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
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  and  need to apply a null space
correction in every iteration of the preconditioner since they are based on , where a null space
property [16, eq.