1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
|
#include <getopt.h>
#include <iomanip>
#include <fstream>
#include "pcg-cpp/include/pcg_random.hpp"
#include "randutils/randutils.hpp"
#include "eigen/Eigen/Dense"
#include "eigen/Eigen/src/Eigenvalues/SelfAdjointEigenSolver.h"
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>;
Vector normalizeVector(const Vector& x) {
return x * sqrt(x.size() / x.squaredNorm());
}
Vector randomVector(unsigned N, Rng& r, Real σ = 1) {
Vector v(N);
for (Real& vᵢ : v) {
vᵢ = r.variate<Real, std::normal_distribution>(0, σ);
}
return v;
}
Vector projectionOfOn(const Vector& v, const Vector& u) {
return (v.dot(u) / u.squaredNorm()) * u;
}
Real wignerCDF(Real λ) {
return 0.5 + (λ * sqrt(4 - pow(λ, 2)) / 4 + atan(λ / sqrt(4 - pow(λ, 2)))) / M_PI;
}
Real wignerInverse(Real p, Real ε = 1e-14) {
Real a = -2;
Real b = 2;
while (b - a > ε) {
Real c = (a + b) / 2;
if ((wignerCDF(a) - p) * (wignerCDF(c) - p) > 0) {
a = c;
} else {
b = c;
}
}
return (a + b) / 2;
}
class QuadraticModel {
public:
Vector J;
unsigned N;
QuadraticModel(unsigned N, Rng& r, bool diag = false) : J(N), N(N) {
if (diag) {
Matrix Jtmp(N, N);
for (unsigned j = 0; j < N; j++) {
for (unsigned i = j; i < N; i++) {
Jtmp(i, j) = r.variate<Real, std::normal_distribution>(0, 1 / sqrt(N));
Jtmp(j, i) = Jtmp(i, j);
}
}
std::cerr << "Beginning diagonalization" << std::endl;
Eigen::SelfAdjointEigenSolver<Matrix> es;
es.compute(Jtmp);
J = es.eigenvalues();
std::cerr << "Finished diagonalization" << std::endl;
} else {
for (Real& Jᵢ : J) {
Jᵢ = wignerInverse(r.uniform(0.0, 1.0));
}
}
}
Real H(const Vector& x) const {
return 0.5 * (J.cwiseProduct(x)).dot(x);
}
Vector ∇H(const Vector& x) const {
Vector ∂H = J.cwiseProduct(x);
return ∂H - projectionOfOn(∂H, x);
}
};
Vector gradientDescent(const QuadraticModel& M, const Vector& x₀, Real E, Real ε = 1e-14) {
Vector xₜ = x₀;
Real Hₜ = M.H(x₀);
Real α = 1.0 / M.N;
Real m;
Vector ∇H;
while (
∇H = (Hₜ / M.N - E) * M.∇H(xₜ) / M.N, m = ∇H.squaredNorm(),
m > ε
) {
Vector xₜ₊₁;
Real Hₜ₊₁;
while (
xₜ₊₁ = normalizeVector(xₜ - α * ∇H), Hₜ₊₁ = M.H(xₜ₊₁),
pow(Hₜ₊₁ / M.N - E, 2) > pow(Hₜ / M.N - E, 2) - α * m
) {
α /= 2;
}
xₜ = xₜ₊₁;
Hₜ = Hₜ₊₁;
α *= 1.25;
}
return xₜ;
}
Vector randomStep(const QuadraticModel& M, const Vector& x₀, Real E, Rng& r, Real Δt = 1e-4) {
Vector η = randomVector(M.N, r, sqrt(2 * Δt));
η -= projectionOfOn(η, x₀);
η -= projectionOfOn(η, M.∇H(x₀));
return gradientDescent(M, normalizeVector(x₀ + η), E);
}
int main(int argc, char* argv[]) {
unsigned N = 10;
Real T = 10;
Real E = 0;
Real Δt = 1e-4;
Real Δw = 1e-2;
bool diag = true;
int opt;
while ((opt = getopt(argc, argv, "N:E:T:t:w:d")) != -1) {
switch (opt) {
case 'N':
N = (unsigned)atof(optarg);
break;
case 'E':
E = atof(optarg);
break;
case 'T':
T = atof(optarg);
break;
case 't':
Δt = atof(optarg);
break;
case 'w':
Δw = atof(optarg);
break;
case 'd':
diag = false;
break;
default:
exit(1);
}
}
Rng r;
QuadraticModel model(N, r, diag);
Vector x₀ = normalizeVector(randomVector(N, r));
x₀ = gradientDescent(model, x₀, E);
std::cout << std::setprecision(15);
Vector x = x₀;
Real timeSinceWrite = Δw;
std::ofstream outfile(std::to_string(N) + "_" + std::to_string(E) + "_" + std::to_string(Δt) + "_" + std::to_string(Δw) + ".dat", std::ios::out | std::ios::binary);
for (Real Jᵢ : model.J) {
float Jif = Jᵢ;
outfile.write((const char*)(&Jif), sizeof(float));
}
for (Real t = 0; t < T; t += Δt) {
x = randomStep(model, x, E, r, Δt);
if (timeSinceWrite >= Δw) {
for (Real xᵢ : x) {
float xif = xᵢ;
outfile.write((const char*)(&xif), sizeof(float));
}
timeSinceWrite = 0;
}
timeSinceWrite += Δt;
}
outfile.close();
return 0;
}
|
