{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "%matplotlib inline\n", "import torch\n", "import numpy as np\n", "import timeit\n", "import matplotlib.pyplot as plt\n", "import matplotlib" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## What will we be doing\n", "\n", "1. Learn 3 techniques to solve ODEs\n", "2. Program them\n", "3. Plot them\n", "\n", "This time we will be using PyTorch so you'll have a chance to experience the package." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# The forward Euler method\n", "\n", "We assume a general nonlinear system of Ordinary Differential Equations of the form\n", "$$ \\dot{\\bf y} = f({\\bf y},t) \\quad \\quad {\\bf y}(0) = {\\bf y})_0. $$\n", "\n", "Using a simple finite difference approximation to the derivative we obtain\n", "$$ {\\frac {{\\bf y}_{j+1} - {\\bf y}_j}{h}} = f({\\bf y}_j,t_j) + {\\cal O}(h) $$\n", "\n", "Yielding the approximation\n", "$$ {\\bf y}_{j+1} = {\\bf y}_j + hf({\\bf y}_j,t_j) $$\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The forward Euler method can be easily coded " ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "def forwardEuler(f,h,n,y0):\n", " \n", " k = y0.shape[0]\n", " Y = torch.zeros(k,n+1)\n", " Y[:,0] = y0\n", " \n", " t = 0.0\n", " for j in range(n):\n", " Y[:,j+1] = Y[:,j] + h*f(Y[:,j],t) \n", " t += h\n", "\n", " return Y" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us test the code" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "# Define our own function\n", "def yfun(y,t):\n", " return -y" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "h = 0.01\n", "n = 1000\n", "y0 = torch.ones(2); y0[1] = -1.0\n", "Y = forwardEuler(yfun,h,n,y0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The analytic solution is $\\exp(-t)$ so this looks good. " ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0.5, 1.0, 'Error in the Numerical vs analytical solution')" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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Sxkr6UNJCSVfUs/wISZskvR28ftHUtjuqm47/AVT15g9v/p7K6sqww3HOuRZLOxFIigI3A8cDI4HTJY2sp+pLZjYqeF3VzLY7nP75+Xxj+CQqI6u44pmbww7HOedaLBNnBKOBhWa2yMwqgfuBcW3QNnRXHvl/dK3el6dW3MOC9UvDDsc551okE4lgEJC6F1wWlNV1sKR3JD0hac9mtkXShZJmSZpVVFSUgbDTF4mIqw//OYYx6clfhh2Oc861SCYSgeopszrzc4ChZrYv8BfgkWa0TRaaTTazQjMrLCgoaHGwmXbkrruzV9dTWFE1k1tm/ifscJxzrtkykQiWAUNS5gcDK1IrmFmxmZUE09OAuKQ+TWnbHtx60g+JVA7hlrnXsXLzjnG24pxzTZWJRDATGCFpuKQcYDwwNbWCpP6SFEyPDvpd15S27UGPLp356Rd/TUJlfOvxn4UdjnPONUvaicDMqoFJwHRgPvCgmc2TNFHSxKDaKcBcSe8AfwbGW1K9bdONKQynjfoie3T6GksqXuW22Y+GHY5zzjWZzOodkt+hFRYW2qxZs8IOYxvrt2zhyHtPwWLreOTkh9m5Z73XvZ1zLhSSZptZYd1y/2ZxBvXq2pWrDv49Car55mMXU5OoCTsk55xrlCeCDBu31ygO6/VtNiQ+4NIZN4QdjnPONcoTQSu46cTzya8ezdOr7uGJBa+GHY5zzm2XJ4JWEI9FufOkq1F1T37y8o9ZumlV2CE551yDPBG0kt0KCrhi/6upZgvjH/0OFdUVYYfknHP18kTQis7c/yCO7fsDim0h35x6ZdjhOOdcvTwRtLLrTziHIfoKczdP57cv3h52OM45tw1PBK1MEvef+is6Ve7FA4v+zD3vPhF2SM45txVPBG0gv0snHvjaX4lUDeaa2VfyzKKZYYfknHOf8UTQRnbp05u/H/s3qMnnkhe+z9zVH4cdknPOAZ4I2tRBQ4fxm4NuIpEwzp52HgvWfxp2SM4554mgrZ28175cvPcfqbIyTnt0AovWLws7JOdclvNEEIILRh/GRV+4hkrbzKmPTmDxxnb3CAbnXAfiiSAk3zn4SCbufjUVton/e+QsPly7OOyQnHNZyhNBiCYdegwTd7+GikQJp/3vLN5c3i4fxeCca+cykggkjZX0oaSFkq6oZ/mZkt4NXq9K2jdl2WJJ70l6W9KO95CBVjbp0KP56ai/Up0wLphxLtMXvhZ2SM65LJN2IpAUBW4GjgdGAqdLGlmn2ifA4Wa2D/AbYHKd5Uea2aj6HpiQDc7YfzTXHToZq+nGZS9/l5vffCDskJxzWSQTZwSjgYVmtsjMKoH7gXGpFczsVTPbEMy+TvIh9S7F2D325M7j7iZaNYxb5/+Wi574LQlLhB2Wcy4LZCIRDAKWpswvC8oacj6Q+jsLBsyQNFvShQ01knShpFmSZhUVFaUV8I6qcMgQpo+/hx7VX+bFNQ9w4gMXsLG8OOywnHMdXCYSgeopq/dByJKOJJkILk8pPtTM9ic5tHSRpMPqa2tmk82s0MwKCwoK0o15h9UvrxtPn/MX9ux0DkvLZ3PU/Sfz0pI5YYflnOvAMpEIlgFDUuYHA9vcGC9pH+B2YJyZrastN7MVwfsa4L8kh5qyWqd4lCmnXcZ5O/+Ryuoqvvvcefz6hVsxqze/OudcWjKRCGYCIyQNl5QDjAemplaQtBPwMHC2mX2UUt5NUl7tNHAsMDcDMbV7kvjhYcdx93FT6FQ1kocW38yxU85m8YblYYfmnOtg0k4EZlYNTAKmA/OBB81snqSJkiYG1X4B9Ab+Vuc20X7Ay5LeAd4EHjezJ9ONqSPZf8hgXjznbvbuPIGVFe9z0iMnc91rd/nZgXMuY9QedyiFhYU2a1bWfeWAR+e+yy9f/RU1nRbQN7Y3N465ir377hp2WM65dkLS7Ppu0/dvFrcj4/bah5fOuZ9RXc5jdcVHnDHtFL79+FVsrigJOzTnXDvmiaCdyeucw7++cQm3HvEgedWjeXXtv/nyfWP58xv3UZOoCTs851w75ImgnfrSzjvz8nl/5/ydbyBRnc9tH/yBg+4+gTveetS/iOacaxZPBO1YJCIu/vIYXj3nEcb0uYzyqmr+9O7POPTuk7jn3WmeEJxzTeIXizuQ9aXlXPnUXby09j4UX09nG8CpI87k+weeRudY57DDc86FrKGLxZ4IOqCizaX89vn7eW7Vv7Gc5UQT+Xyp/0n86JCzGdp9YNjhOedC4okgC5VVVnPDy4/zn4/vpSpnPpjon7Mf39xrPKftNYZYJBZ2iM65NuSJIIuZGf97fy5/nzOFJZXPoVgJ0UR39u5xGOeN+jpHDDsAqb6fjHLOdSSeCBwAazaXcsMrj/Dssmlsic5DkRriiT4c0OdIztr7RL48dD8i8nsInOuIPBG4bbyzYiWTZ07l9TXPUBH/EClBJJHPLt0O4Phdjmb8XkeT1yk37DCdcxniicA1yMyYtXQZ9747nTdXv0Kx5qJoOViUfO3Cnr32Y+yuh3L8rgfRJd4l7HCdcy3kicA1WdHmUu55+0WeWfICS8veoya+FMmSiSGyC7t135sDB45i7IjRDO0+wK8vONdOeCJwLWJmfLC6iIfff4nXVrzJ0rK51MSXIyV/ziKSyKcgviu79RzJ/v1HcujQvdi991C/zuDcDsgTgcuYZRs3Mf2jt3ht+dt8tPF9NtR8DPE1n1dI5NBVA+nfeRi79NyVPQt2ZVT/Xdmr7zA6xTqFF7hzWc4TgWs1ZsZHa4p45dN5vLXqQz7etJA15Uso13IU25xST8SsJ7mRfvTpPIDBeUMY3n0IO/ccyB4FQ9il50ByYjkhbolzHZsnAtfmqmsSzF21gjkrF/BB0ScsLv6UNWUr2FS9iioVodi2P5+tRB6d6ElurBc9cgro1bk3fbr0pl+33gzM78OQ/AKG9exH/9xeRCPRELbKufaroUSQka+WShoL3AREgdvN7Oo6yxUsPwHYAnzTzOY0pa1rv2LRCKMGDWbUoMHAkVstq0kYi9atY37Rp3y8fjmfblrJqi2rWFdeRHHVWtZXrKGo6iNUvgU2brtuMxGxrkQtl7hy6RTpSudoN7rGcukWzyUvJ5funfLp0SmfXl3y6d21Oz065wavbvTs0o38zl3JifoZiHNpJwJJUeBmYAzJB9nPlDTVzN5PqXY8MCJ4HQjcAhzYxLauA4pGxIiCPowo6APsX28dM2NTeTmfrF/Dkk1FrChey6qStRRtWc/68vUUV26ktHoTFYlSSqs3sal6FYmKMoiUo0h1k+IwiyKLI+tEhByidCKmlFckTlRx4pE4sUjyPR7JIScSJx7NISeaQ6doDjnROJ1jOXSKdqJzLE5OsCweiRGLRIlHY8QjUWLRGDnRKDnRGPFI8pUTi5ETjZETjZMTTdbNicXoFJTFI1GikajfneVaTSbOCEYDC81sEYCk+4FxQOrOfBxwtyXHoV6X1EPSAGBYE9q6LCWJHl26sN+goew3aGiT29UkjHWlpawq2cDqko2sLS1mfdkmiitLKK3cQmlVGVuqyyirKqOsppyKmjIqaiqorCmjyiqoSlRQkShjC5swqklQjVEFqsGoBlWjSAMPATIjmoB4NUQTn78iBpEG5iMJiH42b5+XW8ryBEQTQpYsl4FMyWmSdSIky2rr1Nb7rD4p5YhIAmQWlKesy5LlkWBd4vP2yW1M+RvVvgdDzJ/Pb7082aMlS4I+P1teZ92p6U719VVPvXr7/KyObRP7VutqqTRG1Wu3oSV6nnU+R552ccs7r0cmEsEgYGnK/DKSR/2N1RnUxLYASLoQuBBgp512Si9i16FFI6JvXi5983Kxyj4k1q8isT5KYhMkqmpIVJRjpQkSWypJlG7BtmwhUVZKoqwMKysnUV5OorwSq6wiUVWDVde+EsG7YTUJEjWQqDGsBiwB1ACJ9nnUngj2oiaS++pg2oLN+ey9tkHKZtYt23peW+/v6tapna6nvL71bdWuno/a6ournroN1Wu2Fv650+mzfNPaNFrXLxOJoL6Pou52NlSnKW2ThWaTgcmQvFjcnABdO2eGbV5L9YpF1KxcQs3qZVSvWUnNuiJqNm6iZnMJNaVlJLaUk9hSSU15NYmKBDWVRqJK9e8xGqCIoThEYiISj6CYUDSCYhEUixHp3AnFoygWS77H40TiMRSPo5wclBNPTsdzUDwGsRiKxlDs82liMRSLB9PxYFnQJhqHeHI5sZzkumI5Qb0oKALRKFIUolGIRCASRZEoRGMgJeejUVAUopFkP5/VT2kbtPchJ5eJRLAMGJIyPxhY0cQ6OU1o6zqq2h38J/OoWvIh1Us/oWrlcqpXr6Fq7QaqN5ZRXVpFTZmRqGrgC2oyIjkQzYkQ6RIj2jlOvCCXaNcuRPK6EsnNJZqXTyS/O9G8PNQtj0i3fCK5+Si3B5H8HkRye6L8XkS65SV3ys5lmUz8q58JjJA0HFgOjAfOqFNnKjApuAZwILDJzFZKKmpCW9ee1VRTs3Q+lfNnUblgHlVLPqFy+Sqq1myicmMV1VuCsYgUihqxvBjx7p3p0rc30R75RHv2JNangGhBf2J9BxEdMJTowOFEe/dNHv0651os7URgZtWSJgHTSd4CeoeZzZM0MVh+KzCN5K2jC0nePnru9tqmG5MLQSJBzafvUTHnJSrmvUXFx4uoWLqWirWV1FRsfTQf6ybivbrSbc+BxAf0JzZwEPEhOxMbujvxnfck0rOXD1c414b8C2Wu+cyoWfIu5S8/Qdlbb1L24WLKV5RSveXzHX4kDp36dqXTTv3IGT6cnF33IL77KHJ2349Irv+0tXNhaNUvlLmOzaoqqJozgy3PTmXLnHcpW7KRyuLPd/o5PWN03WMnOu8+gk57HUCnAw4nNnQXP6p3rp3wROC2lUhQ/f4LlDw2hS2z36F00UaqS5M7/mgX0WXnfnTf8wt0GX04nQ85jmiv3iEH7JxLhycCB4CVFVP+1L2UzPgfJW8vonxt8mg+2kV03W0wXQ8cTbdjTyFnz/39SN+5DsYTQRazilLKH7+NTY8+zOZ3V1NdFgGMLkPyKDjti+SOO5tO+x3kO37nOjhPBNkmUUPFs/ey6cG7KZ69lKrSCIoauSMHknfc8XQbN4FYQb+wo3TOtSFPBFkisXoRm++8mg1PvETZakBGt9370ufEr5D3jW8T7d4j7BCdcyHxRNCRmVH15lTW33I9G2evJlEVIadXnL7njKH7+ZcR6zcg7AidczsATwQdUaKGssdvZf0/bqf4wzIA8vcfRo9zv0vXo0/yMX/n3FY8EXQkiRrK/nsTRbf8g9JlCSJx6DW2kF6XXEV8p53Djs45t4PyRNARmFH+5G0U/fkvlHxSTbSL6Hv28fSY9Asf+3fONcoTQTtX9fYM1vz6cornlxPJgYLTj6HnD39PNC8v