Coarse-graining support: Support.Multiscale.CubeAverage

Imported from the Apache-2.0 CoarseGraining development at commit c7ddd76c08ade64fed1b8d2ca51be14dfee8deb4.

noncomputable def HCPolySupport.cubeAverage {d : ℕ} (Q : TriadicCube d) (f : Vec d → ℝ) : ℝ

The integral of f on the half-open cube divided by its volume.

Equations

• HCPolySupport.cubeAverage Q f = (HCPolySupport.cubeVolume Q)⁻¹ * ∫ (x : HCPolySupport.Vec d) in HCPolySupport.cubeSet Q, f x

noncomputable def HCPolySupport.cubeAverageVec {d : ℕ} (Q : TriadicCube d) (f : Vec d → Vec d) : Vec d

The vector of coordinatewise averages of a vector field on the cube.

Equations

• HCPolySupport.cubeAverageVec Q f i = HCPolySupport.cubeAverage Q fun (x : HCPolySupport.Vec d) => f x i

noncomputable def HCPolySupport.cubeAverageMat {d : ℕ} (Q : TriadicCube d) (f : Vec d → Mat d) : Mat d

The matrix of entrywise averages of a matrix field on the cube.

Equations

• HCPolySupport.cubeAverageMat Q f i j = HCPolySupport.cubeAverage Q fun (x : HCPolySupport.Vec d) => f x i j

noncomputable def HCPolySupport.cubeProjection {d : ℕ} (Q : TriadicCube d) (j : ℕ) (f : Vec d → ℝ) : Vec d → ℝ

The piecewise constant function equal to the average of f on each depth-j descendant of Q, and zero outside their union.

Equations

• HCPolySupport.cubeProjection Q j f x = ∑ R ∈ HCPolySupport.descendantsAtDepth Q j, if x ∈ HCPolySupport.cubeSet R then HCPolySupport.cubeAverage R f else 0

noncomputable def HCPolySupport.cubeIncrement {d : ℕ} (Q : TriadicCube d) (j : ℕ) (f : Vec d → ℝ) : Vec d → ℝ

The initial cube projection at depth zero, or the difference of consecutive cube projections at a positive depth.

Equations

• HCPolySupport.cubeIncrement Q 0 f = HCPolySupport.cubeProjection Q 0 f
• HCPolySupport.cubeIncrement Q n.succ f = fun (x : HCPolySupport.Vec d) => HCPolySupport.cubeProjection Q (n + 1) f x - HCPolySupport.cubeProjection Q n f x