New code for studying topology of constraint manifolds.

1 files changed, 186 insertions, 0 deletions

diff --git a/topology.cpp b/topology.cpp
new file mode 100644
index 0000000..6048e1e
--- /dev/null
+++ b/topology.cpp
@@ -0,0 +1,186 @@
+#include <getopt.h>
+#include <iomanip>
+#include <random>
+
+#include "pcg-cpp/include/pcg_random.hpp"
+#include "randutils/randutils.hpp"
+
+#include "eigen/Eigen/Dense"
+#include "eigen/unsupported/Eigen/CXX11/Tensor"
+#include "eigen/unsupported/Eigen/CXX11/TensorSymmetry"
+
+using Rng = randutils::random_generator<pcg32>;
+
+using Real = double;
+using Vector = Eigen::Matrix<Real, Eigen::Dynamic, 1>;
+using Matrix = Eigen::Matrix<Real, Eigen::Dynamic, Eigen::Dynamic>;
+
+class Tensor : public Eigen::Tensor<Real, 3> {
+  using Eigen::Tensor<Real, 3>::Tensor;
+
+public:
+  Matrix operator*(const Vector& x) const {
+    std::array<Eigen::IndexPair<int>, 1> ip20 = {Eigen::IndexPair<int>(2, 0)};
+    const Eigen::Tensor<Real, 1> xT = Eigen::TensorMap<const Eigen::Tensor<Real, 1>>(x.data(), x.size());
+    const Eigen::Tensor<Real, 2> JxT = contract(xT, ip20);
+    return Eigen::Map<const Matrix>(JxT.data(), dimension(0), dimension(1));
+  }
+};
+
+Matrix operator*(const Eigen::Matrix<Real, 1, Eigen::Dynamic>& x, const Tensor& J) {
+  std::array<Eigen::IndexPair<int>, 1> ip00 = {Eigen::IndexPair<int>(0, 0)};
+  const Eigen::Tensor<Real, 1> xT = Eigen::TensorMap<const Eigen::Tensor<Real, 1>>(x.data(), x.size());
+  const Eigen::Tensor<Real, 2> JxT = J.contract(xT, ip00);
+  return Eigen::Map<const Matrix>(JxT.data(), J.dimension(1), J.dimension(2));
+}
+
+Vector normalize(const Vector& x) {
+  return x * sqrt(x.size() / x.squaredNorm());
+}
+
+Vector ∂HFromV∂V(const Vector& V, const Matrix& ∂V) {
+  return V.transpose() * ∂V;
+}
+
+Vector VFromABJx(const Vector& b, const Matrix& A, const Matrix& Jx, const Vector& x) {
+  return b + (A + 0.5 * Jx) * x;
+}
+
+class Spherical3Spin {
+private:
+  Tensor J;
+
+public:
+  unsigned N;
+
+  Spherical3Spin(unsigned N, Rng& r) : J(N, N, N), N(N) {
+    Eigen::StaticSGroup<Eigen::Symmetry<0,1>, Eigen::Symmetry<1,2>> sym123;
+
+    for (unsigned i = 0; i < N; i++) {
+      for (unsigned j = i; j < N; j++) {
+        for (unsigned k = j; k < N; k++) {
+          sym123(J, i, j, k) = r.variate<Real, std::normal_distribution>(0, sqrt(2) / N);
+        }
+      }
+    }
+  }
+
+  std::tuple<Real, Vector, Matrix> H_∂H_∂∂H(const Vector& x) const {
+    Matrix ∂∂H = J * x / (6 * N);
+    Vector ∂H = ∂∂H * x;
+    Real   H = ∂H.dot(x);
+    return {H, ∂H, ∂∂H};
+  }
+};
+
+class ConstrainedHeight {
+  private:
+    Vector x0;
+    Real E;
+    Spherical3Spin S;
+
+  public:
+    unsigned N;
+
+  ConstrainedHeight(unsigned N, Real E, Rng& r) : x0(N), E(E), S(N, r), N(N) {
+    for (Real& x0ᵢ : x0) {
+      x0ᵢ = r.variate<Real, std::normal_distribution>();
+    }
+
+    x0 = normalize(x0);
+  }
+
+  Real overlap(const Vector& v) const {
+    return v.head(N).dot(x0) / N;
+  }
+
+  std::tuple<Vector, Matrix> ∂L_∂∂L(const Vector& v) const {
+    Vector x = v.head(N);
+    Real ω₀ = v(N);
+    Real ω₁ = v(N + 1);
+
+    Real H;
+    Vector ∂H∂x;
+    Matrix ∂²H∂²x;
+    std::tie(H, ∂H∂x, ∂²H∂²x) = S.H_∂H_∂∂H(x);
+
+    Vector ∂L∂x = x0 + ω₀ * x + ω₁ * ∂H∂x;
+    Real ∂L∂ω₀ = 0.5 * (x.squaredNorm() - N);
+    Real ∂L∂ω₁ = H - E;
+
+    Vector ∂L(N + 2);
+    ∂L << ∂L∂x, ∂L∂ω₀, ∂L∂ω₁;
+
+    Matrix ∂L²∂x² = ω₀ * Matrix::Identity(N, N) + ω₁ * ∂²H∂²x;
+    Vector ∂²L∂x∂ω₀ = x;
+    Vector ∂²L∂x∂ω₁ = ∂H∂x;
+
+    Matrix ∂∂L(N + 2, N + 2);
+    ∂∂L << ∂L²∂x², ∂²L∂x∂ω₀, ∂²L∂x∂ω₁, ∂²L∂x∂ω₀.transpose(), 0, 0, ∂²L∂x∂ω₁.transpose(), 0, 0;
+
+    return {∂L, ∂∂L};
+  }
+
+  Vector newtonMethod(const Vector& v0, Real γ = 1) {
+    Vector v = v0;
+
+    Vector ∂L;
+    Matrix ∂∂L;
+
+    while (std::tie(∂L, ∂∂L) = ∂L_∂∂L(v), ∂L.squaredNorm() > 1e-10) {
+      v -= γ * ∂∂L.partialPivLu().solve(∂L);
+    }
+
+    return v;
+  }
+};
+
+
+int main(int argc, char* argv[]) {
+  unsigned N = 10;
+  double E = 0;
+  unsigned samples = 10;
+
+  int opt;
+
+  while ((opt = getopt(argc, argv, "N:E:n:")) != -1) {
+    switch (opt) {
+    case 'N':
+      N = (unsigned)atof(optarg);
+      break;
+    case 'E':
+      E = atof(optarg);
+      break;
+    case 'n':
+      samples = atoi(optarg);
+      break;
+    default:
+      exit(1);
+    }
+  }
+
+  Rng r;
+
+  ConstrainedHeight M(N, E, r);
+
+  Vector x(N);
+  for (Real& xᵢ : x) {
+    xᵢ = r.variate<Real, std::normal_distribution>();
+  }
+  x = normalize(x);
+
+  Vector v0(N + 2);
+  v0.head(N) = x;
+  v0(N) = r.variate<Real, std::normal_distribution>();
+  v0(N+1) = r.variate<Real, std::normal_distribution>();
+
+  std::cout << std::setprecision(15);
+
+  for (unsigned sample = 0; sample < samples; sample++) {
+    ConstrainedHeight M(N, E, r);
+    Vector v = M.newtonMethod(v0, 0.05);
+    std::cout << M.overlap(v) << std::endl;
+  }
+
+  return 0;
+}