7NCcc+2EJ4J2KrF+GeuvmsjapxeAid4nHULvK671b/k655rNE0F7k0hQevcvWfW3B6ksjpC792D6XX0zObvsEXZkzrl2yhNBO1KzdC5rfnQeG9/eTLxnJ4Zc91NyvzI+7LCcc+2cJ4L2wIzSO3/GipsforpU9DrxIAp+8zciXbuGHZlzrgPwRLCDs01rWHPxKax/rYic3p0ZetN1dP3SmLDDcs51IA08CLZpJPWS9JSkBcF7z3rqDJH0nKT5kuZJ+kHKsl9JWi7p7eB1QjrxdDRVb/6PJV89nPWvFdHjqH0Y/vTrngSccxmXViIArgCeMbMRwDPBfF3VwKVm9gXgIOAiSSNTlv/JzEYFr2lpxtMxmFEy+ccsOv8yKjbCoF98nwF/e4BIly5hR+ac64DSHRoaBxwRTN8FPA9cnlrBzFYCK4PpzZLmA4OA99Psu0OyqnLW/+RU1jy+gE79ujH4tnvIGTGy8YbOOddC6Z4R9At29LU7/L7bqyxpGLAf8EZK8SRJ70q6o76hpZS2F0qaJWlWUVFRmmHvmGzDClae8WXWPLaQvH13YthjL3oScM61ukYTgaSnJc2t5zWuOR1JygX+A1xsZsVB8S3ALsAokmcN1zfU3swmm1mhmRUWFBQ0p+t2oWbpPD79v2PY9F4JfU45gkH3PekPeXfOtYlGh4bM7EC3HmQAAA20SURBVJiGlklaLWmAma2UNABY00C9OMkkcK+ZPZyy7tUpdW4DHmtO8B1F9fxXWHrB+ZSvh4GXT6T7uReHHZJzLoukOzQ0FZgQTE8AHq1bQcnfPP4HMN/MbqizLPUH8b8GzE0znnanauZjLJlwHhUbxZBrfu5JwDnX5tJNBFcDYyQtAMYE80gaKKn2DqBDgbOBo+q5TfSPkt6T9C5wJHBJmvG0K5VvTGXxty+luiLCTrfcQO5JZ4YdknMuC6V115CZrQOOrqd8BXBCMP0yUO+TUMzs7HT6b8+qZk3j04t+RKImwtB/TKZz4ZfDDsk5l6X8m8UhqHprBksmXkJNZYSdbrvFk4BzLlTpDg25ZqpeMItPJ34vmQRu/QtdDjwi7JCcc1nOzwjaUKJoMcsuOJuqErHTX/5Il0MavCHLOefajJ8RtBEr3cDyCSdRttoY+PNL6HrUV8MOyTnnAE8EbaOmmtUXnkjJoir6fftU8sd/O+yInHPuM54I2sCGq85hw+wN9DphNL0u+U3Y4Tjn3FY8EbSyLf++nlX/nkO33Qvoe+0dYYfjnHPb8ETQiqree45lv5tMPD/OoDv/i6LRsENyzrlteCJoJVa8luWTvkuiOsKQv/+DaK/eYYfknHP18kTQGswouuw0ylbDwCsuotO+o8OOyDnnGuSJoBWU/us3rHtxOT2+vAf5Z00KOxznnNsuTwQZVv3RTJb/6V5yenei3433hB2Oc841yr9ZnEFWVcHKH5xPolLsdPvNRLp1Czsk55xrlJ8RZNCma79LySdV9D33ZDof8KWww3HOuSbxRJAhVe+9wOopL9N1WD49f/j7sMNxzrkmSysRSOol6SlJC4L3eh8+L2lx8ACatyXNam77HZ1VV7Lqx9/HEmLAjbehiOdX51z7ke4e6wrgGTMbATwTzDfkSDMbZWaFLWy/wyq+/nuUfFJJ3wknkbPHPmGH45xzzZJuIhgH3BVM3wWc3MbtQ1ez+D1W3/c8XYbk0vOyq8MOxznnmi3dRNDPzFYCBO99G6hnwAxJsyVd2IL2SLpQ0ixJs4qKitIMO3PWXPkdaipF/z/c4ENCzrl2qdHbRyU9DfSvZ9GVzejnUDNbIakv8JSkD8zsxWa0x8wmA5MBCgsLrTltW0vZY5PZOHstvcbs44+bdM61W40mAjNr8DFaklZLGmBmKyUNANY0sI4VwfsaSf8FRgMvAk1qvyOy8hJWXXMj0a4R+vzm1rDDcc65Fkt3LGMqMCGYngA8WreCpG6S8mqngWOBuU1tv6PaeP3FlBcZ/SadR7RHr7DDcc65Fks3EVwNjJG0ABgTzCNpoKRpQZ1+wMuS3gHeBB43sye3135Hl1izmKKHXqLLTt3IP/fSsMNxzrm0pPUTE2a2Dji6nvIVwAnB9CJg3+a039Gt+80kasoi9Lvy10gKOxznnEuL3+bSTFXzX2fdswvJGzWALoefGHY4zjmXNk8EzbT2qksxg76/vjHsUJxzLiM8ETRDxcv/ZePb6+h11N7k7O7fIHbOdQyeCJph7Q1/IBKD3j/3swHnXMfhiaCJKl74N8XvF9Pz2EJi/QaFHY5zzmWMJ4ImKvrTNUTi0PuKa8MOxTnnMsoTQROUP3s/mz8opefY0UQLBoQdjnPOZZQngiZYe+N1ROJG78v9bMA51/F4ImhExSuPsvmjUnqO/SLRPv3CDsc55zLOE0Ej1t18LYoavfzxk865DsoTwXZUffAmxW+vpcehuxEbMCTscJxzrlV4ItiODTf+EjPodcmvwg7FOedajSeCBtSsWsSGVxaRt3d/cr6wf9jhOOdcq/FE0ICNN/2MRFWE3t//cdihOOdcq/JEUA+r2ML6GXPoOrQbXb50QtjhOOdcq/JEUI+Sf11LdanoeeYZYYfinHOtLq1EIKmXpKckLQjee9ZTZ3dJb6e8iiVdHCz7laTlKct2iMPvDQ8+TCxX5I2fFHYozjnX6tI9I7gCeMbMRgDPBPNbMbMPzWyUmY0CDgC2AP9NqfKn2uVmNq1u+7ZW8er/KP20kp7HHYRycsIOxznnWl26iWAccFcwfRdwciP1jwY+NrMlafbbajbcdhOKGD0u+nnYoTjnXJtINxH0M7OVAMF730bqjwem1CmbJOldSXfUN7RUS9KFkmZJmlVUVJRe1A2oWbOUTTOXkb/fIGIDh7dKH845t6NpNBFIelrS3Hpe45rTkaQc4CTg3ynFtwC7AKOAlcD1DbU3s8lmVmhmhQUFBc3pusmKb/sdiWrR84Lvtcr6nXNuRxRrrIKZHdPQMkmrJQ0ws5WSBgBrtrOq44E5ZrY6Zd2fTUu6DXisaWG3AjM2Tn+JTgUxOh/RrBznnHPtWrpDQ1OBCcH0BODR7dQ9nTrDQkHyqPU1YG6a8bRY+Uv/pXxNgh5jj0BSWGE451ybSzcRXA2MkbQAGBPMI2mgpM/uAJLUNVj+cJ32f5T0nqR3gSOBS9KMp8U23nUrihj55/s3iZ1z2aXRoaHtMbN1JO8Eqlu+AjghZX4L0Lueemen03+mJEo2UDzzU3L37Eusv//KqHMuu/g3i4GSu6+nplL0GH9W2KE451yb80QAbJw6jVgudBt3XtihOOdcm8v6RFA1/01KF2+hxxH7oVhaI2XOOdcuZX0iKL7nr4DoPuH7YYfinHOhyO5EYMamF2bTeUAOOXsfFHY0zjkXiqxOBBVvPEnF2gTdjz407FCccy40WZ0INk25DWTkn31x2KE451xosjYRWE0Nxa/Op9vwXGJDdws7HOecC03WJoLyp6ZQtRnyx44JOxTnnAtV1iaCTQ/dgyJG3pk/CDsU55wLVVYmAquqpHj2YnL36E20d/+ww3HOuVBlZSIom34vNWUi77hjww7FOedCl5WJYPPUh1DEyD11YtihOOdc6LIuEVgiQfGcRXTbtTvRXv3CDsc550KXdYmg/Pn/UF0CeUcdEXYozjm3Q8i6RLD5kSkgI/cbPizknHOQZiKQdKqkeZISkgq3U2+spA8lLZR0RUp5L0lPSVoQvPdMJ57GWCLB5jc/oOvQbsQGDm/Nrpxzrt1I94xgLvB14MWGKkiKAjeTfHj9SOB0SSODxVcAz5jZCOCZYL7VVM58isqNRv7hh7RmN845166klQjMbL6ZfdhItdHAQjNbZGaVwP3AuGDZOOCuYPou4OR04mlM8UN3AUbueB8Wcs65Wm1xjWAQsDRlfllQBtDPzFYCBO99G1qJpAslzZI0q6ioqEWBxAcMpPsXBxEfvmeL2jvnXEfU6CO5JD0N1Pf12yvN7NEm9KF6yqwJ7bZuYDYZmAxQWFjY7PYAPX54HT1a0tA55zqwRhOBmR2TZh/LgCEp84OBFcH0akkDzGylpAHAmjT7cs4510xtMTQ0ExghabikHGA8MDVYNhWYEExPAJpyhuGccy6D0r199GuSlgEHA49Lmh6UD5Q0DcDMqoFJwHRgPvCgmc0LVnE1MEbSAmBMMO+cc64NyaxFw+2hKiwstFmzZoUdhnPOtSuSZpvZNt/5yrpvFjvnnNuaJwLnnMtyngiccy7LeSJwzrks1y4vFksqApa0sHkfYG0Gw2kPfJuzg29zdkhnm4eaWUHdwnaZCNIhaVZ9V807Mt/m7ODbnB1aY5t9aMg557KcJwLnnMty2ZgIJocdQAh8m7ODb3N2yPg2Z901Auecc1vLxjMC55xzKTwROOdclsuqRCBprKQPJS2U1KrPR94RSBoi6TlJ8yXNk/SDsGNqC5Kikt6S9FjYsbQFST0kPSTpg+BvfXDYMbU2SZcE/6bnSpoiqXPYMWWapDskrZE0N6Wsl6SnJC0I3ntmoq+sSQSSosDNwPHASOB0SSPDjarVVQOXmtkXgIOAi7JgmwF+QPInz7PFTcCTZrYHsC8dfNslDQK+DxSa2V5AlORzTjqafwJj65RdATxjZiOAZ4L5tGVNIgBGAwvNbJGZVQL3A+NCjqlVmdlKM5sTTG8muYMYtP1W7ZukwcCJwO1hx9IWJOUDhwH/ADCzSjPbGG5UbSIGdJEUA7ry+VMPOwwzexFYX6d4HHBXMH0XcHIm+sqmRDAIWJoyv4wOvlNMJWkYsB/wRriRtLobgR8DibADaSM7A0XAncFw2O2SuoUdVGsys+XAdcCnwEpgk5nNCDeqNtPPzFZC8kAP6JuJlWZTIlA9ZVlx76ykXOA/wMVmVhx2PK1F0leANWY2O+xY2lAM2B+4xcz2A0rJ0HDBjioYFx8HDAcGAt0knRVuVO1bNiWCZcCQlPnBdMDTybokxUkmgXvN7OGw42llhwInSVpMcujvKEn3hBtSq1sGLDOz2jO9h0gmho7sGOATMysysyrgYeCQkGNqK6slDQAI3tdkYqXZlAhmAiMkDZeUQ/Li0tSQY2pVkkRy7Hi+md0Qdjytzcx+YmaDzWwYyb/vs2bWoY8UzWwVsFTS7kHR0cD7IYbUFj4FDpLUNfg3fjQd/AJ5iqnAhGB6AvBoJlYay8RK2gMzq5Y0CZhO8i6DO8xsXshhtbZDgbOB9yS9HZT91MymhRiTy7zvAfcGBziLgHNDjqdVmdkbkh4C5pC8M+4tOuBPTUiaAhwB9JG0DPglcDXwoKTzSSbEUzPSl//EhHPOZbdsGhpyzjlXD08EzjmX5TwROOdclvNE4JxzWc4TgXPOZTlPBM45l+U8ETjnXJb7f8L33/mst7R9AAAAAElFTkSuQmCC\n", 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "t = np.arange(0,n+1)*h\n", "# The solutions \n", "plt.figure(1)\n", "plt.plot(t,Y[0,:],t,Y[1,:], t, np.exp(-t),t, -np.exp(-t));\n", "plt.title('Numerical vs analytical solution')\n", "# The error\n", "plt.figure(2)\n", "plt.plot(t,Y[0,:].numpy()-np.exp(-t),t,Y[1,:].numpy()+np.exp(-t));\n", "plt.title('Error in the Numerical vs analytical solution')\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "But let us run this again with a few choices of step size $h$." ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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\n", 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gbwQWmtl8MysF1gDr4ndwzs13zs1zzs0Dfgp8wjk34YeEDNuOLpmSCrjoE9D6e9izOYOViYiM3oiB7pzrB27GG72yHXjQObfNzG4ys5syXWAmDduOLpXlH/Ha1OksXUQmmOJ0dnLOrQfWJ6xL+gWoc+5D4y8rexaEK2kdTaCXT4YVH/WGMB54GWYszFxxIiKjULB3isY01Vaxs+N48nZ0qaz8OBSXw+O3Za4wEZFRKvhAbwxXcbIvRTu6VKrCcOEN8PxaONw28v4iIllQ8IHelO6cLokuvtl7fPJ7PlckIjI2BR/oac+6mGhKA5y3BjbdB125v0lKRKTgA31GVSmTK0pGf4YO8ObPQH83PH2n/4WJiIxSwQe6mdEYrhz9GTp4I1wWXwMb7oHuI/4XJyIyCgUf6OBddmntSGOSrmTe/DnoOQIbv+9vUSIio6RAx/ti9EBXD0dODNOOLpU550PT27xm0n0n/S9ORCRNCnTivhg9MIbLLuCdpR/vhGf/1ceqRERGR4HOOIYuxpxxCTRcBH/+DgyM4SxfRMQHCnSgPt12dKnE2tQdaYMt/+ZvcSIiaVKg47Wjmz+jMv1pdJNZ+HaYeS48/m2IRPwrTkQkTQr0qMbayvTa0aViBqs+Cwdegh2/9K8wEZE0KdCjGsNVvHbweHrt6FJZ/E6YtkBt6kQkJxToUU21VUQc6bWjSyVUBG/6DOzdDLv+6F9xIiJpUKBHjbodXSpL10D1HG++dBGRLFKgR426HV0qxWVwyc3w6p+gbYMPlYmIpEeBHjWmdnSpXHADVEzTWbqIZJUCPU5jbdXo2tGlUlYFF30cXvo17N82/uOJiKRBgR6nMVw5+nZ0qaz4GJRWeePSRUSyQIEeZ0zt6FKpmArN/xW2/gwO7Rr/8URERqBAjzPuOV0SXfxJCJV4c7yIiGSYAj3OmNvRpVI9C5a9DzbfD0f3+nNMEZEUFOhxZlSVMr2ylN9v34/z607PSz4NkQE1kxaRjFOgxzEzPvWXTfy59SDrntvjz0GnzYdz3w0tP4QTh/w5pohIEgr0BB+4eB5LG6bwd4+8wOETvf4c9M2fhb7jsOFuf44nIpKEAj1BUcj4+rXncvhkH19bv92fg9aeDWdeBU/dCT3H/DmmiEgCBXoSi+fU8LFVC3iwpZ0ndx7056CrPgfdh2HTvf4cT0QkgQI9hVsuXcjcaZP40s+30N03jil1Y+qbYf5bvCGMb7w2/uOJiCRQoKdQUVrE/7l2CbsOHOeOP7b6c9DLvu71HL33aoW6iPgurUA3s8vN7EUzazWzW5Nsf5+ZPR/9ecLMlvpfavatWhjm2mV13PkfO3lpvw/XvmctgQ/+AnqOwr1XwRuvjv+YIiJRIwa6mRUBtwNXAIuB681sccJurwD/yTl3HvBVIDDDOf7XVWdTWVbMXz+0xZ85XuacDzesg94u+OFVcOiV8R9TRIT0ztBXAK3OuV3OuV5gLbA6fgfn3BPOuTeii08B9f6WmTvTq8r40pVns+m1N3hg425/Djp7KXxwnTeU8d6rNdeLiPginUCvA9riltuj61L5CPDrZBvM7EYzazGzls7OzvSrzLF3X1jPxQum8431O9jvx8RdALPPgxsegb4TXqgf3OnPcUWkYKUT6JZkXdJrD2b2F3iB/sVk251zdzvnmp1zzeFwOP0qc8zM+Np159IzEOErj/g4v/msc71Q7+9WqIvIuKUT6O1AQ9xyPTDkvngzOw+4B1jtnPNp8PbEMX9GJbdcupD1W/bx+xf2+3fgWUu8UB/o8b4oVaiLyBilE+gbgYVmNt/MSoE1wLr4HcxsLvAQ8AHn3Ev+lzkxfGzVAhbNrOLLv9hKV0+/fweeeQ7c8EtvSOMPr4QDPg2TFJGCMmKgO+f6gZuBR4HtwIPOuW1mdpOZ3RTd7cvAdOAOM9tsZi0ZqziHSotDfP2689h7tJtv/vZFfw8+czF86JfgBrwz9QMv+3t8EQk8822a2FFqbm52LS35mft/8/BWfvz0a/z8E29iacMUfw/esQPuuxos5J21hxf5e3wRyWtmtsk515xsm+4UHYMvXH4m4eoybn1oC30DEX8PXnuWF+TOeWfqnT7/S0BEAkuBPgY15SV85Zpz2L73KD94PAM3BtWeBR/6FZh5o186dvj/HiISOAr0MbrsnFm8ffFMvv37l2g7dML/Nwgv8s7UzbxLMB0+TeUrIoGlQB8jM+PvVp9DkRlfenirfy3r4oUXRc/Ui7wz9f0v+P8eIhIYCvRxmD25gi9cdiaPvdTpX8u6RDMWeqFeVAL3vQP2+3hjk4gEigJ9nDLSsi7RjKZoqJd6ob5va2beR0TymgJ9nIpCxjeuO5cjfrasS2Z6ozdOvagsGupbMvdeIpKXFOg+OHt2DR97i9ey7omdBzL3RrFQL6mA+65RqIvIIAp0n5xuWbfVn5Z1qZwK9Unemfre5zP3XiKSVxToPikv8VrWvXLgOLf71bIulWkLvFAvrfJCfc/mzL6fiOQFBbqPVi0Mc92yOu7yq2XdcKbN90K9rBp+tFqhLiIKdL996aqzqfKzZd1wps6LhnoN/Oga2PNsZt9PRCY0BbrPpleV8aWrFrPptTe4f4NPLeuGEwv18snemfpzP4EBH6f2FZG8oUDPgHddUMcljdP5+1/72LJuOFPP8Map19TDz2+Ef1oGG/4F+k5m/r1FZMJQoGeAmfG1a8+l1++WdcOZMhduehzW3A+VtbD+v8O3l8Bj/wgnD2enBhHJKQV6hsybUcmnoy3rfudny7rhhEJw1lXw0d97Z+xzzod//6oX7L/7Mhzbl506RCQnFOgZ9LFVCzhzZrX/LetGYgbz3gzv/xn8t8dg4dvhiX+C286FR25R31KRgFKgZ1BpcYivXXcu+zLRsi5ds5fCe34IN7fA+e+DzffD95rh3z4Me5/LTU0ikhEK9Ay78IypfOCiM7j3iVfZ3JbDa9nTG+Edt8FntsAln4aXfwf//Bb4v9fBK3/yOiSJSF5ToGfBFy47k9rqMv46Ey3rRqt6Frz9K/DZrXDpl2Hf814DjXveBtt/CZEc1yciY6ZAz4Lq8hK+cs0Stu89yvcz0bJuLCqmwKrPe2fsV30TjnfCT94Hd1zkXZYZ6Mt1hSIySgr0LLl8ySz+avFMvvHrHfyXf36Sh599PbOTeKWrpAKWfxQ+9Qy86/teI42HPw7fOR+euhN6j+e6QhFJk2WkdVoampubXUtLS07eO1e6evr50ZOv8pONbbx28ASTK0q47oI6rl8xl0Uzq3Ndnsc57/r649+G3U9AxTRYeROs+BhMmpbr6kQKnpltcs41J92mQM++SMTx1K6D3L9hN49u20ffgOOCuVNYs2IuV583m0mlxbku0bP7KS/YX/oNlFTCkmth7sVQvxymL/TGvYtIVinQJ7CDXT089MzrPLBxN7s6j1NdVszqZXNYs3wuS+om57o8z/5t8Ofvwou/hp4j3rryKVDfDPUroGE51F3ozScjIhmlQM8Dzjk2vvoGazfs5ldb9tLTH+G8+smsWT6Xa86fQ1XZBDhrj0TgwEvQvhHaN0DbRujcATjAIHyWF+71K6Bhhc7iRTJAgZ5njpzo4+fPtrN2Yxs79h1jUmkR7zhvDtevnMvS+smYWa5LPK37CLy+yQv39g1e2HfHzuInQ12zF+71y70zep3Fi4yLAj1POed4tu0wazfs5pHn9nKyb4CzZlXz3pVzWX1+HZMrSnJd4lCRCBxsjZ7BRwO+YzuDzuLrYyG/AmYs0lm8yCgo0APgWHcfv9i8h7Ubd7P19aOUl4S48tzZvHfFXC48Y+rEOmtP1H3UO4tv33g65Lujd83GzuLrLvRmjKyZDdVzvMfyKd68NCJyigI9YLa0H+GBjbtZt3kPXT39NNVWsWZ5A++6oJ6plaW5Lm9kznln8W0bopdpWqDjBXAJd6kWV3h3ttbMgerZ0bCP/sTWVc+G4jz4zCI+UaAH1PGefn71/F7u37CbzW2HKSkyGqZOIlxdxsyacmqry6it8Z6Hq8uorS5nZk0ZVWXFE++Mvr/Hm9732F44uifhca/3eGwv9CdpGDJpxuDAjw/72Bl/xVRd2pFAGHegm9nlwHeAIuAe59w3ErZbdPuVwAngQ865Z4Y7pgLdXzv2HeXhZ/fQdugEHce62X+0h45j3XT3DZ2bpaKkiJk1XsCHa8qYWV1ObU0ZtfG/CKrLqamYYMHvHJx8Iy7k93i/BBJ/ARzvTP76kkoorYSyKiiN/px6Xuk13D71vApKq1M8j/4UTYCRR1JwxhXoZlYEvAS8HWgHNgLXO+deiNvnSuBTeIG+EviOc27lcMdVoGeec45jPf10HO2m42gPHcd62H+0e9BjZ/T5id6h0xCUFYeiQe+d2U+uKKW0yCgpClFaHDr1WFoUoqTIKIk+P7WtKERJsbetLLpu8Gti+3rHDJlhBiEzQsbYf5n090LX/sEhf/Iw9HZBzzFvOoPeLujp8h5PPY+uJ81/tRaXR8N9EhSVRn9KRng+0vYkz60IQkVxjyHvZ9C66PpQKM39Y/uFAPO+q7DY81D0u4sUzyfSL/kCNFygp3OKsQJodc7tih5sLbAaeCFun9XAj5z32+EpM5tiZrOdc3vHWbuMg5lRU15CTXkJTbXDTy3QFQ3+2Jl9Z1zodxztYce+Yxw92U/fQIS+gQi9/RH6I5m/XGcGRizkvYWQgXE69GO/BE49Er++ipAtwgYd04a8BwBFEKqIUE4Pk+hhkjvJJE5SQTeT3EkqOUmF62YS0fWum0m9Jynv7aaYfkrop9j1U0IfxZykxPVRTH/Ctuiy66cotp4JMKfPKEUw3KAfcISij3bqOdE/+dg+sefepqHrXNzf1OB1seentzlS/2JJdpzTy/ESt1mK/eJfkvx9U9WTbP2eBf+Zle//36neYczSCfQ6oC1uuR3vLHykfeqAQYFuZjcCNwLMnTt3tLVKBlWVFVMVrmJBuCrt10Qijr6IF+59A+5U0PdGQ7+v39E7MEBv/+ltfQPe9mSviR3TARHniDgg+ujwHiPOQfTROU5tc4PWecdwzhGJRF8TdSoQXGw57n9bl7jP6W3d0R+XsE