Got the log-fourier method working, but integrator is not working correctly

3 files changed, 111 insertions, 65 deletions

diff --git a/fourier.cpp b/fourier.cpp
index cf45ad1..bb1c92f 100644
--- a/fourier.cpp
+++ b/fourier.cpp
@@ -1,5 +1,5 @@
 #include "fourier.hpp"
-#include <boost/multiprecision/fwd.hpp>
+#include <fftw3.h>
 
 inline Real fP(unsigned p, Real q) {
   return 0.5 * pow(q, p);
@@ -100,15 +100,15 @@ void FourierTransform::writeToA(unsigned i, Real ai) {
   a[i] = ai;
 }
 
-LogarithmicFourierTransform::LogarithmicFourierTransform(unsigned N, Real k, Real Δω, Real Δτ, Real Δs) : N(N), k(k), Δτ(Δτ), Δω(Δω), Δs(Δs) {
+LogarithmicFourierTransform::LogarithmicFourierTransform(unsigned N, Real k, Real Δτ, unsigned pad) : N(N), pad(pad), k(k), Δτ(Δτ) {
   τₛ = -0.5 * N;
   ωₛ = -0.5 * N;
-  sₛ = -0.5 * N;
-  a = reinterpret_cast<Complex*>(fftw_alloc_complex(N));
-  â = reinterpret_cast<Complex*>(fftw_alloc_complex(N));
+  sₛ = -0.5 * pad * N;
+  a = reinterpret_cast<Complex*>(fftw_alloc_complex(pad*N));
+  â = reinterpret_cast<Complex*>(fftw_alloc_complex(pad*N));
   fftw_import_wisdom_from_filename("fftw.wisdom");
-  a_to_â = fftw_plan_dft_1d(N, reinterpret_cast<fftw_complex*>(a), reinterpret_cast<fftw_complex*>(â), 1, 0);
-  â_to_a = fftw_plan_dft_1d(N, reinterpret_cast<fftw_complex*>(â), reinterpret_cast<fftw_complex*>(a), 1, 0);
+  a_to_â = fftw_plan_dft_1d(pad*N, reinterpret_cast<fftw_complex*>(a), reinterpret_cast<fftw_complex*>(â), FFTW_BACKWARD, 0);
+  â_to_a = fftw_plan_dft_1d(pad*N, reinterpret_cast<fftw_complex*>(â), reinterpret_cast<fftw_complex*>(a), FFTW_BACKWARD, 0);
   fftw_export_wisdom_to_filename("fftw.wisdom");
 }
 
@@ -125,11 +125,11 @@ Real LogarithmicFourierTransform::τ(unsigned n) const {
 }
 
 Real LogarithmicFourierTransform::ω(unsigned n) const {
-  return Δω * (n + ωₛ);
+  return Δτ * (n + ωₛ);
 }
 
 Real LogarithmicFourierTransform::s(unsigned n) const {
-  return Δs * (n + sₛ);
+  return (n + sₛ) * 2*M_PI / (pad * N * Δτ);
 }
 
 Real LogarithmicFourierTransform::t(unsigned n) const {
@@ -149,29 +149,57 @@ Complex gamma(Complex z) {
   return exp(logΓ.val + 1i * argΓ.val);
 }
 
-std::vector<Complex> LogarithmicFourierTransform::fourier(const std::vector<Real>& c, bool symmetric){ 
+std::vector<Complex> LogarithmicFourierTransform::fourier(const std::vector<Real>& c, bool symmetric) {
   std::vector<Complex> ĉ(N);
   std::vector<Real> σs = {1};
   if (symmetric){
     σs.push_back(-1);
   }
   for (Real σ : σs) {
-    for (unsigned n = 0; n < N; n++) {
-      a[n] = c[n] * exp((1 - k) * τ(n));
+    for (unsigned n = 0; n < pad*N; n++) {
+      if (n < N) {
+        a[n] = c[n] * exp((1 - k) * τ(n));
+      } else {
+        a[n] = 0;
+      }
     }
     fftw_execute(a_to_â);
-    for (unsigned n = 0; n < N; n++) {
-      â[(n + N / 2) % N] *= pow(-1i * σ, 1i * (s(n)) - k) * gamma(k - 1i * (s(n)));
+    for (unsigned n = 0; n < pad*N; n++) {
+      â[(pad*N / 2 + n) % (pad*N)] *= pow(1i * σ, 1i * s(n) - k) * gamma(k - 1i * s(n));
     }
     fftw_execute(â_to_a);
     for (unsigned n = 0; n < N; n++) {
-      ĉ[n] += exp(-k * ω(n) * 4 * M_PI) * a[n] / pow(2 * M_PI, 1);
+      ĉ[n] += exp(-k * ω(n)) * a[(pad - 1)*N+n].real() / (Real)(pad*N);
     }
   }
 
