pointsource_algs: comparison src/kernels/hat

-:efa60bc4f743
+:b087e3eab191
 Bounds,
 LocalAnalysis,
 GlobalAnalysis,
 Bounded,
 };
-use alg_tools::mapping::{Apply, Differentiable};
+use alg_tools::mapping::{
+Mapping,
+Instance,
+DifferentiableImpl,
+Differential,
+};
 use alg_tools::maputil::array_init;
 use crate::types::Lipschitz;
 use super::base::*;
 use super::ball_indicator::CubeIndicator;
 ///         -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}
 /// $$
+// Hence
+// $$
+//     (h\*h)'(y) =
+//     \begin{cases}
+//         2 (y+1)^2 & -1<y\leq -\frac{1}{2}, \\\\
+//         -6 y^2-4 y & -\frac{1}{2}<y\leq 0, \\\\
+//         6 y^2-4 y & 0<y<\frac{1}{2}, \\\\
+//         -2 (y-1)^2 & \frac{1}{2}\leq y<1. \\\\
+//     \end{cases}
+// $$
+// as well as
+// $$
+//     (h\*h)''(y) =
+//     \begin{cases}
+//         4 (y+1) & -1<y\leq -\frac{1}{2}, \\\\
+//         -12 y-4 & -\frac{1}{2}<y\leq 0, \\\\
+//         12 y-4 & 0<y<\frac{1}{2}, \\\\
+//         -4 (y-1) & \frac{1}{2}\leq y<1. \\\\
+//     \end{cases}
+// $$
+// This is maximised at y=±1/2 with value 2, and minimised at y=0 with value -4.
+// Now observe that
+// $$
+//     [∇f(x\_1, …, x\_n)]_j = \frac{4}{σ} (h\*h)'(x\_j/σ) \prod\_{j ≠ i} \frac{4}{σ} (h\*h)(x\_i/σ)
+// $$
 #[derive(Copy,Clone,Debug,Serialize,Eq)]
 pub struct HatConv<S : Constant, const N : usize> {
 /// The parameter $σ$ of the kernel.
 pub radius : S,
 }
 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>
+impl<'a, S, const N : usize> Mapping<Loc<S::Type, N>> for HatConv<S, N>
 where S : Constant {
-type Output = S::Type;
+type Codomain = S::Type;
-#[inline]
-fn apply(&self, y : &'a Loc<S::Type, N>) -> Self::Output {
+#[inline]
+fn apply<I : Instance<Loc<S::Type, N>>>(&self, y : I) -> Self::Codomain {
 let σ = self.radius();
-y.product_map(|x| {
+y.cow().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<S, const N : usize> Lipschitz<L1> for HatConv<S, N>
 // = [ψ_1(x_1)-ψ_1(y_1)] ∏_{i=2}^N ψ_i(x_i)
 //   + ψ_1(y_1)[ ∏_{i=2}^N ψ_i(x_i) - ∏_{i=2}^N ψ_i(y_i)]
 // = ∑_{j=1}^N [ψ_j(x_j)-ψ_j(y_j)]∏_{i > j} ψ_i(x_i) ∏_{i < j} ψ_i(y_i)
 // Thus
 // |∏_{i=1}^N ψ_i(x_i) - ∏_{i=1}^N ψ_i(y_i)|
-// ≤ ∑_{j=1}^N |ψ_j(x_j)-ψ_j(y_j)| ∏_{j ≠ i} \max_i |ψ_i|
+// ≤ ∑_{j=1}^N |ψ_j(x_j)-ψ_j(y_j)| ∏_{j ≠ i} \max_j |ψ_j|
 let σ = self.radius();
 let l1d = self.lipschitz_1d_σ1() / (σ*σ);
 let m1d = self.value_1d_σ1(0.0) / σ;
 Some(l1d * m1d.powi(N as i32 - 1))
 }
 self.lipschitz_factor(L1).map(|l1| l1 * <S::Type>::cast_from(N).sqrt())
 }
 }
-impl<'a, S, const N : usize> Differentiable<&'a Loc<S::Type, N>> for HatConv<S, N>
+impl<'a, S, const N : usize> DifferentiableImpl<Loc<S::Type, N>> for HatConv<S, N>