/ifulI3NtchKJowBe7Popc3GOkjxADhFwEI3L6Mfo8xADmvGU7tS6234AXrdH9Uh0jPh6NCDg3NK7d0Pi2JOtC7nSkg/d3Oih6nTu1LvYnEds2aN2pP6TBvwpi+9uQP8XTyzbM30f864Y9RsojJD926vdMvj40eU7KdxiPdAI92Wcb8t9kGvvgnLsbuBu8Sy5pvLdMYKGQURYqoqy4KNeliAjpTZ/bDjTELdcDe8awj4iIZFA6gb4RWGhm882sFFgDrEvYZx3wQfNcBBzR9XMRkewa8ZKLc67fzG4GHsUbtvgD59w2M7spuv0uYD3eCJdWvGGLH85cySIikkxaA2mdc+vxQjt+3V1xzx3wSX9LExGR0dCtcyIiAaFAFxEJCAW6iEhAKNBFRAIiZ7Mtmlkn8NoYXz4DOOBjOfmkUD97oX5uKNzPXqifG4b/7Gc458LJNuQs0MfDzFpSTU4TdIX62Qv1c0PhfvZC/dww9s+uSy4iIgGhQBcRCYh8DfS7c11ADhXqZy/Uzw2F+9kL9XPDGD97Xl5DFxGRofL1DF1ERBIo0EVEAiLvAt3MLjezF82s1cxuzXU92WBmDWb2RzPbbmbbzOyWXNeUbWZWZGbPmtkvc11LtkRbOf7UzHZE/+4vznVN2WJmn43+t77VzB4ws/Jc15QJZvYDM+sws61x66aZ2e/M7OXo49R0j5dXgR5tWH07cAWwGLjezBbntqqs6Ac+75w7G7gI+GSBfO54twDbc11Eln0H+I1z7ixgKQXy+c2sDvg00OycW4I3bfea3FaVMfcClyesuxX4g3NuIfCH6HJa8irQiWtY7ZzrBWINqwPNObfXOfdM9PkxvP+x63JbVfaYWT1wFXBPrmvJFjOrAd4CfB/AOdfrnDuc26qyqhioMLNiYBIB7YDmnHsMOJSwejVwX/T5fcA70z1evgV6qmbUBcPM5gHLgKdzW0lW3Qb8D2JdhwvDAqAT+GH0UtM9ZlaZ66KywTn3OvCPwG68RvNHnHO/zW1VWRPY1tkAAAF1SURBVDUz1vEt+lib7gvzLdDTakYdVGZWBfwM+Ixz7miu68kGM7sa6HDObcp1LVlWDFwA3OmcWwYcZxT/9M5n0WvGq4H5wByg0szen9uq8kO+BXrBNqM2sxK8MP+xc+6hXNeTRW8CrjGzV/Eusf2lmf1rbkvKinag3TkX+5fYT/ECvhC8DXjFOdfpnOsDHgIuyXFN2bTfzGYDRB870n1hvgV6Og2rA8fMDO9a6nbn3LdyXU82Oef+2jlX75ybh/f3/e/OucCfrTnn9gFtZnZmdNWlwAs5LCmbdgMXmdmk6H/7l1IgXwhHrQNuiD6/AfhFui9Mq6foRJGqYXWOy8qGNwEfALaY2ebouv8Z7fUqwfUp4MfRk5ddFEjzdefc02b2U+AZvBFezxLQaQDM7AHgrcAMM2sH/hb4BvCgmX0E75fbe9I+nm79FxEJhny75CIiIiko0EVEAkKBLiISEAp0EZGAUKCLiASEAl1EJCAU6CIiAfH/AR8kxDW7Ep39AAAAAElFTkSuQmCC\n", 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\n", 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\n", 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\n", 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\n", 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "for i in range(8):\n", " h = 0.25*(i+1)\n", " n = np.int(10/h)\n", " y0 = torch.ones(2); y0[1] = -1.0\n", " Y = forwardEuler(yfun,h,n,y0)\n", " t = np.arange(0,n+1)*h\n", " plt.figure(i+1)\n", " plt.plot(t,Y[0,:],t,np.exp(-t))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The solution starts reasonable but then it simply deviates from the analytic solution and starts to diverge. The question is why and how can we fix it." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Stability of the solution\n", "\n", "Using a numerical method to solve an ODE does not always guarantees reasonable results. Maybe the obvious problem is stability of the solution.\n", "\n", "Consider the simplest case of the ODE\n", "$$ \\dot{y} = \\lambda y $$\n", "and assume that $Re(\\lambda) \\le 0$. The solution of this problem is non-increasing.\n", "\n", "Now consider the forward Euler solution to this problem that reads\n", "$$ y_{j+1} = (1 + h \\lambda) y_j. $$.\n", "Since the continuous ODE is non-increasing we need to have the\n", "solution of the discrete ODE to be non-increasing as well.\n", "This can happen only if\n", "$$ | 1 + h \\lambda | \\le 1. $$ \n", "\n", "Remember that $\\lambda$ is a complex number $\\lambda = \\lambda_r + i \\lambda_i$ and the condition above implies that\n", "$$ (1 + h\\lambda_r)^2 + h^2\\lambda_i^2 \\le 1 $$\n", "Reorganizing we obtain that\n", "$$ h(\\lambda_r^2 + \\lambda_i^2) = h|\\lambda |^2 \\le -2\\lambda_r. $$\n", "\n", "\n", "If $Re(\\lambda) < 0$ then this condition tells us that $h$ needs to be small enough. However, note that if $\\lambda$ is purely imaginary then there is {\\bf no step} that yields a non-increasing solution and the forward Euler method fails.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# The backward Euler and the trapezoidal methods\n", "\n", "To deal with the stability issue of the forward Euler it is possible to use a different discretization.