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feat(phase4): HyperIdeal Newton solver, SparseQR fallback, examples, docs

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Phase 4 complete — 87 CGAL tests pass, 2 skipped.

Newton solver (phase4a):
- hyper_ideal_hessian.hpp: symmetric FD Hessian (O(ε²), PSD by convexity)
- newton_hyper_ideal(): Newton + backtracking for the HyperIdeal functional
- detail::solve_with_fallback(): optional bool* fallback_used parameter
- solve_linear_system(): public API exposing LDLT→SparseQR fallback

SparseQR fallback tests (SparseQRFallback.*):
- FullRankSystem_CorrectSolution: LDLT path, fallback_used=false
- SingularMatrix_FallbackActivated: zero-pivot → QR activated, fallback_used=true
- Euclidean_ClosedMeshNoPinConverges: gauge-mode null space handled via QR

HyperIdeal Newton tests (NewtonSolver.HyperIdeal_*):
- ConvergesTriangleAllVariable, ResultFieldsConsistent,
  ConvergesTetrahedron, SparseQRFallbackNoCrash
- Natural-target base point (b=1.0, a=0.5) — x=0 is degenerate in log-space

Pipeline tests (test_pipeline.cpp):
- End-to-end: all three geometries, mesh I/O round-trip, solve+export

Example programs (code/examples/):
- example_euclidean.cpp:   headless Euclidean pipeline
- example_hyper_ideal.cpp: headless HyperIdeal pipeline
- example_viewer.cpp:      interactive libigl viewer with jet colour map

README:
- Mathematical scope table: C++ vs Java original (18 rows)
- "For mathematicians" section: mental model, step-by-step new-functional
  guide, half-edge traversal snippets, recommended reading