 where S : Constant {
-type Output = Loc<S::Type, N>;
+type Derivative = Loc<S::Type, N>;
-#[inline]
-fn differential(&self, y : &'a Loc<S::Type, N>) -> Self::Output {
+#[inline]
+fn differential_impl<I : Instance<Loc<S::Type, N>>>(&self, y0 : I) -> Self::Derivative {
+let y = y0.cow();
 let σ = self.radius();
 let σ2 = σ * σ;
 let vs = y.map(|x| {
 self.value_1d_σ1(x  / σ) / σ
 });
-product_differential(y, &vs, |x| {
+product_differential(&*y, &vs, |x| {
 self.diff_1d_σ1(x  / σ) / σ2
 })
 }
 }
-impl<'a, S, const N : usize> Differentiable<Loc<S::Type, N>> for HatConv<S, N>
-where S : Constant {
+#[replace_float_literals(S::Type::cast_from(literal))]
-type Output = Loc<S::Type, N>;
+impl<'a, F : Float, S, const N : usize> Lipschitz<L2>
-#[inline]
+for Differential<'a, Loc<F, N>, HatConv<S, N>>
-fn differential(&self, y : Loc<S::Type, N>) -> Self::Output {
+where S : Constant<Type=F> {
-self.differential(&y)
+type FloatType = F;
-}
-}
+#[inline]
+fn lipschitz_factor(&self, _l2 : L2) -> Option<F> {
+let h = self.base_fn();
+let σ = h.radius();
+Some(product_differential_lipschitz_factor::<F, N>(
+h.value_1d_σ1(0.0) / σ,
+h.lipschitz_1d_σ1() / (σ*σ),
+h.maxabsdiff_1d_σ1() / (σ*σ),
+h.lipschitz_diff_1d_σ1() / (σ*σ),
+))
+}
+}
 #[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 lipschitz_1d_σ1(&self) -> F {
 // Maximal absolute differential achieved at ±0.5 by diff_1d_σ1 analysis
 2.0
 }
+/// Computes the maximum absolute differential of the kernel for $n=1$ with $σ=1$.
+#[inline]
+fn maxabsdiff_1d_σ1(&self) -> F {
+// Maximal absolute differential achieved at ±0.5 by diff_1d_σ1 analysis
+2.0
+}
+/// Computes the second differential of the kernel for $n=1$ with $σ=1$.
+#[inline]
+#[allow(dead_code)]
+fn diff2_1d_σ1(&self, x : F) -> F {
+let y = x.abs();
+if y >= 1.0 {
+0.0
+} else if y > 0.5 {
+- 16.0 * (y - 1.0)
+} else /* 0 ≤ y ≤ 0.5 */ {
+48.0 * y - 16.0
+}
+}
+/// Computes the differential of the kernel for $n=1$ with $σ=1$.
+#[inline]
+fn lipschitz_diff_1d_σ1(&self) -> F {
+// Maximal absolute second differential achieved at 0 by diff2_1d_σ1 analysis
+16.0
+}
 }
 impl<'a, S, const N : usize> Support<S::Type, N> for HatConv<S, N>
 where S : Constant {
 #[inline]
 self.bounds().upper()
 }
 }
 #[replace_float_literals(F::cast_from(literal))]
-impl<'a, F : Float, R, C, const N : usize> Apply<&'a Loc<F, N>>
+impl<'a, F : Float, R, C, const N : usize> Mapping<Loc<F, N>>
 for Convolution<CubeIndicator<R, N>, HatConv<C, N>>
 where R : Constant<Type=F>,
 C : Constant<Type=F> {
-type Output = F;
+type Codomain = F;
 #[inline]
-fn apply(&self, y : &'a Loc<F, N>) -> F {
+fn apply<I : Instance<Loc<F, N>>>(&self, y : I) -> 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| {
