From Table 6.3, we see that the null space algorithm performs better in the case of random permeability which can be a realistic simulation of an underground situation.

We specify now more precisely how our fundamental cycle null space basis is constructed.

POTHEN, The null space problem I: complexity, SIAM J.

This concept is represented in Figure 1(a) where the channel [h.sub.k] of the kth unselected user is projected onto the null space Sp[(H(S)).sup.[perpendicular to]] using (12).

It has been observed in [16, Theorem 2.E] that the same is true for the null space, [I.sup.p] of the semi-norm.

That is, [U.sup.H.sub.l] should lie in the left null space of [H.sub.l,m] [V.sub.M], i.e.,

Compute the matrix H in (3.12) and a basis [u.sub.j] = [([u.sup.(j).sub.1],..., [u.sup.(j).sub.n]).sup.T], j = 1,..., n - k, for the null space of H.

Additionally, one can also see that the power consumption of the null space based scheme seems constant.

One context in which a null space basis is required is constrained optimization when the Karush-Kuhn-Tucker (KKT) system

Alternatively, we can view the algorithm as a way of coarsening the fine grid null space. We can coarsen the null space by summing columns of T associated with nodes in an aggregate.

where [V.sup.H.sub.2] [H.sup.H.sub.12] is adx 3d matrix which has a 2d -dimensional null space. Thus, [u.sub.2] is determined as 2d basis vectors of the null space of [V.sup.H.sub.2] [H.sup.H.sub.12]

Note that these two recent parallel inexact BDDC implementations [1] and [44] need to apply a null space correction in every iteration of the preconditioner since they are based on [16], where a null space property [16, eq.