   return ĉ;
 }
 
+std::vector<Real> LogarithmicFourierTransform::inverse(const std::vector<Complex>& ĉ) {
+  std::vector<Real> c(N);
+  std::vector<Real> σs = {1, -1};
+  for (Real σ : σs) {
+    for (unsigned n = 0; n < pad * N; n++) {
+      if (n < N) {
+        a[n] = ĉ[n] * exp((1 - k) * ω(n));
+      } else {
+        a[n] = 0;
+      }
+    }
+    fftw_execute(a_to_â);
+    for (unsigned n = 0; n < pad*N; n++) {
+      â[(pad*N / 2 + n) % (pad*N)] *= pow(-1i * σ, 1i * s(n) - k) * gamma(k - 1i * s(n));
+    }
+    fftw_execute(â_to_a);
+    for (unsigned n = 0; n < N; n++) {
+      c[n] += exp(-k * τ(n)) * a[(pad - 1)*N+n].real() / (Real)(pad*N) / (2 * M_PI);
+    }
+  }
+
+  return c;
+}
+
 std::string fourierFile(std::string prefix, unsigned p, unsigned s, Real λ, Real τ₀, Real y, unsigned log2n, Real τₘₐₓ) {
   return prefix + "_" + std::to_string(p) + "_" + std::to_string(s) + "_" + std::to_string(λ) + "_" + std::to_string(τ₀) + "_" + std::to_string(y) + "_" + std::to_string(log2n) + "_" + std::to_string(τₘₐₓ) + ".dat";
 }
diff --git a/fourier.hpp b/fourier.hpp
index 22a57e7..1708667 100644
--- a/fourier.hpp
+++ b/fourier.hpp
@@ -44,15 +44,14 @@ private:
   fftw_plan a_to_â;
   fftw_plan â_to_a;
   unsigned N;
+  unsigned pad;
   Real k;
   Real Δτ;
-  Real Δω;
-  Real Δs;
   Real τₛ;
   Real ωₛ;
   Real sₛ;
 public:
-  LogarithmicFourierTransform(unsigned N, Real k, Real Δω, Real Δτ, Real Δs);
+  LogarithmicFourierTransform(unsigned N, Real k, Real Δτ, unsigned pad = 4);
   ~LogarithmicFourierTransform();
   Real τ(unsigned n) const;
   Real ω(unsigned n) const;
@@ -60,10 +59,7 @@ public:
   Real ν(unsigned n) const;
   Real s(unsigned n) const;
   std::vector<Complex> fourier(const std::vector<Real>& c, bool symmetric);
-  std::vector<Complex> fourier();
   std::vector<Real> inverse(const std::vector<Complex>& ĉ);
-  void writeToA(unsigned i, Real ai);
-  std::vector<Real> convolve(const std::vector<Real>& Γh, const std::vector<Real>& R);
 };
 
 Complex gamma(Complex z);
diff --git a/log-fourier_integrator.cpp b/log-fourier_integrator.cpp
index 247270b..177db46 100644
--- a/log-fourier_integrator.cpp
+++ b/log-fourier_integrator.cpp
@@ -1,27 +1,25 @@
 #include "fourier.hpp"
 #include <getopt.h>
 #include <iostream>
-#include <fstream>
 
 int main(int argc, char* argv[]) {
   unsigned p = 2;
   unsigned s = 2;
   Real λ = 0.5;
   Real τ₀ = 0;
-  Real β₀ = 0;
-  Real yₘₐₓ = 0.5;
-  Real Δy = 0.05;
 
   unsigned log2n = 8;
-  Real τₘₐₓ = 20;
+  Real Δτ = 0.1;
+  Real k = 0.1;
 
-  unsigned maxIterations = 1000;
-  Real ε = 1e-14;
+  Real ε = 1e-16;
   Real γ = 1;
+  Real βₘₐₓ = 0.7;
+  Real Δβ = 0.01;
 
   int opt;
 
-  while ((opt = getopt(argc, argv, "p:s:2:T:t:0:y:d:I:g:l")) != -1) {
+  while ((opt = getopt(argc, argv, "p:s:2:T:t:b:d:g:k:D:")) != -1) {
     switch (opt) {
     case 'p':
       p = atoi(optarg);
@@ -32,27 +30,24 @@ int main(int argc, char* argv[]) {
     case '2':
       log2n = atoi(optarg);
       break;
-    case 'T':
-      τₘₐₓ = atof(optarg);
-      break;
     case 't':
       τ₀ = atof(optarg);
       break;
-    case '0':
-      β₀ = atof(optarg);
-      break;
-    case 'y':
-      yₘₐₓ = atof(optarg);
+    case 'b':
+      βₘₐₓ = atof(optarg);
       break;
     case 'd':
-      Δy = atof(optarg);
-      break;
-    case 'I':
-      maxIterations = (unsigned)atof(optarg);
+      Δβ = atof(optarg);
       break;
     case 'g':
       γ = atof(optarg);
       break;
+    case 'k':
+      k = atof(optarg);
+      break;
+    case 'D':
+      Δτ = atof(optarg);
+      break;
     default:
       exit(1);
     }
@@ -60,10 +55,7 @@ int main(int argc, char* argv[]) {
 
   unsigned N = pow(2, log2n);
 