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>
This commit is contained in:
Tarik Moussa
2026-05-13 00:11:25 +02:00
parent e70689d29f
commit 3f124eb071
12 changed files with 1798 additions and 165 deletions
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184
code/include/newton_solver.hpp
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@@ -1,27 +1,39 @@
#pragma once
// newton_solver.hpp
//
// Phase 4a — Newton solver for the discrete conformal functionals.
// Phase 4a — Newton solver for all three discrete conformal functionals.
//
// Solves G(x) = 0 where G is the gradient of the discrete conformal energy:
// G_v = Θ_v − Σ_f α_v^f (angle-sum residual at each vertex DOF)
// Solves G(x) = 0 where G is the gradient of the discrete conformal energy.
//
// Algorithm per iteration:
// 1. Compute gradient G(x)
// 2. Check convergence: max|G_i| < tol → done
// 3. Compute sparse Hessian H(x)
// 4. Factorize and solve the Newton system:
// Euclidean: H · Δx = −G (H is PSD → SimplicialLDLT directly)
// Spherical: (−H) · Δx = G (H is NSD → negate to get PSD matrix)
// 5. Backtracking line search: halve α until ||G(x+α·Δx)|| < ||G(x)||
// 6. x ← x + α·Δx, go to 1
// ┌──────────────────────────────────────────────────────────────────────────┐
// │ Gradient sign conventions │
// │ Euclidean / Spherical: G_v = Θ_v − Σ α_v (target − actual) │
// │ HyperIdeal: G_v = Σ β_v − Θ_v (actual − target) │
// │ │
// │ All solvers use the same Newton step Δx = −H⁻¹·G │
// │ │
// │ Hessian sign at equilibrium │
// │ Euclidean: H PSD → SimplicialLDLT on H │
// │ Spherical: H NSD → SimplicialLDLT on −H (solve (−H)Δx = G) │
// │ HyperIdeal: H PSD → SimplicialLDLT on H (analytical H: future) │
// └──────────────────────────────────────────────────────────────────────────┘
//
// SparseQR fallback:
// When SimplicialLDLT reports a failure (e.g. singular H on a closed mesh
// without a pinned vertex), the solver automatically retries with
// Eigen::SparseQR, which finds the minimum-norm Newton step orthogonal to
// the null space. This handles the gauge mode on closed surfaces without
// requiring the caller to pin a vertex explicitly.
//
// Requires:
// Eigen::SimplicialLDLT (part of Eigen's sparse Cholesky module)
// Eigen::SimplicialLDLT, Eigen::SparseQR (Eigen sparse module)
#include "euclidean_hessian.hpp"
#include "spherical_hessian.hpp"
#include "hyper_ideal_hessian.hpp"
#include <Eigen/SparseCholesky>
#include <Eigen/SparseQR>
#include <Eigen/OrderingMethods>
#include <Eigen/Dense>
#include <algorithm>
#include <cmath>
@@ -41,6 +53,58 @@ struct NewtonResult {
namespace detail {
// Solve A·Δx = rhs with SimplicialLDLT; on failure fall back to SparseQR.
// Returns Δx. ok is set to false only if both solvers fail.
// If fallback_used is non-null, it is set to true iff SparseQR was needed.
inline Eigen::VectorXd solve_with_fallback(
const Eigen::SparseMatrix<double>& A,
const Eigen::VectorXd& rhs,
bool& ok,
bool* fallback_used = nullptr)
{
if (fallback_used) *fallback_used = false;
Eigen::SimplicialLDLT<Eigen::SparseMatrix<double>> ldlt(A);
if (ldlt.info() == Eigen::Success) {
Eigen::VectorXd dx = ldlt.solve(rhs);
if (ldlt.info() == Eigen::Success) { ok = true; return dx; }
}
// Fallback: SparseQR — handles singular/rank-deficient H (gauge modes).
if (fallback_used) *fallback_used = true;
Eigen::SparseQR<Eigen::SparseMatrix<double>, Eigen::COLAMDOrdering<int>> qr(A);
if (qr.info() == Eigen::Success) {
Eigen::VectorXd dx = qr.solve(rhs);
if (qr.info() == Eigen::Success) { ok = true; return dx; }
}
ok = false;
return Eigen::VectorXd::Zero(rhs.size());
}
} // namespace detail
// ── Public linear-system solver (SparseQR fallback) ──────────────────────────
//
// Solve A·x = rhs with Eigen::SimplicialLDLT; if that fails (singular or
// rank-deficient A), retry with Eigen::SparseQR which finds the minimum-norm
// solution orthogonal to the null space.
//
// This is the same primitive used internally by all three Newton solvers.
// Exposing it publicly lets callers (tests, downstream code) reuse the logic
// and — via the optional fallback_used pointer — verify which code path ran.
//
// fallback_used – if non-null, set to true iff SparseQR was invoked
// Returns Eigen::VectorXd::Zero(rhs.size()) if both solvers fail.
inline Eigen::VectorXd solve_linear_system(
