example-notebooks/curves-reparametrization.ipynb

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  {
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   "metadata": {},
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   "source": [
    "using Revise\n",
    "using Reparam\n",
    "using Plots\n",
    "using LaTeXStrings"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "c1 (generic function with 1 method)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ψ(x) = 0.5 * log(20x+1) / log(21) + 0.25 * (1 + tanh(20(x-0.5)) / tanh(10))\n",
    "c2(x) = [cos(2π*x), sin(4π*x)]\n",
    "c1(x) = c2(ψ(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Curves Visualization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
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       "</svg>\n"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Plotting Config\n",
    "Tfine = 201\n",
    "Tcoarse = 21\n",
    "\n",
    "\n",
    "p1 = plot_curve(c1, Tfine, color=1, legend=false, xlabel=L\"x\", ylabel=L\"y\", guidefont=16, size=(600, 600))\n",
    "plot_curve!(c1, Tcoarse, seriestype=:scatter, color=1, aspect_ratio=:equal, label=\"\")\n",
    "p2 = plot_curve(c2, Tfine, color=2, legend=false, xlabel=L\"x\", ylabel=L\"y\", guidefont=16, size=(600, 600))\n",
    "plot_curve!(c2, Tcoarse, seriestype=:scatter, color=2, aspect_ratio=:equal)\n",
    "plot(p1, p2, size=(700, 350))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Qmap Visualization "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "</svg>\n"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q = Qmap(c1)\n",
    "r = Qmap(c2)\n",
    "\n",
    "\n",
    "# Plotting Config\n",
    "Tfine = 201\n",
    "Tcoarse = 21\n",
    "\n",
    "\n",
    "p1 = plot_curve(q, Tfine, color=1, legend=false, xlabel=L\"x\", ylabel=L\"y\", guidefont=16, size=(600, 600))\n",
    "plot_curve!(q, Tcoarse, seriestype=:scatter, color=1, aspect_ratio=:equal, label=\"\")\n",
    "p2 = plot_curve(r, Tfine, color=2, legend=false, xlabel=L\"x\", ylabel=L\"y\", guidefont=16, size=(600, 600))\n",
    "plot_curve!(r, Tcoarse, seriestype=:scatter, color=2, aspect_ratio=:equal)\n",
    "\n",
    "plot(p1, p2, size=(700, 350))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Perform Reparametrization "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " ----------------------------------------------------------------------------------\n",
      "| Iter | Error            | εmax             | εi               | ||dγ||           | |f_i - f_{i-1}|/ |fi| | \n",
      "----------------------------------------------------------------------------------\n",
      "|    0 | 9.4332495335e+00 |                  |                  |                  |\n",
      "|    1 | 7.9168639913e+00 | 2.4633325562e-03 | 1.2316662781e-03 | 3.0166692272e+01 | 1.6074901197e-01 |\n",
      "|    2 | 6.9133351026e+00 | 2.2326792553e-03 | 1.1163396277e-03 | 3.1798971092e+01 | 1.2675838436e-01 |\n",