\n", "To this end we use a different approximation to the derivative in order to obtain a stable method\n", "\n", "The backward Euler method uses the backward difference formula. in order to\n", "approximate the ODE\n", "\\begin{eqnarray}\n", "{\\bf y}_{j+1} - {\\bf y}_j = h f({\\bf y}_{j+1},t_{j+1})\n", "\\end{eqnarray}\n", "\n", "This is an {\\bf implicit} method. The function $\\bf y_{j+1}$ at times $t_{j+1}$ needs to\n", "be solved from the equation. This, in many cases cannot be done analytically\n", "and numerical methods need to be used to solve the problem.\n", "Given this added complexity one may ask, why do we need such a technique?\n", "\n", "A quick repeat of the stability analysis above yields that the method is unconditionaly stable.\n", "\n", "\n", "The trapezoidal method use the trapezoidal method of integration to obtain\n", "\\begin{eqnarray}\n", "{\\bf y}_{j+1} - {\\bf y}_j = {\\frac h2}\\left( f({\\bf y}_{j+1},t_{j+1}) + f({\\bf y}_{j},t_{j})\\right)\n", "\\end{eqnarray}\n", "\n", "This is an implicit method as well. However, it is second order and has some nice properties that we will resolve next.\n", "\n", "\n", "# Class assignment\n", "Use the same analysis we have done for the forward Euler method to show that the backward Euler method and the trapezoidal methods are unconditionally stable." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Using Implicit Methods to solve a linear system of ODEs\n", "\n", "Coding an implicit ODE solver require us to solve a nonlinear system of equations and therefore requires more numerics than we have now. However, for linear systems it is possible to use implicit methods and to write efficient codes for that.\n", "\n", "To this end assume that\n", "$$ \\dot {\\bf y} = {\\bf A}(t) {\\bf y} + f(t)$$\n", "The backward Euler method for this problem reads\n", "$$ \\left({\\bf I} - h {\\bf A}(t_{j+1}) \\right) {\\bf y}_{j+1} = {\\bf y}_{j} + hf(t_{j+1}) $$\n", "\n", "Solving the system we obtain\n", "$$ {\\bf y}_{j+1} = \\left({\\bf I} - h {\\bf A}(t_{j+1}) \\right)^{-1} \\left( {\\bf y}_{j} + hf(t_{j+1}) \\right) $$\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Quick review - solving linear systems\n", "\n", "Solving linear systems is one of the cornerstones of scientific computing and it appears in numerous applications in science and engineering. In general, the goal is to find a vector ${\\bf x}$ that solves the linear system\n", "$$ {\\bf A} {\\bf x} = {\\bf b} $$\n", "\n", "For example, the solution of the system \n", "$$ \\begin{pmatrix} -2 & 1 \\\\ 1 & -2 \\end{pmatrix}\n", "\\begin{pmatrix} {\\bf x}_1 \\\\ {\\bf x}_2 \\end{pmatrix} = \n", "\\begin{pmatrix} 1 \\\\ 2 \\end{pmatrix} $$\n", "is ${\\bf x} = -[\\frac 43, \\frac 53]$.\n", "\n", "There are a number of methods to solve linear systems. Commonly the matrix ${\\bf A}$ is decomposed into a lower trangular\n", "and an upper triangular systems\n", "$$ {\\bf A} = {\\bf L}{\\bf U}. $$\n", "\n", "For the example above\n", "$$ \\begin{pmatrix} -2 & 1 \\\\ 1 & -2 \\end{pmatrix} =\n", "\\begin{pmatrix} 1 & 0 \\\\ -0.5 & 1 \\end{pmatrix} \\begin{pmatrix} -2 & 1 \\\\ 0 & -1.5 \\end{pmatrix} $$\n", "\n", "We can then solve the system by forward/backward substitutions\n", "$$ {\\bf A}{\\bf x} = {\\bf L}{\\bf U}{\\bf x} ={\\bf b} $$\n", "Then, solve (by forward substitution)\n", "$$ {\\bf L}{\\bf z} ={\\bf b} \\quad {\\rm where} \\quad {\\bf z} = {\\bf U}{\\bf x}$$\n", "and finally solve (by backward substitution\n", "$$ {\\bf U}{\\bf x} = {\\bf z} $$ \n", "\n", "It is important to remember that not all linear systems are solvable. For example, the system\n", "$$ \\begin{pmatrix} 1 & 1 \\\\ 1 & 1 \\end{pmatrix}\n", "\\begin{pmatrix} {\\bf x}_1 \\\\ {\\bf x}_2 \\end{pmatrix} = \n", "\\begin{pmatrix} 1 \\\\ 2 \\end{pmatrix} $$\n", "Has no solution (why?)