+y.cow().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
 self.value_1d_σ1(x / σ, β / σ)
 })
 }
 }
-impl<'a, F : Float, R, C, const N : usize> Apply<Loc<F, N>>
+#[replace_float_literals(F::cast_from(literal))]
+impl<'a, F : Float, R, C, const N : usize> DifferentiableImpl<Loc<F, N>>
 for Convolution<CubeIndicator<R, N>, HatConv<C, N>>
 where R : Constant<Type=F>,
 C : Constant<Type=F> {
-type Output = F;
+type Derivative = Loc<F, N>;
 #[inline]
-fn apply(&self, y : Loc<F, N>) -> F {
+fn differential_impl<I : Instance<Loc<F, N>>>(&self, y0 : I) -> Loc<F, N> {
-self.apply(&y)
+let y = y0.cow();
-}
-}
-#[replace_float_literals(F::cast_from(literal))]
-impl<'a, F : Float, R, C, const N : usize> Differentiable<&'a Loc<F, N>>
-for Convolution<CubeIndicator<R, N>, HatConv<C, N>>
-where R : Constant<Type=F>,
-C : Constant<Type=F> {
-type Output = Loc<F, N>;
-#[inline]
-fn differential(&self, y : &'a Loc<F, N>) -> Loc<F, N> {
 let Convolution(ref ind, ref hatconv) = self;
 let β = ind.r.value();
 let σ = hatconv.radius();
 let σ2 = σ * σ;
 let vs = y.map(|x| {
 self.value_1d_σ1(x / σ, β / σ)
 });
-product_differential(y, &vs, |x| {
+product_differential(&*y, &vs, |x| {
 self.diff_1d_σ1(x  / σ, β / σ) / σ2
 })
 }
 }
-impl<'a, F : Float, R, C, const N : usize> Differentiable<Loc<F, N>>
-for Convolution<CubeIndicator<R, N>, HatConv<C, N>>
-where R : Constant<Type=F>,
-C : Constant<Type=F> {
-type Output = Loc<F, N>;
-#[inline]
-fn differential(&self, y : Loc<F, N>) -> Loc<F, N> {
-self.differential(&y)
-}
-}
 /// Integrate $f$, whose support is $[c, d]$, on $[a, b]$.
 /// If $b > d$, add $g()$ to the result.
 #[inline]
 #[replace_float_literals(F::cast_from(literal))]
 )
 )
 }
 }
+/*
+impl<'a, F : Float, R, C, const N : usize> Lipschitz<L2>
+for Differential<Loc<F, N>, Convolution<CubeIndicator<R, N>, HatConv<C, N>>>
+where R : Constant<Type=F>,
+C : Constant<Type=F> {
+type FloatType = F;
+#[inline]
+fn lipschitz_factor(&self, _l2 : L2) -> Option<F> {
+dbg!("unimplemented");
+None
+}
+}
+*/
 impl<F : Float, R, C, const N : usize>
 Convolution<CubeIndicator<R, N>, HatConv<C, N>>
 where R : Constant<Type=F>,
 C : Constant<Type=F> {
 }
 #[cfg(test)]
 mod tests {
 use alg_tools::lingrid::linspace;
-use alg_tools::mapping::Apply;
+use alg_tools::mapping::Mapping;
 use alg_tools::norms::Linfinity;
 use alg_tools::loc::Loc;
 use crate::kernels::{BallIndicator, CubeIndicator, Convolution};
 use super::HatConv;

comparison: src/kernels/hat_convolution.rs

src/kernels/hat_convolution.rs

Mercurial > repos > pointsource_algs / file comparison

comparison: src/kernels/hat_convolution.rs

src/kernels/hat_convolution.rs