Enum SphereWahbaKernel 

pub enum SphereWahbaKernel {
    Sobolev,
    Pseudo,
    SobolevTruncated {
        lmax: u16,
    },
    PseudoTruncated {
        lmax: u16,
    },
}

Which reproducing kernel to use for SphereMethod::Wahba.

Both options yield positive-definite reproducing kernels on S² with the same family of penalty_order m ∈ {1, 2, 3, 4}. They define different RKHS, however:

• Sobolev (default, more correct): true Wahba/Sobolev kernel K_m(γ) = (1/4π) Σ_{l ≥ 1} (2l+1) · [l(l+1)]^{-m} · P_l(cos γ), the reproducing kernel of H^m(S²) under the Laplace–Beltrami inner product. Penalty quadratic form recovers ‖f‖²_{H^m} = Σ_l [l(l+1)]^m · |f̂_l|². For m=1, 2, 3 this is evaluated via the closed forms from Beatson & zu Castell (2018) “Thinplate Splines on the Sphere” (SIGMA 14 (2018), 083) using elementary functions plus the di/trilogarithm. For m=4 we fall back to the spectral Legendre series (96 terms ⇒ truncation error ≲ 1e-12).

• Pseudo: Wahba 1981’s “pseudo-spline” kernel with Legendre weights 2 / [(l+1)(l+2)···(l+m+1)] (decaying as l^{-(m+1)}, different from Sobolev’s l^{-2m}). Faster to evaluate (one elementary polynomial in sin(γ/2), log), and matches mgcv’s bs="sos" exactly. At m=4 the pseudo-spline produces numerically tiny kernel values (K(p,p) ≈ 3e-4) and the basis pipeline historically collapsed the smooth contribution to zero — the cure is the REML scale-invariance fix in the solver, not the kernel itself.

Default is Sobolev because it matches the canonical “Wahba spline of order m on S²” interpretation. Use Pseudo for exact mgcv compatibility.

Variants

Sobolev

True Sobolev H^m(S²) reproducing kernel via closed-form (elementary + Li_n) for m=1,2,3 and a deep (L=4096..96) spectral series for m=4. This is the user-facing default.

Pseudo

Wahba 1981 closed-form pseudo-spline (mgcv bs="sos" compatible).

SobolevTruncated

Finite truncated-spectral Sobolev kernel evaluated by the same Legendre 3-term recurrence the GPU s2_wahba_legendre_colmajor kernel runs in registers:

K_L^{Sobolev}(γ) = Σ_{ℓ=1..L} c_ℓ · P_ℓ(cos γ), c_0 = 0, c_ℓ = (2ℓ+1) / (4π · [ℓ(ℓ+1)]^m).

This is the explicit parity target for the GPU kernel: at any finite lmax the closed-form (or m=4-spectral-at-fixed-deep-L) variants differ by the documented truncation error, while the GPU matches this variant exactly to roundoff. Single-source: both CPU and GPU use the same Legendre recurrence and the same c_ℓ array.

Fields

lmax: u16

Largest Legendre degree retained (≥ 1). Practical range 5..200; GPU kernel uses LMAX as a compile-time #define.

PseudoTruncated

Finite truncated-spectral Wahba-1981 pseudo-spline kernel:

K_L^{Pseudo}(γ) = Σ_{ℓ=1..L} c_ℓ · P_ℓ(cos γ), c_0 = 0, c_ℓ = 2 / (4π · Π_{k=1..m+1}(ℓ + k)).

Same role as Self::SobolevTruncated for the pseudo branch.

Fields

lmax: u16

Largest Legendre degree retained (≥ 1).

Trait Implementations

impl Clone for SphereWahbaKernel

fn clone(&self) -> SphereWahbaKernel

Returns a duplicate of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl Copy for SphereWahbaKernel

impl Debug for SphereWahbaKernel

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl Default for SphereWahbaKernel

fn default() -> SphereWahbaKernel

Returns the “default value” for a type.

impl<'de> Deserialize<'de> for SphereWahbaKernel

fn deserialize<__D>(__deserializer: __D) -> Result<Self, __D::Error>

where __D: Deserializer<'de>,

Deserialize this value from the given Serde deserializer.

impl Eq for SphereWahbaKernel

impl PartialEq for SphereWahbaKernel

fn eq(&self, other: &SphereWahbaKernel) -> bool

Tests for self and other values to be equal, and is used by ==.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl Serialize for SphereWahbaKernel

fn serialize<__S>(&self, __serializer: __S) -> Result<__S::Ok, __S::Error>

where __S: Serializer,

Serialize this value into the given Serde serializer.

impl StructuralPartialEq for SphereWahbaKernel