Interpolation and Related Coarsening Techniques for the Algebraic Multigrid Method

Abstract

Let A be a symmetric and positive definite matrix and b ∊ R ^r. We consider the system

$$Au = b$$

with the (unique) exact solution u ∊ R ^r. For q ≥ 2 let C ₁, C₂, …, C _q be a sequence of nonvoid subsets of {1,..., r} such that

$$\left\{ {1, \cdots ,r} \right\} = {C_1} \supset {C_2} \supset \cdots \supset {C_q}$$

$$\left| {{C_m}} \right| = {n_{m,}}m = 1, \cdots ,q$$

$$r = {n_1} > {n_2} > \cdots > {n_q} \ge 1$$

where by |C _m | we denoted the number of elements in the set C _m . Furthermore, for m = 1, 2,..., q − 1 we consider the matrices A ¹ − A and A ^m+1 and the linear operators

$$I_{m + 1}^m:{R^{{n_{m + 1}}}} \to {R^{{n_m}}},I_m^{m + 1}:{R^{{n_m}}} \to {R^{{n_{m + 1}}}}$$

with the properties: I ^m _m+1 has full rank,

$$I_m^{m + 1} = {(I_{m + 1}^m)^t}$$

$${A^{m + 1}} = I_m^{m + 1}{A^m}I_{m + 1}^m$$

We also define the coarse grid correction operators T ^m by

$${T^m} = {I_m} - I_{m + 1}^m{({A^{m + 1}})^{ - 1}}I_m^{m + 1}{A^m}$$

and the smoothing process

$$u_{new}^m = {G^m}u_{old}^m + \left( {{I_m} - {G^m}} \right){\left( {{A^m}} \right)^{ - 1}}{b^m}$$

where I _m is the identity and

$${A^m}{u^m} = {b^m}$$

are the systems corresponding to the coarse levels.