The proposed algorithm designs an efficient regular partitioning topology by performing link measurements when the nodes are awake for event transmission or reception with minimal extra energy expenditure.
The proposed control law is implemented as a piecewise-affine function defined on a regular simplicial partition, and has two main positive features.
We assume that (pi_{h}={kappa}) is a regular simplex partition of Ω and satisfies (overline{Omega}=bigcupoverline{kappa}) (see [20]).
Let ({mathcal{T}_{h}}) be a regular rectangle partition of Ω, then there exists a positive constant C such that begin{aligned} &h_{T} Vert Anabla v_{h}cdot n Vert ^{2}_{0,partial T} leq C vert v_{h} vert ^{2}_{1,T},quad forall v_{h}in V_{h}, &h_{T} Vert Bnabla v_{h}cdot n Vert ^{2}_{0,partial T} leq C vert v_{h} vert ^{2}_{1,T}, quad forall v_{h}in V_{h}.
Let T h be regular partition of Ω. Associated with T h is a finite-dimensional subspace V h of C , such that χ | τ are polynomials of order m ( m ≥ 1 ) ∀ χ ∈ V h and τ ∈ T h.
So we have K δ = A. The a m, ℓ, k 's are the renormalized coefficients of g ℓ ( m ) on the regular partition of size K.
In this work, we consider regular partitions begin{aligned}& a = x _{0} < x _{1} < cdots< x _{m} < cdots< x _{M} = b, quad m in { 0, 1, ldots, M }, & c = y _{0} < y _{1} < cdots< y _{n} < cdots< y _{N} = d, quad n in { 0, 1, ldots, N }, end{aligned} of the intervals ([a, b]) and ([c, d]), respectively, each of them with norm given by Δx and Δy.
When regular partitions are considered, the resulting estimator satisfies an oracle inequality similar to the oracle inequality established in Theorem 2 for the kernel rule combined with the GL method.
