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Overlap remapping is an L2 projection, conservative and positive

Statement

Assume Countable Choice. Under the finite partition hypotheses of Intersection-volume conservative remapping, let $P_{0} (T)$ be the target cellwise-constant subspace of $L^{2} (Ω)$ . For every source piecewise-constant $v$ , $R v$ is the $L^{2}$ orthogonal projection of $v$ onto $P_{0} (T)$ , hence $∥ R v ∥_{2} \leq ∥ v ∥_{2}$ . It preserves total mass, $\int_{Ω} R v = \int_{Ω} v$ , and if all $v_{i} \geq 0$ then every target value $(R v)_{j} \geq 0$ .

Facts & Assumptions

Given: Countable Choice; a finite-measure domain, finite source and target partitions up to null sets, positive target volumes, and source cell values $v_{i}$ .

[F1]

Intersection-volume conservative remapping: $w_{j i} = ∣ T_{j} \cap S_{i} ∣$ and $(R v)_{j} = ∣ T_{j} ∣^{- 1} \sum_{i} w_{j i} v_{i}$ .

[F2]

Holder's inequality for integrals, including the endpoint cases: Cauchy–Schwarz makes products of $L^{2}$ functions integrable and gives the $L^{2}$ inner-product inequality.

[F3]

The Lebesgue integral is linear on $L^{1} (μ)$ : the integral is linear on integrable functions.

[F4]

Assuming countable choice, $L (R^{n})$ is a sigma-algebra containing every elementary set and $λ_{n}$ is a complete measure extending elementary volume: under Countable Choice, Lebesgue measure is a countably additive measure.

[A1]

Countable Choice is assumed for [F4] (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

1.1F1F4A1given

By [F4], Lebesgue measure is countably additive and hence finitely additive. The partition hypotheses therefore give $\sum_{i} w_{j i} = ∣ T_{j} ∣$ for every target cell and $\sum_{j} w_{j i} = ∣ S_{i} ∣$ for every source cell; omitted partition boundaries are null and do not change these measures. The same finite additivity gives $\int_{T_{j}} v = \sum_{i} w_{j i} v_{i}$ .

2.1F1F3F4A1step 1.1algebra

For any target cellwise-constant $z$ , sum over target cells: $\int_{Ω} (v - R v) z = \sum_{j} z_{j} (\sum_{i} w_{j i} v_{i} - ∣ T_{j} ∣ (R v)_{j}) = 0$ by [F1] and step 1.1. Thus $v - R v$ is orthogonal to the target subspace.

3.1F2step 2.1algebra

Orthogonality with $z = R v$ gives $∥ v ∥_{2}^{2} = ∥ R v ∥_{2}^{2} + ∥ v - R v ∥_{2}^{2}$ , so $∥ R v ∥_{2} \leq ∥ v ∥_{2}$ . All terms are finite because the partitions are finite, cells have finite volume, and $v$ is piecewise constant.

4.1F1F3F4A1step 1.1algebra∎

By the column-sum identity in step 1.1, $\int R v = \sum_{j} ∣ T_{j} ∣ (R v)_{j} = \sum_{i} ∣ S_{i} ∣ v_{i} = \int v$ . If each $v_{i} \geq 0$ , every overlap weight is nonnegative, so the defining finite weighted average is nonnegative.

Depends on

Used by

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35 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Devon Powell and Tom Abel, An exact general remeshing scheme applied to physically conservative voxelization (standard reference, not scraped)
Zhengru Zhang, Moving Mesh Method with Conservative Interpolation Based on L2-Projection, Commun. Comput. Phys. 1 (2006), 930–944 (standard reference, not scraped)