--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/kernels/hat_convolution.rs Thu Dec 01 23:07:35 2022 +0200
@@ -0,0 +1,450 @@
+//! Implementation of the convolution of two hat functions,
+//! and its convolution with a [`CubeIndicator`].
+use numeric_literals::replace_float_literals;
+use serde::Serialize;
+use alg_tools::types::*;
+use alg_tools::norms::*;
+use alg_tools::loc::Loc;
+use alg_tools::sets::Cube;
+use alg_tools::bisection_tree::{
+ Support,
+ Constant,
+ Bounds,
+ LocalAnalysis,
+ GlobalAnalysis,
+ Bounded,
+};
+use alg_tools::mapping::Apply;
+use alg_tools::maputil::array_init;
+
+use super::base::*;
+use super::ball_indicator::CubeIndicator;
+
+/// Hat convolution kernel.
+///
+/// This struct represents the function
+/// $$
+/// f(x\_1, …, x\_n) = \prod\_{i=1}^n \frac{4}{σ} (h\*h)(x\_i/σ)
+/// $$
+/// where the “hat function” $h(y)= \max(0, 1 - |2y|)$.
+/// The factor $4/σ$ normalises $∫ f d x = 1$.
+/// We have
+/// $$
+/// (h*h)(y) =
+/// \begin{cases}
+/// \frac{2}{3} (y+1)^3 & -1<y\leq -\frac{1}{2}, \\\\
+/// -2 y^3-2 y^2+\frac{1}{3} & -\frac{1}{2}<y\leq 0, \\\\
+/// 2 y^3-2 y^2+\frac{1}{3} & 0<y<\frac{1}{2}, \\\\
+/// -\frac{2}{3} (y-1)^3 & \frac{1}{2}\leq y<1. \\\\
+/// \end{cases}
+/// $$
+#[derive(Copy,Clone,Debug,Serialize,Eq)]
+pub struct HatConv<S : Constant, const N : usize> {
+ /// The parameter $σ$ of the kernel.
+ pub radius : S,
+}
+
+impl<S1, S2, const N : usize> PartialEq<HatConv<S2, N>> for HatConv<S1, N>
+where S1 : Constant,
+ S2 : Constant<Type=S1::Type> {
+ fn eq(&self, other : &HatConv<S2, N>) -> bool {
+ self.radius.value() == other.radius.value()
+ }
+}
+
+impl<'a, S, const N : usize> HatConv<S, N> where S : Constant {
+ /// Returns the $σ$ parameter of the kernel.
+ #[inline]
+ pub fn radius(&self) -> S::Type {
+ self.radius.value()
+ }
+}
+
+impl<'a, S, const N : usize> Apply<&'a Loc<S::Type, N>> for HatConv<S, N>
+where S : Constant {
+ type Output = S::Type;
+ #[inline]
+ fn apply(&self, y : &'a Loc<S::Type, N>) -> Self::Output {
+ let σ = self.radius();
+ y.product_map(|x| {
+ self.value_1d_σ1(x / σ) / σ
+ })
+ }
+}
+
+impl<'a, S, const N : usize> Apply<Loc<S::Type, N>> for HatConv<S, N>
+where S : Constant {
+ type Output = S::Type;
+ #[inline]
+ fn apply(&self, y : Loc<S::Type, N>) -> Self::Output {
+ self.apply(&y)
+ }
+}
+
+
+#[replace_float_literals(S::Type::cast_from(literal))]
+impl<'a, F : Float, S, const N : usize> HatConv<S, N>
+where S : Constant<Type=F> {
+ /// Computes the value of the kernel for $n=1$ with $σ=1$.