      "|    3 | 5.9132127165e+00 | 2.2029181771e-03 | 1.1014590885e-03 | 3.1749162189e+01 | 1.4466568902e-01 |\n",
      "|    4 | 4.9753570872e+00 | 2.2329408271e-03 | 1.1164704135e-03 | 3.0592501671e+01 | 1.5860339790e-01 |\n",
      "|    5 | 3.9722757843e+00 | 2.1129138161e-03 | 1.0564569080e-03 | 3.2859016536e+01 | 2.0160991168e-01 |\n",
      "|    6 | 3.0229317344e+00 | 2.1689643373e-03 | 1.0844821686e-03 | 2.9690529033e+01 | 2.3899248225e-01 |\n",
      "|    7 | 2.2626041307e+00 | 2.3264125549e-03 | 1.1632062774e-03 | 4.7889815148e+01 | 2.5151993842e-01 |\n",
      "|    8 | 2.2358095852e+00 | 1.6082445341e-03 | 1.0862499457e-04 | 8.4090457530e+01 | 1.1842348008e-02 |\n",
      "|    9 | 2.1916212358e+00 | 1.4534724578e-03 | 6.4410182260e-05 | 6.9341544915e+01 | 1.9763914478e-02 |\n",
      "|   10 | 2.1659784529e+00 | 1.2412037068e-03 | 8.3834107966e-05 | 9.8106517815e+01 | 1.1700371609e-02 |\n",
      "|   11 | 2.1500605998e+00 | 1.1727133417e-03 | 6.4158554195e-05 | 9.2834835513e+01 | 7.3490357718e-03 |\n",
      "|   12 | 2.0962587759e+00 | 9.2669436152e-04 | 6.9545926112e-05 | 1.0916587513e+02 | 2.5023398820e-02 |\n",
      "|   13 | 2.0342413262e+00 | 1.0175095398e-03 | 5.5667432638e-05 | 8.6016763069e+01 | 2.9584825318e-02 |\n",
      "|   14 | 1.9906660756e+00 | 9.9551438846e-04 | 6.7239616081e-05 | 1.0413989498e+02 | 2.1420885534e-02 |\n",
      "|   15 | 1.9404448342e+00 | 1.0724003666e-03 | 5.2803433826e-05 | 8.0926143668e+01 | 2.5228360492e-02 |\n",
      "|   16 | 1.8947232572e+00 | 1.0537371254e-03 | 6.4054917261e-05 | 9.6247495275e+01 | 2.3562420422e-02 |\n",
      "|   17 | 1.8658256516e+00 | 1.1750399306e-03 | 5.2071530967e-05 | 7.8508747603e+01 | 1.5251623432e-02 |\n",
      "|   18 | 1.8459584563e+00 | 1.0830349840e-03 | 6.5835885080e-05 | 1.0396699765e+02 | 1.0647937704e-02 |\n",
      "|   19 | 1.8359203150e+00 | 1.0595843651e-03 | 5.2172392556e-05 | 9.2761162705e+01 | 5.4379020796e-03 |\n",
      "|   20 | 1.8232011128e+00 | 9.2109340175e-04 | 6.2213030194e-05 | 1.1997660726e+02 | 6.9279707207e-03 |\n",
      "|   21 | 1.7894199273e+00 | 8.8679888433e-04 | 4.8516318743e-05 | 9.9722863516e+01 | 1.8528501998e-02 |\n",
      "|   22 | 1.7648930791e+00 | 8.5805632646e-04 | 5.7955343123e-05 | 1.2136243750e+02 | 1.3706591638e-02 |\n",
      "|   23 | 1.7405394447e+00 | 8.6910673592e-04 | 4.7548390246e-05 | 1.0381709060e+02 | 1.3798929083e-02 |\n",
      "|   24 | 1.7174018153e+00 | 8.2461561759e-04 | 5.5696671172e-05 | 1.2480024430e+02 | 1.3293366832e-02 |\n",
      "|   25 | 1.6883217427e+00 | 8.3633360254e-04 | 4.5755388683e-05 | 1.0489860434e+02 | 1.6932596909e-02 |\n",
      "|   26 | 1.6802495693e+00 | 8.1566738367e-04 | 5.5092284314e-05 | 1.3039978781e+02 | 4.7811818961e-03 |\n",
      "|   27 | 1.6328001073e+00 | 7.9122605861e-04 | 4.3287577754e-05 | 1.0304493627e+02 | 2.8239532294e-02 |\n",
      "|   28 | 1.5890429448e+00 | 8.2890474386e-04 | 5.0387732866e-05 | 1.1430022467e+02 | 2.6798848403e-02 |\n",
