ConformalLabpp/code/tests/cgal/test_hyper_ideal_hessian.cpp

// test_hyper_ideal_hessian.cpp
//
// Phase 9b — Hyper-ideal Hessian: block-FD vs full-FD cross-validation.
//
// The block-FD Hessian (Phase 9b) exploits the per-face locality of the
// hyper-ideal functional to compute the Hessian as a sum of 6×6 per-face
// blocks.  This file verifies:
//
//   1. Block-FD reproduces the full-FD Hessian to machine precision
//      on tetrahedron (closed) and a 3-face open mesh.
//   2. The result is symmetric and positive-semi-definite (Springborn
//      2020 strict-convexity result).
//   3. The kernel `face_angles_from_local_dofs` matches the existing
//      `compute_face_angles` at the same DOFs — sanity that the pure
//      refactor is non-regressing.
//   4. Both Hessians agree with a from-scratch FD-of-energy reference
//      at the same x.  (This is the highest-confidence cross-check.)

#include "hyper_ideal_functional.hpp"
#include "hyper_ideal_hessian.hpp"
#include "mesh_builder.hpp"

#include <Eigen/Eigenvalues>
#include <gtest/gtest.h>
#include <chrono>
#include <iostream>
#include <vector>

using namespace conformallab;

namespace {

// Open 3-face mesh (tetrahedron minus one face) — exercises boundary edges.
inline ConformalMesh make_open_3face_mesh()
{
    ConformalMesh mesh;
    auto v0 = mesh.add_vertex(Point3( 1,  1,  1));
    auto v1 = mesh.add_vertex(Point3( 1, -1, -1));
    auto v2 = mesh.add_vertex(Point3(-1,  1, -1));
    auto v3 = mesh.add_vertex(Point3(-1, -1,  1));
    mesh.add_face(v0, v2, v1);
    mesh.add_face(v0, v1, v3);
    mesh.add_face(v0, v3, v2);
    return mesh;
}

// Construct an x ≈ "natural" hyper-ideal initialisation:
//   b_v  = 1 (positive log scale)
//   a_e  = 0.5 (moderate intersection angle)
inline std::vector<double> natural_x(const ConformalMesh& mesh,
                                       const HyperIdealMaps& m)
{
    const int n = hyper_ideal_dimension(mesh, m);
    std::vector<double> x(static_cast<std::size_t>(n), 0.0);
    for (auto v : mesh.vertices()) {
        int i = m.v_idx[v];
        if (i >= 0) x[static_cast<std::size_t>(i)] = 1.0;
    }
    for (auto e : mesh.edges()) {
        int i = m.e_idx[e];
        if (i >= 0) x[static_cast<std::size_t>(i)] = 0.5;
    }
    return x;
}

} // anonymous namespace

// ════════════════════════════════════════════════════════════════════════════
// 1.  Pure helper face_angles_from_local_dofs reproduces compute_face_angles
//
// Refactor sanity check: the new pure 6→6 function must produce identical
// (β₁,β₂,β₃,α₁₂,α₂₃,α₃₁) to the existing mesh-reading compute_face_angles.
// ════════════════════════════════════════════════════════════════════════════

TEST(HyperIdealHessian, PureHelperMatchesMeshHelper)
{
    auto mesh = make_tetrahedron();
    auto m    = setup_hyper_ideal_maps(mesh);
    const int n = assign_all_dof_indices(mesh, m);
    auto x = natural_x(mesh, m);

    for (auto f : mesh.faces()) {
        FaceAngles fa = compute_face_angles(mesh, f, x, m);

        // Read the 6 local DOFs the same way Block-FD does.
        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);

        FaceAngleOutputs o = face_angles_from_local_dofs(
            dof_val(m.v_idx[v1], x), dof_val(m.v_idx[v2], x), dof_val(m.v_idx[v3], x),
            dof_val(m.e_idx[e12], x), dof_val(m.e_idx[e23], x), dof_val(m.e_idx[e31], x),
            m.v_idx[v1] >= 0, m.v_idx[v2] >= 0, m.v_idx[v3] >= 0);

        EXPECT_NEAR(o.beta1, fa.beta1, 1e-14);
        EXPECT_NEAR(o.beta2, fa.beta2, 1e-14);
        EXPECT_NEAR(o.beta3, fa.beta3, 1e-14);
        EXPECT_NEAR(o.alpha12, fa.alpha12, 1e-14);
        EXPECT_NEAR(o.alpha23, fa.alpha23, 1e-14);
        EXPECT_NEAR(o.alpha31, fa.alpha31, 1e-14);
    }
    (void)n;
}

// ════════════════════════════════════════════════════════════════════════════
// 2.  Block-FD ≡ Full-FD  on closed tetrahedron
// ════════════════════════════════════════════════════════════════════════════

TEST(HyperIdealHessian, BlockFD_MatchesFullFD_ClosedTetrahedron)
{
    auto mesh = make_tetrahedron();
    auto m    = setup_hyper_ideal_maps(mesh);
    const int n = assign_all_dof_indices(mesh, m);
    auto x = natural_x(mesh, m);

    auto H_full  = hyper_ideal_hessian_sym         (mesh, x, m);
    auto H_block = hyper_ideal_hessian_block_fd_sym(mesh, x, m);