+ #[inline]
+ fn value_1d_σ1(&self, x : F) -> F {
+ let y = x.abs();
+ if y >= 1.0 {
+ 0.0
+ } else if y > 0.5 {
+ - (8.0/3.0) * (y - 1.0).powi(3)
+ } else /* 0 ≤ y ≤ 0.5 */ {
+ (4.0/3.0) + 8.0 * y * y * (y - 1.0)
+ }
+ }
+}
+
+impl<'a, S, const N : usize> Support<S::Type, N> for HatConv<S, N>
+where S : Constant {
+ #[inline]
+ fn support_hint(&self) -> Cube<S::Type,N> {
+ let σ = self.radius();
+ array_init(|| [-σ, σ]).into()
+ }
+
+ #[inline]
+ fn in_support(&self, y : &Loc<S::Type,N>) -> bool {
+ let σ = self.radius();
+ y.iter().all(|x| x.abs() <= σ)
+ }
+
+ #[inline]
+ fn bisection_hint(&self, cube : &Cube<S::Type, N>) -> [Option<S::Type>; N] {
+ let σ = self.radius();
+ cube.map(|c, d| symmetric_peak_hint(σ, c, d))
+ }
+}
+
+#[replace_float_literals(S::Type::cast_from(literal))]
+impl<S, const N : usize> GlobalAnalysis<S::Type, Bounds<S::Type>> for HatConv<S, N>
+where S : Constant {
+ #[inline]
+ fn global_analysis(&self) -> Bounds<S::Type> {
+ Bounds(0.0, self.apply(Loc::ORIGIN))
+ }
+}
+
+impl<S, const N : usize> LocalAnalysis<S::Type, Bounds<S::Type>, N> for HatConv<S, N>
+where S : Constant {
+ #[inline]
+ fn local_analysis(&self, cube : &Cube<S::Type, N>) -> Bounds<S::Type> {
+ // The function is maximised/minimised where the 2-norm is minimised/maximised.
+ let lower = self.apply(cube.maxnorm_point());
+ let upper = self.apply(cube.minnorm_point());
+ Bounds(lower, upper)
+ }
+}
+
+#[replace_float_literals(C::Type::cast_from(literal))]
+impl<'a, C : Constant, const N : usize> Norm<C::Type, L1>
+for HatConv<C, N> {
+ #[inline]
+ fn norm(&self, _ : L1) -> C::Type {
+ 1.0
+ }
+}
+
+#[replace_float_literals(C::Type::cast_from(literal))]
+impl<'a, C : Constant, const N : usize> Norm<C::Type, Linfinity>
+for HatConv<C, N> {
+ #[inline]
+ fn norm(&self, _ : Linfinity) -> C::Type {
+ self.bounds().upper()
+ }
+}
+
+#[replace_float_literals(F::cast_from(literal))]
+impl<'a, F : Float, R, C, const N : usize> Apply<&'a Loc<F, N>>
+for Convolution<CubeIndicator<R, N>, HatConv<C, N>>
+where R : Constant<Type=F>,
+ C : Constant<Type=F> {
+
+ type Output = F;
+
+ #[inline]
+ fn apply(&self, y : &'a Loc<F, N>) -> F {
+ let Convolution(ref ind, ref hatconv) = self;
+ let β = ind.r.value();
+ let σ = hatconv.radius();
+
+ // This is just a product of one-dimensional versions
+ y.product_map(|x| {
+ // With $u_σ(x) = u_1(x/σ)/σ$ the normalised hat convolution
+ // we have
+ // $$
+ // [χ_{-β,β} * u_σ](x)
+ // = ∫_{x-β}^{x+β} u_σ(z) d z
+ // = (1/σ)∫_{x-β}^{x+β} u_1(z/σ) d z
+ // = ∫_{(x-β)/σ}^{(x+β)/σ} u_1(z) d z
+ // = [χ_{-β/σ, β/σ} * u_1](x/σ)
+ // $$
+ self.value_1d_σ1(x / σ, β / σ)
+ })
+ }
+}
+
+impl<'a, F : Float, R, C, const N : usize> Apply<Loc<F, N>>
+for Convolution<CubeIndicator<R, N>, HatConv<C, N>>
+where R : Constant<Type=F>,
+ C : Constant<Type=F> {
+
+ type Output = F;
+
+ #[inline]
+ fn apply(&self, y : Loc<F, N>) -> F {
+ self.apply(&y)
+ }
+}
+
+
+#[replace_float_literals(F::cast_from(literal))]
+impl<F : Float, C, R, const N : usize> Convolution<CubeIndicator<R, N>, HatConv<C, N>>
+where R : Constant<Type=F>,
+ C : Constant<Type=F> {
+ #[inline]
+ pub fn value_1d_σ1(&self, x : F, β : F) -> F {
+ // The integration interval
+ let a = x - β;
+ let b = x + β;
+
+ #[inline]
+ fn pow4<F : Float>(x : F) -> F {
+ let y = x * x;
+ y * y
+ }
+
+ /// Integrate $f$, whose support is $[c, d]$, on $[a, b]$.