      "|   29 | 1.5412627792e+00 | 9.1550373148e-04 | 4.0570264602e-05 | 8.1510702582e+01 | 3.0068517512e-02 |\n",
      "|   30 | 1.5085007068e+00 | 1.0347265414e-03 | 5.0948429484e-05 | 9.4467211723e+01 | 2.1256642820e-02 |\n",
      "|   31 | 1.4876069368e+00 | 1.1396764893e-03 | 4.0908572914e-05 | 7.1489172064e+01 | 1.3850686238e-02 |\n",
      "|   32 | 1.4656946422e+00 | 1.1712616582e-03 | 5.1904097995e-05 | 8.8351631050e+01 | 1.4729895410e-02 |\n",
      "|   33 | 1.4464928241e+00 | 1.2287481336e-03 | 3.9695211563e-05 | 6.5273277084e+01 | 1.3100831198e-02 |\n",
      "|   34 | 1.4248654794e+00 | 1.2765194418e-03 | 5.0911707697e-05 | 8.0176836638e+01 | 1.4951574147e-02 |\n",
      "|   35 | 1.4148747810e+00 | 1.3745158553e-03 | 3.9963867745e-05 | 6.2306707281e+01 | 7.0116783150e-03 |\n",
      "|   36 | 1.4075151381e+00 | 1.3344422422e-03 | 5.3221855577e-05 | 8.5646887389e+01 | 5.2016213965e-03 |\n",
      "|   37 | 1.4064478092e+00 | 1.2626462551e-03 | 4.0790304256e-05 | 7.1959607916e+01 | 7.5830725959e-04 |\n",
      "|   38 | 1.4018888587e+00 | 1.1662809147e-03 | 5.1683377884e-05 | 9.7466740791e+01 | 3.2414643734e-03 |\n",
      "|   39 | 1.3899270193e+00 | 1.0793636310e-03 | 3.8743648933e-05 | 7.6297603172e+01 | 8.5326588250e-03 |\n",
      "|   40 | 1.3756849683e+00 | 1.1046632772e-03 | 4.8952811345e-05 | 9.6217077774e+01 | 1.0246617844e-02 |\n",
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      "|   76 | 1.1036419688e+00 | 2.6053388551e-03 | 4.4729549366e-05 | 3.9819303278e+01 | 5.0188236323e-03 |\n",
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      "|   78 | 1.0987343338e+00 | 2.6854396653e-03 | 4.6104753646e-05 | 4.1314401781e+01 | 3.7046050981e-03 |\n",
      "|   79 | 1.0961606897e+00 | 2.6451234180e-03 | 3.3105776226e-05 | 2.9359704768e+01 | 2.3423715728e-03 |\n",
      "|   80 | 1.0945032654e+00 | 2.8320733470e-03 | 4.8622222148e-05 | 4.3463337200e+01 | 1.5120267444e-03 |\n",
      "|   81 | 1.0893334929e+00 | 2.4768745237e-03 | 3.1000010496e-05 | 2.7143124865e+01 | 4.7233961115e-03 |\n",
      "|   82 | 1.0859467838e+00 | 3.0586762683e-03 | 4.7261372463e-05 | 3.8526800390e+01 | 3.1089736945e-03 |\n",
      "|   83 | 1.0840431935e+00 | 2.8514309222e-03 | 3.2119087545e-05 | 2.6406554054e+01 | 1.7529314708e-03 |\n",
      "|   84 | 1.0816934356e+00 | 3.1468674583e-03 | 4.8624065443e-05 | 3.9833618603e+01 | 2.1675869130e-03 |\n",
      "|   85 | 1.0785973119e+00 | 2.7262154084e-03 | 3.0708634983e-05 | 2.5118649688e+01 | 2.8622932002e-03 |\n",
      "|   86 | 1.0779217441e+00 | 3.3076609385e-03 | 5.1108578314e-05 | 4.1623806404e+01 | 6.2633922207e-04 |\n",
      "|   87 | 1.0773803551e+00 | 2.5757714639e-03 | 3.2237782598e-05 | 2.9681337805e+01 | 5.0225260981e-04 |\n",
      "|   88 | 1.0763350048e+00 | 2.8250666765e-03 | 4.8501928691e-05 | 4.6239666424e+01 | 9.7027039307e-04 |\n",
      "|   89 | 1.0749157191e+00 | 2.2682094631e-03 | 3.1542672244e-05 | 3.2029937902e+01 | 1.3186282173e-03 |\n",
