ConformalLabpp/code/include/hyper_ideal_hessian.hpp

#pragma once
// hyper_ideal_hessian.hpp
//
// Phase 4a — Hessian of the hyper-ideal discrete conformal functional.
// Phase 9b — Block-finite-difference Hessian (intermediate optimisation).
//
// ┌──────────────────────────────────────────────────────────────────────────┐
// │  Implementation strategy                                                  │
// │                                                                           │
// │  The hyper-ideal functional involves angle functions (ζ, σ, α, β)        │
// │  composed through several nested layers (lij → ζ13/14/15 → β/α).        │
// │  Deriving closed-form Hessian entries analytically through all these      │
// │  layers is feasible but lengthy and is deferred to a future PR.           │
// │                                                                           │
// │  TWO Hessian implementations are provided here:                           │
// │                                                                           │
// │    1.  `hyper_ideal_hessian` — full finite-difference baseline.           │
// │        Cost  ≈  n × (cost of full gradient evaluation)                    │
// │             =  O(n · F)         where n = #DOFs and F = #faces.           │
// │        Used for correctness reference and small meshes.                   │
// │                                                                           │
// │    2.  `hyper_ideal_hessian_block_fd` — block-local finite-difference,    │
// │        Phase 9b.  Exploits the fact that each face contributes to the     │
// │        gradient through exactly 6 DOFs (3 vertex b_i + 3 edge a_e).       │
// │        Cost  ≈  F × 6 × (cost of a single face-angle evaluation)          │
// │             =  O(36 · F).                                                 │
// │        Speed-up factor ≈ n / 36, i.e. typically 10–50× on V > 200.        │
// │                                                                           │
// │  Both produce the same Hessian to O(ε²) and pass identical PSD checks.    │
// │  The block-FD variant is the production default; the full-FD variant is   │
// │  kept for cross-validation tests.                                         │
// │                                                                           │
// │  An analytic Hessian via Schläfli-type differentiation through the chain  │
// │     (bᵢ, aₑ) → lᵢⱼ → ζ₁₃/ζ₁₄/ζ₁₅ → αᵢⱼ / βᵢ                            │
// │  is deferred to a future PR (Phase 9b-analytic).  Speed-up would be       │
// │  another ~6×, taking the cost to O(F).                                    │
// │                                                                           │
// │  The hyper-ideal energy is strictly convex (Springborn 2020), so H is    │
// │  positive semi-definite everywhere and Eigen::SimplicialLDLT applies      │
// │  directly to either Hessian variant.                                      │
// └──────────────────────────────────────────────────────────────────────────┘
//
// Note on the Java reference: HyperIdealFunctional.java line 295-298 declares
//   public boolean hasHessian() { return false; }
// — i.e. the upstream Java implementation supplies NO Hessian, analytic or
// numerical.  Both `hyper_ideal_hessian` and `hyper_ideal_hessian_block_fd`
// are conformallab++ additions beyond Java parity.

#include "hyper_ideal_functional.hpp"
#include <Eigen/Sparse>
#include <vector>
#include <cmath>
#include <cstdint>

namespace conformallab {

// ── Full finite-difference Hessian (baseline, Phase 4a) ──────────────────────
//
// Returns the n×n sparse Hessian, where n = hyper_ideal_dimension(mesh, m).
// eps: finite-difference step size (default 1e-5 gives ~1e-10 relative error).
//
// Cost: n × full-gradient evaluations ≈ O(n·F).  Use for small meshes or
// as a correctness reference for the block-FD variant below.
inline Eigen::SparseMatrix<double> hyper_ideal_hessian(
    ConformalMesh&             mesh,
    const std::vector<double>& x,
    const HyperIdealMaps&      m,
    double                     eps = 1e-5)
{
    const int n = hyper_ideal_dimension(mesh, m);
    std::vector<Eigen::Triplet<double>> trips;
    trips.reserve(static_cast<std::size_t>(n * n));

    std::vector<double> xp = x, xm = x;

    for (int j = 0; j < n; ++j) {
        const std::size_t sj = static_cast<std::size_t>(j);
        xp[sj] = x[sj] + eps;
        xm[sj] = x[sj] - eps;

        auto Gp = evaluate_hyper_ideal(mesh, xp, m, /*energy=*/false).gradient;
        auto Gm = evaluate_hyper_ideal(mesh, xm, m, /*energy=*/false).gradient;

        xp[sj] = xm[sj] = x[sj];   // restore

        for (int i = 0; i < n; ++i) {
            double val = (Gp[static_cast<std::size_t>(i)]
                        - Gm[static_cast<std::size_t>(i)]) / (2.0 * eps);
            if (std::abs(val) > 1e-15)
                trips.emplace_back(i, j, val);
        }
    }

    Eigen::SparseMatrix<double> H(n, n);
    H.setFromTriplets(trips.begin(), trips.end());
    return H;
}

// ── Symmetrised Hessian (full-FD variant) ────────────────────────────────────
//
// The FD Hessian is symmetric in exact arithmetic; floating-point rounding
// can introduce tiny asymmetries.  This helper returns (H + Hᵀ)/2.
inline Eigen::SparseMatrix<double> hyper_ideal_hessian_sym(
    ConformalMesh&             mesh,
    const std::vector<double>& x,
    const HyperIdealMaps&      m,
    double                     eps = 1e-5)
{
    auto H = hyper_ideal_hessian(mesh, x, m, eps);
    Eigen::SparseMatrix<double> Ht = H.transpose();
    return (H + Ht) * 0.5;
}

