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.