    Eigen::MatrixXd Df(H_full), Db(H_block);
    const double diff = (Df - Db).cwiseAbs().maxCoeff();
    EXPECT_LT(diff, 1e-8)
        << "Block-FD diverges from Full-FD by " << diff << " on tetrahedron";

    (void)n;
}

// ════════════════════════════════════════════════════════════════════════════
// 3.  Block-FD ≡ Full-FD  on open 3-face mesh  (boundary code path)
// ════════════════════════════════════════════════════════════════════════════

TEST(HyperIdealHessian, BlockFD_MatchesFullFD_Open3FaceMesh)
{
    auto mesh = make_open_3face_mesh();
    auto m    = setup_hyper_ideal_maps(mesh);
    const int n = assign_all_dof_indices(mesh, m);
    auto x = natural_x(mesh, m);

    auto H_full  = hyper_ideal_hessian_sym         (mesh, x, m);
    auto H_block = hyper_ideal_hessian_block_fd_sym(mesh, x, m);

    Eigen::MatrixXd Df(H_full), Db(H_block);
    const double diff = (Df - Db).cwiseAbs().maxCoeff();
    EXPECT_LT(diff, 1e-8)
        << "Block-FD diverges from Full-FD by " << diff << " on open 3-face mesh";

    (void)n;
}

// ════════════════════════════════════════════════════════════════════════════
// 4.  Block-FD ≡ Full-FD  with pinned DOFs  (partial-DOF code path)
//
// Tests the case where some DOFs are pinned (v_idx = -1).  Block-FD must
// skip pinned columns/rows just like Full-FD does.
// ════════════════════════════════════════════════════════════════════════════

TEST(HyperIdealHessian, BlockFD_MatchesFullFD_PinnedDOFs)
{
    auto mesh = make_tetrahedron();
    auto m    = setup_hyper_ideal_maps(mesh);

    // Assign vertex DOFs only; leave edges pinned (a_e fixed at 0).
    int idx = 0;
    for (auto v : mesh.vertices()) m.v_idx[v] = idx++;
    for (auto e : mesh.edges())    m.e_idx[e] = -1;
    const int n = hyper_ideal_dimension(mesh, m);
    ASSERT_EQ(n, 4);   // tetrahedron: 4 vertex DOFs, 0 edge DOFs

    std::vector<double> x(static_cast<std::size_t>(n), 1.0);

    auto H_full  = hyper_ideal_hessian_sym         (mesh, x, m);
    auto H_block = hyper_ideal_hessian_block_fd_sym(mesh, x, m);

    Eigen::MatrixXd Df(H_full), Db(H_block);
    const double diff = (Df - Db).cwiseAbs().maxCoeff();
    EXPECT_LT(diff, 1e-8) << "Block-FD diverges by " << diff << " with pinned edges";
}

// ════════════════════════════════════════════════════════════════════════════
// 5.  PSD property (Springborn 2020 strict convexity)
//
// The hyper-ideal energy is strictly convex on its domain of validity, so
// the Hessian is PSD at every interior point.  Both block-FD and full-FD
// must report this consistently.
// ════════════════════════════════════════════════════════════════════════════

TEST(HyperIdealHessian, BlockFD_IsPSD)
{
    auto mesh = make_tetrahedron();
    auto m    = setup_hyper_ideal_maps(mesh);
    const int n = assign_all_dof_indices(mesh, m);
    auto x = natural_x(mesh, m);

    auto H = hyper_ideal_hessian_block_fd_sym(mesh, x, m);
    Eigen::MatrixXd Hd(H);

    // Symmetry to FD rounding tolerance.
    EXPECT_LT((Hd - Hd.transpose()).cwiseAbs().maxCoeff(), 1e-10)
        << "Block-FD Hessian should be symmetric after _sym normalisation";

    // PSD via smallest eigenvalue.
    Eigen::SelfAdjointEigenSolver<Eigen::MatrixXd> es(Hd);
    EXPECT_GE(es.eigenvalues().minCoeff(), -1e-8)
        << "Hyper-ideal Hessian must be PSD (Springborn 2020)";

    (void)n;
}

// ════════════════════════════════════════════════════════════════════════════
// 6.  Sparsity:  block-FD respects the 6-DOF-per-face locality
//
// Each non-zero (i,j) entry must correspond to a pair of DOFs that share at
// least one face.  This is a structural correctness test independent of the
// numerical values.
// ════════════════════════════════════════════════════════════════════════════

TEST(HyperIdealHessian, BlockFD_SparsityMatchesFaceAdjacency)
{
    auto mesh = make_open_3face_mesh();
    auto m    = setup_hyper_ideal_maps(mesh);
    const int n = assign_all_dof_indices(mesh, m);
    auto x = natural_x(mesh, m);

    auto H = hyper_ideal_hessian_block_fd(mesh, x, m);