+ /// If $b > d$, add $g()$ to the result.
+ #[inline]
+ fn i<F: Float>(a : F, b : F, c : F, d : F, f : impl Fn(F) -> F,
+ g : impl Fn() -> F) -> F {
+ if b < c {
+ 0.0
+ } else if b <= d {
+ if a <= c {
+ f(b) - f(c)
+ } else {
+ f(b) - f(a)
+ }
+ } else /* b > d */ {
+ g() + if a <= c {
+ f(d) - f(c)
+ } else if a < d {
+ f(d) - f(a)
+ } else {
+ 0.0
+ }
+ }
+ }
+
+ // Observe the factor 1/6 at the front from the antiderivatives below.
+ // The factor 4 is from normalisation of the original function.
+ (4.0/6.0) * i(a, b, -1.0, -0.5,
+ // (2/3) (y+1)^3 on -1 < y ≤ - 1/2
+ // The antiderivative is (2/12)(y+1)^4 = (1/6)(y+1)^4
+ |y| pow4(y+1.0),
+ || i(a, b, -0.5, 0.0,
+ // -2 y^3 - 2 y^2 + 1/3 on -1/2 < y ≤ 0
+ // The antiderivative is -1/2 y^4 - 2/3 y^3 + 1/3 y
+ |y| y*(-y*y*(y*3.0 + 4.0) + 2.0),
+ || i(a, b, 0.0, 0.5,
+ // 2 y^3 - 2 y^2 + 1/3 on 0 < y < 1/2
+ // The antiderivative is 1/2 y^4 - 2/3 y^3 + 1/3 y
+ |y| y*(y*y*(y*3.0 - 4.0) + 2.0),
+ || i(a, b, 0.5, 1.0,
+ // -(2/3) (y-1)^3 on 1/2 < y ≤ 1
+ // The antiderivative is -(2/12)(y-1)^4 = -(1/6)(y-1)^4
+ |y| -pow4(y-1.0),
+ || 0.0
+ )
+ )
+ )
+ )
+ }
+}
+
+impl<F : Float, R, C, const N : usize>
+Convolution<CubeIndicator<R, N>, HatConv<C, N>>
+where R : Constant<Type=F>,
+ C : Constant<Type=F> {
+
+ #[inline]
+ fn get_r(&self) -> F {
+ let Convolution(ref ind, ref hatconv) = self;
+ ind.r.value() + hatconv.radius()
+ }
+}
+
+impl<F : Float, R, C, const N : usize> Support<F, N>
+for Convolution<CubeIndicator<R, N>, HatConv<C, N>>
+where R : Constant<Type=F>,
+ C : Constant<Type=F> {
+
+ #[inline]
+ fn support_hint(&self) -> Cube<F, N> {
+ let r = self.get_r();
+ array_init(|| [-r, r]).into()
+ }
+
+ #[inline]
+ fn in_support(&self, y : &Loc<F, N>) -> bool {
+ let r = self.get_r();
+ y.iter().all(|x| x.abs() <= r)
+ }
+
+ #[inline]
+ fn bisection_hint(&self, cube : &Cube<F, N>) -> [Option<F>; N] {
+ // It is not difficult to verify that [`HatConv`] is C^2.