-  Real Δτ = 1e-2;
-  Real Δω = 1e-2;
-  Real Δs = 1e-2;
-  Real k = 0.1;
+  LogarithmicFourierTransform fft(N, k, Δτ, 4);
 
   Real Γ₀ = 1.0;
   Real μ = Γ₀;
@@ -73,9 +65,10 @@ int main(int argc, char* argv[]) {
 
   std::vector<Real> C(N);
   std::vector<Real> R(N);
+  std::vector<Complex> Ct(N);
+  std::vector<Complex> Rt(N);
 
-  LogarithmicFourierTransform fft(N, k, Δω, Δτ, Δs);
-
+  // start from the exact solution for β = 0
   for (unsigned n = 0; n < N; n++) {
     if (τ₀ > 0) {
       C[n] = Γ₀ * (exp(-μ * fft.t(n)) - μ * τ₀ * exp(-fft.t(n) / τ₀)) / (μ - pow(μ, 3) * pow(τ₀, 2));
@@ -83,32 +76,61 @@ int main(int argc, char* argv[]) {
       C[n] = Γ₀ * exp(-μ * fft.t(n)) / μ;
     }
     R[n] = exp(-μ * fft.t(n));
+
+    Ct[n] = 2 * Γ₀ / (pow(μ, 2) + pow(fft.ν(n), 2)) / (1 + pow(τ₀ * fft.ν(n), 2));
+    Rt[n] = 1.0 / (μ + 1i * fft.ν(n));
   }
 
-  std::vector<Complex> Ct = fft.fourier(C, true);
-  std::vector<Complex> Rt = fft.fourier(R, false);
+  Real β = 0;
+  while (β < βₘₐₓ) {
+  Real ΔC = 100;
+  while (ΔC > ε) {
+    std::vector<Real> RddfC(N);
+    std::vector<Real> dfC(N);
+    for (unsigned n = 0; n < N; n++) {
+      RddfC[n] = R[n] * ddf(λ, p, s, C[n]);
+      dfC[n] = df(λ, p, s, C[n]);
+    }
+    std::vector<Complex> RddfCt = fft.fourier(RddfC, false);
+    std::vector<Complex> dfCt = fft.fourier(dfC, true);
 
-  /*
-  for (unsigned n = 0; n < N; n++) {
-    std::cout << fft.t(n) << " " << C[n] << std::endl;
+    std::vector<Complex> Rtnew(N);
+    std::vector<Complex> Ctnew(N);
+
+    for (unsigned n = 0; n < N; n++) {
+      Rtnew[n] = (1.0 + pow(β, 2) * RddfCt[n] * Rt[n]) / (μ + 1i * fft.ν(n));
+      Ctnew[n] = (2 * Γ₀ * std::conj(Rt[n]) / (1 + pow(τ₀ * fft.ν(n), 2)) + pow(β, 2) * (RddfCt[n] * Ct[n] + dfCt[n] * std::conj(Rt[n]))) / (μ + 1i * fft.ν(n));
+    }
+
+    std::vector<Real> Cnew = fft.inverse(Ctnew);
+    std::vector<Real> Rnew = fft.inverse(Rtnew);
+
+    ΔC = 0;
+    for (unsigned i = 0; i < N; i++) {
+      ΔC += std::norm(Ct[i] - Ctnew[i]);
+      ΔC += std::norm(Rt[i] - Rtnew[i]);
+    }
+    ΔC = sqrt(ΔC) / (2*N);
+
+    for (unsigned i = 0; i < N; i++) {
+      C[i] += γ * (Cnew[i] - C[i]);
+      R[i] += γ * (Rnew[i] - R[i]);
+      Ct[i] += γ * (Ctnew[i] - Ct[i]);
+      Rt[i] += γ * (Rtnew[i] - Rt[i]);
+    }
+
+    μ *= C[0];
+
+//    std::cerr << ΔC << std::endl;
   }
-  */
 
-  for (unsigned n = 0; n < N; n++) {
-    std::cout << fft.ν(n) << " " << Ct[n].real() << std::endl;
+  std::cerr << β << " " << μ << " " << Ct[0].real() << std::endl;
+   β += Δβ;
   }
 
-  /*
-  // start from the exact solution for τ₀ = 0
-  for (unsigned i = 0; i < N + 1; i++) {
-    Real ω = i * Δω;
-    Ct[i] = 2 * Γ₀ / (pow(μ, 2) + pow(ω, 2)) / (1 + pow(τ₀ * ω, 2));
-    Rt[i] = 1.0 / (μ + 1i * ω);
-    Γt[i] = Γ₀ / (1 + pow(τ₀ * ω, 2));
+  for (unsigned i = 0; i < N; i++) {
+    std::cout << fft.t(i) << " " << Rt[i].imag() << std::endl;
   }
-  C = fft.inverse(Ct);
-  R = fft.inverse(Rt);
-  */
 
   return 0;
 }