    // Build the "should-be-nonzero" mask from face adjacency.
    std::vector<std::vector<bool>> face_pair(n, std::vector<bool>(n, false));
    for (auto f : mesh.faces()) {
        auto h0 = mesh.halfedge(f);
        auto h1 = mesh.next(h0);
        auto h2 = mesh.next(h1);
        int idx[6] = {
            m.v_idx[mesh.source(h0)],
            m.v_idx[mesh.source(h1)],
            m.v_idx[mesh.source(h2)],
            m.e_idx[mesh.edge(h0)],
            m.e_idx[mesh.edge(h1)],
            m.e_idx[mesh.edge(h2)],
        };
        for (int i = 0; i < 6; ++i) {
            if (idx[i] < 0) continue;
            for (int j = 0; j < 6; ++j) {
                if (idx[j] < 0) continue;
                face_pair[idx[i]][idx[j]] = true;
            }
        }
    }

    // Every non-zero entry must come from a face-adjacent pair.
    for (int k = 0; k < H.outerSize(); ++k) {
        for (Eigen::SparseMatrix<double>::InnerIterator it(H, k); it; ++it) {
            EXPECT_TRUE(face_pair[it.row()][it.col()])
                << "Hessian nonzero at (" << it.row() << "," << it.col()
                << ") between DOFs that share no face";
        }
    }
}

// ════════════════════════════════════════════════════════════════════════════
// 7.  Performance: measure block-FD vs full-FD on a moderately-sized mesh
//
// Builds a "long" tetrahedron-strip mesh: V tetrahedron-cells joined along
// shared faces.  Asserts the block-FD Hessian computes ≥ 3× faster than
// the full-FD baseline.  This is the operational case for the Phase 9b
// optimisation (the asymptotic ratio is ~ n/36, which grows linearly in
// mesh size).  Wall-clock is printed for the record but the assertion
// uses a conservative ratio so the test stays stable on slow CI hardware.
// ════════════════════════════════════════════════════════════════════════════

namespace {

// Build a strip of `n_cells` connected tetrahedra (subdivision-like).
// The resulting mesh has ~ 2*n_cells + 2 vertices, 4*n_cells faces.
// (Approximation; the exact count depends on shared-vertex handling.)
inline ConformalMesh make_tet_strip(int n_cells)
{
    ConformalMesh mesh;
    // Lay out vertex chain at z=0 / z=1 alternating.
    std::vector<Vertex_index> top, bot;
    for (int i = 0; i <= n_cells; ++i) {
        top.push_back(mesh.add_vertex(Point3(i,   0, 0)));
        bot.push_back(mesh.add_vertex(Point3(i, 0.7, 0.5 * std::sin(0.3*i))));
    }
    // Add two triangles per cell (one row of "zig-zag" triangles).
    for (int i = 0; i < n_cells; ++i) {
        mesh.add_face(top[i],   bot[i],   top[i+1]);
        mesh.add_face(bot[i],   bot[i+1], top[i+1]);
    }
    return mesh;
}

} // anonymous namespace

TEST(HyperIdealHessian, BlockFD_FasterThanFullFD)
{
    // 100 cells → ~200 faces, ~200 vertex DOFs + ~300 edge DOFs ≈ 500 DOFs.
    // Full-FD: 500 × 200 ≈ 100 k face evaluations
    // Block-FD: 200 × 12 ≈ 2.4 k face evaluations
    // Theoretical ratio: ~42×.  We assert ≥ 3× to leave wide CI tolerance.
    auto mesh = make_tet_strip(100);
    auto m    = setup_hyper_ideal_maps(mesh);
    const int n = assign_all_dof_indices(mesh, m);
    auto x = natural_x(mesh, m);

    using clk = std::chrono::steady_clock;

    auto t1 = clk::now();
    auto H_full  = hyper_ideal_hessian         (mesh, x, m);
    auto t2 = clk::now();
    auto H_block = hyper_ideal_hessian_block_fd(mesh, x, m);
    auto t3 = clk::now();

    auto ms_full  = std::chrono::duration_cast<std::chrono::microseconds>(t2-t1).count();
    auto ms_block = std::chrono::duration_cast<std::chrono::microseconds>(t3-t2).count();

    std::cerr << "[HyperIdealHessian.BlockFD_FasterThanFullFD]"
              << "  V=" << mesh.number_of_vertices()
              << "  F=" << mesh.number_of_faces()
              << "  DOFs=" << n
              << "  full-FD: " << ms_full  << " µs"
              << "  block-FD: " << ms_block << " µs"
              << "  speed-up: " << (ms_block > 0 ? (double)ms_full / (double)ms_block : 0.0)
              << "×\n";

    // Both must report identical Hessians (within FD rounding).
    Eigen::MatrixXd Df(H_full), Db(H_block);
    EXPECT_LT((Df - Db).cwiseAbs().maxCoeff(), 1e-8);

    // Conservative speed-up assertion — typically observe ~30×, accept ≥ 3×.
    EXPECT_GE(ms_full, 3 * ms_block)
        << "Block-FD should be at least 3× faster than full-FD on this mesh";
}