+ // Therefore, so is [`Convolution<CubeIndicator<R, N>, HatConv<C, N>>`] so that a finer
+ // subdivision for the hint than this is not particularly useful.
+ let r = self.get_r();
+ cube.map(|c, d| symmetric_peak_hint(r, c, d))
+ }
+}
+
+impl<F : Float, R, C, const N : usize> GlobalAnalysis<F, Bounds<F>>
+for Convolution<CubeIndicator<R, N>, HatConv<C, N>>
+where R : Constant<Type=F>,
+ C : Constant<Type=F> {
+ #[inline]
+ fn global_analysis(&self) -> Bounds<F> {
+ Bounds(F::ZERO, self.apply(Loc::ORIGIN))
+ }
+}
+
+impl<F : Float, R, C, const N : usize> LocalAnalysis<F, Bounds<F>, N>
+for Convolution<CubeIndicator<R, N>, HatConv<C, N>>
+where R : Constant<Type=F>,
+ C : Constant<Type=F> {
+ #[inline]
+ fn local_analysis(&self, cube : &Cube<F, N>) -> Bounds<F> {
+ // The function is maximised/minimised where the absolute value is minimised/maximised.
+ let lower = self.apply(cube.maxnorm_point());
+ let upper = self.apply(cube.minnorm_point());
+ //assert!(upper >= lower);
+ if upper < lower {
+ let Convolution(ref ind, ref hatconv) = self;
+ let β = ind.r.value();
+ let σ = hatconv.radius();
+ eprintln!("WARNING: Hat convolution {β} {σ} upper bound {upper} < lower bound {lower} on {cube:?} with min-norm point {:?} and max-norm point {:?}", cube.minnorm_point(), cube.maxnorm_point());
+ Bounds(upper, lower)
+ } else {
+ Bounds(lower, upper)
+ }
+ }
+}
+
+
+/// This [`BoundedBy`] implementation bounds $u * u$ by $(ψ * ψ) u$ for $u$ a hat convolution and
+/// $ψ = χ_{[-a,a]^N}$ for some $a>0$.
+///
+/// This is based on the general formula for bounding $(uχ) * (uχ)$ by $(ψ * ψ) u$,
+/// where we take $ψ = χ_{[-a,a]^N}$ and $χ = χ_{[-σ,σ]^N}$ for $σ$ the width of the hat
+/// convolution.
+#[replace_float_literals(F::cast_from(literal))]
+impl<F, C, S, const N : usize>
+BoundedBy<F, SupportProductFirst<AutoConvolution<CubeIndicator<S, N>>, HatConv<C, N>>>
+for AutoConvolution<HatConv<C, N>>
+where F : Float,
+ C : Constant<Type=F>,
+ S : Constant<Type=F> {
+
+ fn bounding_factor(
+ &self,
+ kernel : &SupportProductFirst<AutoConvolution<CubeIndicator<S, N>>, HatConv<C, N>>
+ ) -> Option<F> {
+ // We use the comparison $ℱ[𝒜(ψ v)] ≤ L_1 ℱ[𝒜(ψ)u] ⟺ I_{v̂} v̂ ≤ L_1 û$ with
+ // $ψ = χ_{[-w, w]}$ satisfying $supp v ⊂ [-w, w]$, i.e. $w ≥ σ$. Here $v̂ = ℱ[v]$ and
+ // $I_{v̂} = ∫ v̂ d ξ. For this relationship to be valid, we need $v̂ ≥ 0$, which is guaranteed
+ // by $v̂ = u_σ$ being an autoconvolution. With $u = v$, therefore $L_1 = I_v̂ = ∫ u_σ(ξ) d ξ$.