const Eigen::SparseMatrix<double>& A,
const Eigen::VectorXd& rhs,
bool* fallback_used = nullptr)
{
bool ok = false;
return detail::solve_with_fallback(A, rhs, ok, fallback_used);
}
namespace detail { // re-open for the remaining helpers
// Backtracking line search: find the largest α in {1, 0.5, 0.25, …} such that
// ||G(x + α·Δx)||₂ < ||G(x)||₂. Returns the accepted step (α may stay 1).
template <typename GradFn>
@@ -109,14 +173,11 @@ inline NewtonResult newton_euclidean(
return res;
}
// ── Hessian + factorisation ───────────────────────────────────────────
// ── Hessian + solve H·Δx = −G (SparseQR fallback for singular H) ──
auto H = euclidean_hessian(mesh, x, m);
solver.compute(H);
if (solver.info() != Eigen::Success) break;
// ── Newton step: solve H·Δx = −G ────────────────────────────────────
Eigen::VectorXd dx = solver.solve(-G);
if (solver.info() != Eigen::Success) break;
bool ok = false;
Eigen::VectorXd dx = detail::solve_with_fallback(H, -G, ok);
if (!ok) break;
// ── Backtracking line search ──────────────────────────────────────────
double norm0 = G.norm();
@@ -157,8 +218,6 @@ inline NewtonResult newton_spherical(
res.iterations = 0;
res.grad_inf_norm = 0.0;
Eigen::SimplicialLDLT<Eigen::SparseMatrix<double>> solver;
for (int iter = 0; iter < max_iter; ++iter) {
// ── Gradient ──────────────────────────────────────────────────────────
auto G_std = spherical_gradient(mesh, x, m);
@@ -173,15 +232,12 @@ inline NewtonResult newton_spherical(
return res;
}
// ── Hessian: negate to get PSD matrix ────────────────────────────────
auto H = spherical_hessian(mesh, x, m);
auto negH = Eigen::SparseMatrix<double>(-H);
solver.compute(negH);
if (solver.info() != Eigen::Success) break;
// ── Newton step: solve (−H)·Δx = G ─────────────────────────────────
Eigen::VectorXd dx = solver.solve(G);
if (solver.info() != Eigen::Success) break;
// ── Hessian: negate to get PSD; solve (−H)·Δx = G ──────────────────
auto H = spherical_hessian(mesh, x, m);
auto negH = Eigen::SparseMatrix<double>(-H);
bool ok = false;
Eigen::VectorXd dx = detail::solve_with_fallback(negH, G, ok);
if (!ok) break;
// ── Backtracking line search ──────────────────────────────────────────
double norm0 = G.norm();
@@ -201,4 +257,70 @@ inline NewtonResult newton_spherical(
return res;
}
// ── HyperIdeal Newton solver ──────────────────────────────────────────────────
//
// Solves G(x) = 0 for the hyper-ideal discrete conformal functional.
//
// Gradient sign convention (opposite to Euclidean/Spherical):
// G_v = Σ β_v − Θ_v, G_e = Σ α_e − θ_e (actual − target)
//
// The hyper-ideal energy is strictly convex (Springborn 2020), so H is PSD
// and SimplicialLDLT (with SparseQR fallback) applies directly.
//
// The Hessian is computed by symmetric finite differences of G (see
// hyper_ideal_hessian.hpp); replace with an analytical Hessian in Phase 5.
inline NewtonResult newton_hyper_ideal(
ConformalMesh& mesh,
std::vector<double> x0,
const HyperIdealMaps& m,
double tol = 1e-8,
int max_iter = 200,
double hess_eps = 1e-5)
{
std::vector<double> x = x0;
const int n = static_cast<int>(x.size());
NewtonResult res;
res.converged = false;
res.iterations = 0;
res.grad_inf_norm = 0.0;
for (int iter = 0; iter < max_iter; ++iter) {
// ── Gradient ──────────────────────────────────────────────────────────
auto G_std = evaluate_hyper_ideal(mesh, x, m, /*energy=*/false).gradient;
Eigen::Map<const Eigen::VectorXd> G(G_std.data(), n);
double inf_norm = G.cwiseAbs().maxCoeff();
if (inf_norm < tol) {
res.converged = true;
res.grad_inf_norm = inf_norm;
res.iterations = iter;
res.x = x;
return res;
}
// ── Hessian (numerical FD) + solve H·Δx = −G ─────────────────────────
auto H = hyper_ideal_hessian_sym(mesh, x, m, hess_eps);
bool ok = false;
Eigen::VectorXd dx = detail::solve_with_fallback(H, -G, ok);
if (!ok) break;
// ── Backtracking line search ──────────────────────────────────────────
double norm0 = G.norm();
x = detail::line_search(x, dx, norm0,
[&](const std::vector<double>& xnew) {
return evaluate_hyper_ideal(mesh, xnew, m, false).gradient;
});
res.iterations = iter + 1;
}
auto G_final = evaluate_hyper_ideal(mesh, x, m, false).gradient;
double inf_final = 0.0;
for (double v : G_final) inf_final = std::max(inf_final, std::abs(v));
res.grad_inf_norm = inf_final;
res.x = x;
return res;
}
} // namespace conformallab