pointsource_algs: comparison src/kernels/hat

-:bd13c2ae3450
+:56c8adc32b09
 Bounds,
 LocalAnalysis,
 GlobalAnalysis,
 Bounded,
 };
-use alg_tools::mapping::Apply;
+use alg_tools::mapping::{Apply, Differentiable};
 use alg_tools::maputil::array_init;
+use crate::types::Lipschitz;
 use super::base::*;
 use super::ball_indicator::CubeIndicator;
 /// Hat convolution kernel.
 ///
 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>
+where S : Constant {
+type FloatType = S::Type;
+#[inline]
+fn lipschitz_factor(&self, L1 : L1) -> Option<Self::FloatType> {
+// For any ψ_i, we have
+// ∏_{i=1}^N ψ_i(x_i) - ∏_{i=1}^N ψ_i(y_i)
+// = [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|
+let σ = self.radius();
+Some((self.lipschitz_1d_σ1() / (σ*σ)) * (self.value_1d_σ1(0.0) / σ))
+}
+}
+impl<S, const N : usize> Lipschitz<L2> for HatConv<S, N>
+where S : Constant {
+type FloatType = S::Type;
+#[inline]
+fn lipschitz_factor(&self, L2 : L2) -> Option<Self::FloatType> {
+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>
+where S : Constant {
+type Output = Loc<S::Type, N>;
+#[inline]
+fn differential(&self, y : &'a Loc<S::Type, N>) -> Self::Output {
+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| {
+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 {
+type Output = Loc<S::Type, N>;
+#[inline]
+fn differential(&self, y : Loc<S::Type, N>) -> Self::Output {
+self.differential(&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 $σ=1$.
 } 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)
 }
+}
+/// Computes the differential of the kernel for $n=1$ with $σ=1$.
+#[inline]
+fn diff_1d_σ1(&self, x : F) -> F {
+let y = x.abs();
+if y >= 1.0 {
+0.0
+} else if y > 0.5 {
+- 8.0 * (y - 1.0).powi(2)
+} else /* 0 ≤ y ≤ 0.5 */ {
+(24.0 * y - 16.0) * y
+}
+}
+/// Computes the Lipschitz factor 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
 }
 }
 impl<'a, S, const N : usize> Support<S::Type, N> for HatConv<S, N>
 where S : Constant {
 fn apply(&self, y : Loc<F, N>) -> F {
 self.apply(&y)
 }
 }
+#[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| {
+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))]
+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
+}
+}
+}
 #[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> {
+/// Calculates the value of the 1D hat convolution further convolved by a interval indicator.
+/// As both functions are piecewise polynomials, this is implemented by explicit integral over
+/// all subintervals of polynomiality of the cube indicator, using easily formed
+/// antiderivatives.
 #[inline]
 pub fn value_1d_σ1(&self, x : F, β : F) -> F {
 // The integration interval
 let a = x - β;
 let b = x + β;
 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
+// (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
 )
 )
 )
 )
 }
+/// Calculates the derivative of the 1D hat convolution further convolved by a interval
+/// indicator. The implementation is similar to [`Self::value_1d_σ1`], using the fact that
+/// $(θ * ψ)' = θ * ψ'$.
+#[inline]
+pub fn diff_1d_σ1(&self, x : F, β : F) -> F {
+// The integration interval
+let a = x - β;
+let b = x + β;
+// The factor 4 is from normalisation of the original function.
+4.0 * i(a, b, -1.0, -0.5,
+// (2/3) (y+1)^3  on  -1 < y ≤ -1/2
+|y| (2.0/3.0) * (y + 1.0).powi(3),
+|| i(a, b, -0.5, 0.0,
+// -2 y^3 - 2 y^2 + 1/3  on  -1/2 < y ≤ 0
+|y| -2.0*(y - 1.0) * y * y + (1.0/3.0),
+|| i(a, b, 0.0, 0.5,
+// 2 y^3 - 2 y^2 + 1/3 on 0 < y < 1/2
+|y| 2.0*(y - 1.0) * y * y + (1.0/3.0),
+|| i(a, b, 0.5, 1.0,
+// -(2/3) (y-1)^3  on  1/2 < y ≤ 1
+|y| -(2.0/3.0) * (y - 1.0).powi(3),
+|| 0.0
+)
+)
+)
+)
+}
 }
 impl<F : Float, R, C, const N : usize>
 Convolution<CubeIndicator<R, N>, HatConv<C, N>>
 where R : Constant<Type=F>,

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