\n", "\n", "Similarly, the linear system\n", "$$ \\begin{pmatrix} 1 & 1 \\\\ 1 & 1+\\epsilon \\end{pmatrix}\n", "\\begin{pmatrix} {\\bf x}_1 \\\\ {\\bf x}_2 \\end{pmatrix} = \n", "\\begin{pmatrix} 1 \\\\ 2 \\end{pmatrix} $$\n", "has a solution, but as $\\epsilon \\rightarrow 0$ the solution becomes unstable and in finite precision cannot be obtained\n", "in any reliable way. This is often refers to as ill-conditioning.\n", "\n", "Most scientific software packages has implemented some solver for systems of equations. In PyTorch this is done by the command solve, demonstrated below. \n" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "lines_to_next_cell": 2 }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "5 8.186776767615811e-08\n" ] } ], "source": [ "n = 5 # The size of the linear system\n", "A = torch.randn(n) + torch.eye(n)\n", "b = torch.randn(n)\n", "b = b.unsqueeze(1) # note - solve expects a matrix for b\n", "x, LU = torch.solve(b,A) \n", "\n", "# Check accuracy\n", "r = torch.matmul(A,x) - b\n", "err = torch.norm(r)/torch.norm(b)\n", "print(n,err.item())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Class assignment\n", "Modify the code above and change $n= 2^j, \\ j=2,\\ldots,12$ and plot the error as a function of $j$. What do you observe?\n" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "lines_to_next_cell": 2 }, "outputs": [], "source": [ "for j in range(2,13):\n", " n = 2**j\n", " ## Your code\n", " \n", "# plt.plot ... " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Back to the backward Euler method\n", "We now return to the backward Euler method. Using the solution of the linear systems we can now easily program the backward Euler method as" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [], "source": [ "def backEuler(y0,fun,Afun,h,n):\n", " k = y0.shape[0]\n", " Y = torch.zeros(k,n+1)\n", " Y[:,0] = y0\n", " \n", " t = 0.0\n", " for j in range(n):\n", " A = Afun(t)\n", " M = torch.eye(k) - h*A\n", " b = Y[:,j] + h*fun(t+h)\n", " x, LU = torch.solve(b.unsqueeze(1), M) \n", " Y[:,j+1] = x.squeeze(1)\n", " t += h\n", "\n", " return Y" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can test the method by defining a matrix and a forcing function and then solve the problem" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [], "source": [ "def forceFun(t):\n", " return torch.zeros(2)\n", "\n", "def AmatFun(t):\n", " A = torch.tensor([[-2.0, 1],[1, -2.0]])\n", " return A" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "h = 0.1; n = np.int(3/h)\n", "Y = backEuler(y0,forceFun,AmatFun,h,n)\n", "\n", "t = np.arange(0,n+1)*h\n", "plt.plot(t,Y[0,:],t,Y[1,:]);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Class assignment\n", "\n", "We claimed that the backward Euler method is unconditionally stable. Demonstrate that by chaging $h$ and viewing the results.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Homework assignment\n", "\n", "Write a similar code to the backward Euler that uses the trapezoidal method.\n", "\n", "Given the problem\n", "$$ \\dot y = A y \\quad y(0) = y_0 $$\n", "with \n", "$$ A = \\begin{pmatrix} 0 & 1 \\\\ -1 & 0 \\end{pmatrix} $$\n", "and $y_0 = [0,1]$ test and compare the 3 methods for $h=10^{-j}, j=[1,2,3,4]$ when integrating\n", "the system in the interval $t = [0,10]$\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "jupytext": { "formats": "ipynb,py:percent", "notebook_metadata_filter": "all,-language_info,-toc,-latex_envs" }, "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.6" } }, "nbformat": 4, "nbformat_minor": 2 }