// ── Block-FD Hessian (Phase 9b) ──────────────────────────────────────────────
//
// Computes the Hessian by FD on each face's 6×6 local block.  The 6 local
// DOFs of a face f are:
//      (b_{v1}, b_{v2}, b_{v3}, a_{e12}, a_{e23}, a_{e31}).
// For each face we recompute the 6 output angles (β₁,β₂,β₃,α₁₂,α₂₃,α₃₁)
// at x ± ε along each local axis and read off the 6×6 Jacobian.  The result
// scatters into the global Hessian via the DOF-index lookup.
//
// Why this is correct:
// ─────────────────────
// The global gradient decomposes by face:
//     G_b_v = Σ_{f ∋ v} β_v(f)  −  Θ_v
//     G_a_e = Σ_{f ∋ e} α_e(f)  −  θ_e
// Since β and α at face f depend ONLY on the 6 local DOFs of f, the
// Hessian also decomposes:
//     ∂G_x/∂y = Σ_{f: x,y ∈ local(f)}  ∂(β or α)/∂y  at f.
// So accumulating per-face 6×6 blocks reproduces the full Hessian.
//
// Cost: F × 12 face-angle evaluations  (6 DOFs × 2 directions).
// On a tetrahedron (F=4, n≈10):       48 face evaluations
//                                     vs full-FD ≈ 80         → ~1.7× speed-up.
// On cathead.obj (F=248, n≈400):     2976 face evaluations
//                                     vs full-FD ≈ 99,200    → ~33× speed-up.
// On brezel.obj (F=13824, n≈14000): 165 888 face evaluations
//                                     vs full-FD ≈ 193 M     → ~1166× speed-up.
inline Eigen::SparseMatrix<double> hyper_ideal_hessian_block_fd(
    ConformalMesh&             mesh,
    const std::vector<double>& x,
    const HyperIdealMaps&      m,
    double                     eps = 1e-5)
{
    const int n = hyper_ideal_dimension(mesh, m);
    std::vector<Eigen::Triplet<double>> trips;
    trips.reserve(36 * mesh.number_of_faces());

    for (auto f : mesh.faces()) {
        Halfedge_index h0 = mesh.halfedge(f);
        Halfedge_index h1 = mesh.next(h0);
        Halfedge_index h2 = mesh.next(h1);

        Vertex_index v1 = mesh.source(h0);
        Vertex_index v2 = mesh.source(h1);
        Vertex_index v3 = mesh.source(h2);
        Edge_index   e12 = mesh.edge(h0);
        Edge_index   e23 = mesh.edge(h1);
        Edge_index   e31 = mesh.edge(h2);

        // Local DOF indices: (b1, b2, b3, a12, a23, a31).  Pinned slots = -1.
        const int idx[6] = {
            m.v_idx[v1], m.v_idx[v2], m.v_idx[v3],
            m.e_idx[e12], m.e_idx[e23], m.e_idx[e31]
        };
        const bool v1b = idx[0] >= 0;
        const bool v2b = idx[1] >= 0;
        const bool v3b = idx[2] >= 0;

        // Local DOF values (0 for pinned).
        const double vals[6] = {
            dof_val(idx[0], x), dof_val(idx[1], x), dof_val(idx[2], x),
            dof_val(idx[3], x), dof_val(idx[4], x), dof_val(idx[5], x)
        };

        // For each free local DOF, evaluate the 6 outputs at ±ε.
        // We never perturb a pinned DOF (its column would be physically zero
        // because it is not part of the DOF vector at all).
        for (int j = 0; j < 6; ++j) {
            if (idx[j] < 0) continue;

            double vp[6], vm[6];
            for (int k = 0; k < 6; ++k) { vp[k] = vm[k] = vals[k]; }
            vp[j] += eps;
            vm[j] -= eps;

            auto Op = face_angles_from_local_dofs(
                vp[0], vp[1], vp[2], vp[3], vp[4], vp[5], v1b, v2b, v3b);
            auto Om = face_angles_from_local_dofs(
                vm[0], vm[1], vm[2], vm[3], vm[4], vm[5], v1b, v2b, v3b);

            const double Gp[6] = {
                Op.beta1, Op.beta2, Op.beta3,
                Op.alpha12, Op.alpha23, Op.alpha31
            };
            const double Gm[6] = {
                Om.beta1, Om.beta2, Om.beta3,
                Om.alpha12, Om.alpha23, Om.alpha31
            };

            for (int i = 0; i < 6; ++i) {
                if (idx[i] < 0) continue;   // pinned: contributes nothing
                const double val = (Gp[i] - Gm[i]) / (2.0 * eps);
                if (std::abs(val) > 1e-15)
                    trips.emplace_back(idx[i], idx[j], val);
            }
        }
    }

    Eigen::SparseMatrix<double> H(n, n);
    H.setFromTriplets(trips.begin(), trips.end());
    return H;
}

// ── Symmetrised block-FD Hessian ─────────────────────────────────────────────
//
// The per-face block is symmetric to within FD rounding, but the
// accumulation may amplify tiny asymmetries.  This helper returns
// (H + Hᵀ)/2, identical in spirit to `hyper_ideal_hessian_sym`.
inline Eigen::SparseMatrix<double> hyper_ideal_hessian_block_fd_sym(
    ConformalMesh&             mesh,
    const std::vector<double>& x,
    const HyperIdealMaps&      m,
    double                     eps = 1e-5)
{
    auto H = hyper_ideal_hessian_block_fd(mesh, x, m, eps);
    Eigen::SparseMatrix<double> Ht = H.transpose();
    return (H + Ht) * 0.5;
}

} // namespace conformallab