+ let SupportProductFirst(AutoConvolution(ref ind), hatconv2) = kernel;
+ let σ = self.0.radius();
+ let a = ind.r.value();
+ let bounding_1d = 4.0 / (3.0 * σ);
+
+ // Check that the cutting indicator of the comparison
+ // `SupportProductFirst<AutoConvolution<CubeIndicator<S, N>>, HatConv<C, N>>`
+ // is wide enough, and that the hat convolution has the same radius as ours.
+ if σ <= a && hatconv2 == &self.0 {
+ Some(bounding_1d.powi(N as i32))
+ } else {
+ // We cannot compare
+ None
+ }
+ }
+}
+
+/// This [`BoundedBy`] implementation bounds $u * u$ by $u$ for $u$ a hat convolution.
+///
+/// This is based on Example 3.3 in the manuscript.
+#[replace_float_literals(F::cast_from(literal))]
+impl<F, C, const N : usize>
+BoundedBy<F, HatConv<C, N>>
+for AutoConvolution<HatConv<C, N>>
+where F : Float,
+ C : Constant<Type=F> {
+
+ /// Returns an estimate of the factor $L_1$.
+ ///
+ /// Returns `None` if `kernel` does not have the same width as hat convolution that `self`
+ /// is based on.
+ fn bounding_factor(
+ &self,
+ kernel : &HatConv<C, N>
+ ) -> Option<F> {
+ if kernel == &self.0 {
+ Some(1.0)
+ } else {
+ // We cannot compare
+ None
+ }
+ }
+}
+
+#[cfg(test)]
+mod tests {
+ use alg_tools::lingrid::linspace;
+ use alg_tools::mapping::Apply;
+ use alg_tools::norms::Linfinity;
+ use alg_tools::loc::Loc;
+ use crate::kernels::{BallIndicator, CubeIndicator, Convolution};
+ use super::HatConv;
+
+ /// Tests numerically that [`HatConv<f64, 1>`] is monotone.
+ #[test]
+ fn hatconv_monotonicity() {
+ let grid = linspace(0.0, 1.0, 100000);
+ let hatconv : HatConv<f64, 1> = HatConv{ radius : 1.0 };
+ let mut vals = grid.into_iter().map(|t| hatconv.apply(Loc::from(t)));
+ let first = vals.next().unwrap();
+ let monotone = vals.fold((first, true), |(prev, ok), t| (prev, ok && prev >= t)).1;
+ assert!(monotone);
+ }
+
+ /// Tests numerically that [`Convolution<CubeIndicator<f64, 1>, HatConv<f64, 1>>`] is monotone.
+ #[test]
+ fn convolution_cubeind_hatconv_monotonicity() {
+ let grid = linspace(-2.0, 0.0, 100000);
+ let hatconv : Convolution<CubeIndicator<f64, 1>, HatConv<f64, 1>>
+ = Convolution(BallIndicator { r : 0.5, exponent : Linfinity },
+ HatConv{ radius : 1.0 } );
+ let mut vals = grid.into_iter().map(|t| hatconv.apply(Loc::from(t)));
+ let first = vals.next().unwrap();
+ let monotone = vals.fold((first, true), |(prev, ok), t| (prev, ok && prev <= t)).1;
+ assert!(monotone);
+
+ let grid = linspace(0.0, 2.0, 100000);
+ let hatconv : Convolution<CubeIndicator<f64, 1>, HatConv<f64, 1>>
+ = Convolution(BallIndicator { r : 0.5, exponent : Linfinity },
+ HatConv{ radius : 1.0 } );
+ let mut vals = grid.into_iter().map(|t| hatconv.apply(Loc::from(t)));
+ let first = vals.next().unwrap();
+ let monotone = vals.fold((first, true), |(prev, ok), t| (prev, ok && prev >= t)).1;
+ assert!(monotone);
+ }
+}