Two-level Toeplitz preconditioning: approximation results for matrices and functions
Dimitrios Noutsos, Stefano Serra‐Capizzano, Paris Vassalos · SIAM Journal on Scientific Computing · 2006
Large 2-level Toeplitz systems arise in a variety of applications (see, e.g., [R. H. Chan and M. Ng, SIAM Rev., 38 (1996), pp. 427-482]) for which efficient numerical methods for their solution are required. Some successful numerical techniques need the explicit knowledge of the generating function f of the considered system $T_{n}(f){\rm x}=b$, an assumption that usually is not fulfilled in real applications. In this paper we analyze and complete the procedure proposed in [D. Noutsas, S. Serra Capizzano, and P. Vassalos, Numer. Linear Algebra Appl., 12 (2005), pp. 231-239] for the 2-level case. In such a way, from the knowledge of the coefficients of $T_{n}(f)$, we determine optimal preconditioning strategies for the solution of our systems. Finally, some numerical experiments are performed and discussed in connection with our theoretical analysis.