A coarse grid correction (CGC) approach is proposed to enhance the efficiency of the matrix exponential and $\varphi$ matrix function evaluations. The approach is intended for iterative methods computing the matrix-vector products with these functions. It is based on splitting the vector by which the matrix function is multiplied into a smooth part and a remaining part. The smooth part is then handled on a coarser grid, whereas the computations on the original grid are carried out with a relaxed stopping criterion tolerance. Estimates on the error are derived for the two-grid and multigrid variants of the proposed CGC algorithm. Numerical experiments demonstrate the efficiency of the algorithm, when employed in combination with Krylov subspace and Chebyshev polynomial expansion methods.
翻译:提议粗略网格校正(CGC)方法,以提高矩阵指数和美元等值矩阵功能评价的效率,该方法旨在用这些功能来计算矩阵-矢量产品,其基础是将矢量分离成一个平滑部分和其余部分,然后将矩阵函数乘以一个平滑部分,平滑部分在粗略网格上处理,而原网格的计算则以宽松的停止标准容忍度进行。为拟议的CGC算法的二格和多格变方计算出误差的估计数。数字实验表明,与Krylov子空间和Chebyshev多元扩张方法结合使用算法时,算法的效率。