程及二维稳态对流扩散方程验证了其在高维问题中的适用性。
摘要：
本文提出了一种新颖的基于张量的公式，用于求解涉及Kronecker和的线性系统。此类系统与Sylvester方程的矩阵形式和张量形式直接相关。新的基于张量的公式证明了一个众所周知的事实：如果矩阵的谱之和不包含零，则Sylvester张量方程具有唯一解。我们通过在具有Dirichlet边界条件的矩形域上高效求解二维和三维离散Poisson方程以及二维稳态对流扩散方程，展示了该方法的有效性。结果表明，该方法非常适合高维问题。
英文摘要：
A novel tensor-based formula for solving the linear systems involving Kronecker sum is proposed. Such systems are directly related to the matrix and tensor forms of Sylvester equation. The new tensor-based formula demonstrates the well-known fact that a Sylvester tensor equation has a unique solution if the sum of spectra of the matrices does not contain zero. We have showcased the effectiveness of the method by efficiently solving the 2D and 3D discretized Poisson equations, as well as the 2D steady-state convection-diffusion equation, on a rectangular domain with Dirichlet boundary conditions. The results suggest that this approach is well-suited for high-dimensional problems.
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