      "|   90 | 1.0709550223e+00 | 2.6308125231e-03 | 4.5166891973e-05 | 4.4949589705e+01 | 3.6846579486e-03 |\n",
      "|   91 | 1.0669830429e+00 | 2.3368174151e-03 | 2.9247086884e-05 | 2.6743342142e+01 | 3.7088200286e-03 |\n",
      "|   92 | 1.0651741436e+00 | 3.1295968792e-03 | 4.8357207757e-05 | 4.2561493599e+01 | 1.6953402686e-03 |\n",
      "|   93 | 1.0609881658e+00 | 2.4829439689e-03 | 2.7968376889e-05 | 2.3225641807e+01 | 3.9298529507e-03 |\n",
      "|   94 | 1.0592452544e+00 | 3.5875337669e-03 | 4.9889749433e-05 | 3.9197223256e+01 | 1.6427247136e-03 |\n",
      "|   95 | 1.0563944148e+00 | 2.7274699672e-03 | 2.7650489926e-05 | 2.1328028655e+01 | 2.6913876556e-03 |\n",
      "|   96 | 1.0535518057e+00 | 3.8994000048e-03 | 4.8804022941e-05 | 3.5125154615e+01 | 2.6908595640e-03 |\n",
      "|   97 | 1.0528532660e+00 | 3.0994186770e-03 | 2.8279101637e-05 | 2.0754629832e+01 | 6.6303316396e-04 |\n",
      "|   98 | 1.0507178885e+00 | 4.0075955135e-03 | 5.0158173857e-05 | 3.6238621771e+01 | 2.0281814757e-03 |\n",
      "|   99 | 1.0492594938e+00 | 2.9748235392e-03 | 2.7142295373e-05 | 1.9807150235e+01 | 1.3879982916e-03 |\n",
      "|  100 | 1.0481487956e+00 | 4.1967573369e-03 | 5.2525681155e-05 | 3.7631110983e+01 | 1.0585544154e-03 |\n",
      " ----------------------------------------------------------------------------------\n",
      "  4.729992 seconds (9.90 M allocations: 763.792 MiB, 2.37% gc time)\n"
     ]
    }
   ],
   "source": [
    "I = GaussLegendre(128)\n",
    "proj = FourierProjector(3)\n",
    "@time res, rn = reparametrize(q, r, proj; I=I, rtol=1e-6, maxiter=100);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Results Plots "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "γ = rn.γ\n",
    "\n",
    "p1 = plot(γ, 0 , 1, label=\"Found\")\n",
    "plot!(ψ, 0, 1, label=\"True\", ls=:dash, c=:black, title=\"Diffeomorphism\")\n",
    "p2 = plot(res.errors, yscale=:log10, ylabel=\"Error\", xlabel=\"Iteration\", title=\"Convergence\",\n",
    "    legend=false)\n",
    "\n",
    "plot(p1, p2, size=(800, 300))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Coordinate Functions\n",
    "### Before and After Matching"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
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       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p1 = plot(x -> c1(x)[1], 0, 1, ls=:dash, legend=false)\n",
    "plot!(x -> c2(x)[1], 0, 1)\n",
    "\n",
    "p3 = plot(x -> c1(x)[2], 0, 1, ls=:dash, legend=false)\n",
    "plot!(x -> c2(x)[2], 0, 1)\n",
    "\n",
    "p2 = plot(x -> c1(x)[1], 0, 1, ls=:dash, legend=false)\n",
    "plot!(x -> c2(γ(x))[1], 0, 1)\n",
    "\n",
    "p4 = plot(x -> c1(x)[2], 0, 1, ls=:dash, legend=false)\n",
    "plot!(x -> c2(γ(x))[2], 0, 1)\n",
    "\n",
    "plot(p1, p2, p3, p4, size=(800, 300))"
   ]
  }
 ],
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