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#include <complex>
#include <cstdlib>
#include <functional>
#include <getopt.h>
#include <random>
#include "pcg-cpp/include/pcg_random.hpp"
#include "randutils/randutils.hpp"
#include <eigen3/Eigen/LU>
#include <eigen3/Eigen/Dense>
#include "complex_normal.hpp"
#include "p-spin.hpp"
#include "stereographic.hpp"
using Rng = randutils::random_generator<pcg32>;
Vector normalize(const Vector& z) {
return z * sqrt(z.size()) / sqrt((Scalar)(z.transpose() * z));
}
template <class Distribution, class Generator>
Vector randomVector(unsigned N, Distribution d, Generator& r) {
Vector z(N);
for (unsigned i = 0; i < N; i++) {
z(i) = d(r);
}
return z;
}
std::tuple<double, Vector> gradientDescent(const Tensor& J, const Vector& z0, double ε, double γ0 = 1, double δγ = 2) {
Vector z = z0;
double W;
Vector dW;
std::tie(W, dW) = WdW(J, z);
double γ = γ0;
while (W > ε) {
Vector zNew = normalize(z - γ * dW.conjugate());
double WNew;
Vector dWNew;
std::tie(WNew, dWNew) = WdW(J, zNew);
if (WNew < W) { // If the step lowered the objective, accept it!
z = zNew;
W = WNew;
dW = dWNew;
γ = γ0;
} else { // Otherwise, shrink the step and try again.
γ /= δγ;
}
if (γ < 1e-15) {
std::cerr << "Gradient descent stalled." << std::endl;
exit(1);
}
}
return {W, z};
}
Vector findSaddle(const Tensor& J, const Vector& z0, double ε, double δW = 2, double γ0 = 1, double δγ = 2) {
double W;
std::tie(W, std::ignore) = WdW(J, z0);
Vector z = z0;
Vector ζ = euclideanToStereographic(z);
Vector dH;
Matrix ddH;
std::tie(std::ignore, dH, ddH) = stereographicHamGradHess(J, ζ, z);
while (W > ε) {
// ddH is complex symmetric, which is (almost always) invertible, so a
// partial pivot LU decomposition can be used.
Vector dζ = ddH.partialPivLu().solve(dH);
Vector ζNew = ζ - dζ;
Vector zNew = stereographicToEuclidean(ζNew);
double WNew;
std::tie(WNew, std::ignore) = WdW(J, zNew);
if (WNew < W) { // If Newton's step lowered the objective, accept it!
ζ = ζNew;
z = zNew;
W = WNew;
} else { // Otherwise, do gradient descent until W is a factor δW smaller.
std::tie(W, z) = gradientDescent(J, z, W / δW, γ0, δγ);
ζ = euclideanToStereographic(z);
}
std::tie(std::ignore, dH, ddH) = stereographicHamGradHess(J, ζ, z);
}
return z;
}
std::tuple<double, Vector> langevin(const Tensor& J, const Vector& z0, double T, double γ, unsigned N, Rng& r) {
Vector z = z0;
double W;
std::tie(W, std::ignore) = WdW(J, z);
complex_normal_distribution<> d(0, γ, 0);
for (unsigned i = 0; i < N; i++) {
Vector dz = randomVector(z.size(), d, r.engine());
Vector zNew = normalize(z + dz);
double WNew;
std::tie(WNew, std::ignore) = WdW(J, zNew);
if (exp((W - WNew) / T) > r.uniform(0.0, 1.0)) {
z = zNew;
W = WNew;
}
}
return {W, z};
}
int main(int argc, char* argv[]) {
// model parameters
unsigned N = 10; // number of spins
double T = 1; // temperature
double Rκ = 1; // real part of distribution parameter
double Iκ = 0; // imaginary part of distribution parameter
// simulation parameters
double ε = 1e-4;
double δ = 1e-2; // threshold for determining saddle
double γ = 1e-2; // step size
unsigned t = 1000; // number of Langevin steps
unsigned M = 100;
unsigned n = 100;
int opt;
while ((opt = getopt(argc, argv, "N:M:n:T:e:r:i:g:t:E:")) != -1) {
switch (opt) {
case 'N':
N = (unsigned)atof(optarg);
break;
case 't':
t = (unsigned)atof(optarg);
break;
case 'T':
T = atof(optarg);
break;
case 'e':
δ = atof(optarg);
break;
case 'E':
ε = atof(optarg);
break;
case 'g':
γ = atof(optarg);
case 'r':
Rκ = atof(optarg);
break;
case 'i':
Iκ = atof(optarg);
break;
case 'n':
n = (unsigned)atof(optarg);
break;
case 'M':
M = (unsigned)atof(optarg);
break;
default:
exit(1);
}
}
Scalar κ(Rκ, Iκ);
double σ = sqrt(factorial(p) / (2.0 * pow(N, p - 1)));
Rng r;
Tensor J = generateCouplings<Scalar, PSPIN_P>(N, complex_normal_distribution<>(0, σ, κ), r.engine());
Vector z0 = normalize(randomVector(N, complex_normal_distribution<>(0, 1, 0), r.engine()));
Vector zSaddle = findSaddle(J, z0, ε);
double W;
Vector z = zSaddle;
for (unsigned i = 0; i < n; i++) {
std::tie(W, z) = langevin(J, z, T, γ, M, r);
Vector zNewSaddle = findSaddle(J, z, ε);
Scalar H;
Matrix ddH;
std::tie(H, std::ignore, ddH) = hamGradHess(J, zNewSaddle);
Eigen::SelfAdjointEigenSolver<Matrix> es(ddH.adjoint() * ddH);
std::cout << (zNewSaddle - zSaddle).norm() << " " << real(H) << " " << imag(H) << " " << es.eigenvalues().transpose() << std::endl;
std::cerr << M * (i+1) << " steps taken to move " << (zNewSaddle - zSaddle).norm() << ", saddle information saved." << std::endl;
}
return 0;
}
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