0011/Problem-Algorithm-Solver pattern 2021-07-16 Julia 1.6.ipynb

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    "# Problem-Algorithm-Solver pattern\n",
    "\n",
    "* Gen Kuroki\n",
    "* 2021-07-11\n",
    "* https://discourse.julialang.org/t/function-depending-on-the-global-variable-inside-module/64322/10\n",
    "\n",
    "A code will be more readable and efficient if the global variables that store the parameters of a problem are passed as arguments to functions _everytime_.\n",
    "\n",
    "However, many people find it troublesome to pass the parameters as arguments to the function _everytime_.\n",
    "\n",
    "In order to resolve this misunderstanding, they can try the problem-algorithm-solver pattern explained below.\n",
    "\n",
    "In order to use the `(; a, b, c) = p` syntax, require VERSION ≥ v\"1.7.0-beta\" [#39285](https://github.com/JuliaLang/julia/pull/39285)."
   ]
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    "#VERSION ≥ v\"1.7.0-beta\" #> true"
   ]
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   "source": [
    "## A minimal working example of the problem-algorithm-solver pattern"
   ]
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       "Main.FreeFall"
      ]
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   "source": [
    "module FreeFall\n",
    "\n",
    "using Parameters\n",
    "\n",
    "\"\"\"Problem type\"\"\"\n",
    "Base.@kwdef struct Problem{G, Y0, V0, TS}\n",
    "    g::G = 9.80665\n",
    "    y0::Y0 = 0.0\n",
    "    v0::V0 = 30.0\n",
    "    tspan::TS = (0.0, 8.0)\n",
    "end\n",
    "\n",
    "\"\"\"Algorithm types\"\"\"\n",
    "Base.@kwdef struct EulerMethod{T}  dt::T = 0.1 end\n",
    "Base.@kwdef struct ExactFormula{T} dt::T = 0.1 end\n",
    "default_algorithm(::Problem) = EulerMethod()\n",
    "\n",
    "\"\"\"Solution type\"\"\"\n",
    "struct Solution{Y, V, T, P<:Problem, A} y::Y; v::V; t::T; prob::P; alg::A end\n",
    "\n",
    "\"\"\"Solver function\"\"\"\n",
    "solve(prob::Problem) = solve(prob, default_algorithm(prob))\n",
    "\n",
    "function solve(prob::Problem, alg::ExactFormula)\n",
    "    # (; g, y0, v0, tspan) = prob\n",
    "    # (; dt) = alg\n",
    "    @unpack g, y0, v0, tspan = prob\n",
    "    @unpack dt = alg\n",
    "    t0, t1 = tspan\n",
    "    t = range(t0, t1 + dt/2; step = dt)    \n",
    "    y(t) = y0 + v0*(t - t0) - g*(t - t0)^2/2\n",
    "    v(t) = v0 - g*(t - t0)\n",
    "    Solution(y.(t), v.(t), t, prob, alg)\n",
    "end\n",
    "\n",
    "function solve(prob::Problem, alg::EulerMethod)\n",
    "    # (; g, y0, v0, tspan) = prob\n",
    "    # (; dt) = alg\n",
    "    @unpack g, y0, v0, tspan = prob\n",
    "    @unpack dt = alg\n",
    "    t0, t1 = tspan\n",
    "    t = range(t0, t1 + dt/2; step = dt)    \n",
    "    n = length(t)    \n",
    "    y = Vector{typeof(y0)}(undef, n)\n",
    "    v = Vector{typeof(v0)}(undef, n)\n",
    "    y[1] = y0\n",
    "    v[1] = v0\n",
    "    for i in 1:n-1\n",
    "        v[i+1] = v[i] - g*dt\n",
    "        y[i+1] = y[i] + v[i]*dt\n",
    "    end\n",
    "    Solution(y, v, t, prob, alg)\n",
    "end\n",
    "\n",
    "end"
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    "This module will calculate the free-fall motion under a uniform gravity potential.\n",
    "\n",
    "* Problem: The parameters of problems are stored in objects of type `FreeFall.Problem`.\n",
    "\n",
    "* Algorithm: The parameters of algorithms are stored in objects of types `FreeFall.EulerMethod` and of `FreeFall.ExactFormula`.\n",
    "\n",
    "* Solver function: The function `FreeFall.solve(prob::FreeFall.Problem, alg)` returns the object of type `FreeFall.Solution` obtained by solving the problem `prob` with the algorithm `alg`.\n",
    "\n",
    "Note that the `(; a, b, c) = p` syntax of Julia ≥ v1.7, is very useful for extracting the contents of the object that contains parameters. (You can also do the same thing with `@unpack` in the exellent well-known package [Parameters.jl](https://github.com/mauro3/Parameters.jl).)"
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   "source": [
    "### Example 1: Compare a numerical solution and an exact one"
   ]
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      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots\n",
    "\n",
    "earth = FreeFall.Problem()\n",
    "sol_euler = FreeFall.solve(earth)\n",
    "sol_exact = FreeFall.solve(earth, FreeFall.ExactFormula())\n",
    "\n",
    "plot(sol_euler.t, sol_euler.y; label=\"Euler's method (dt = $(sol_euler.alg.dt))\", ls=:auto)\n",
    "plot!(sol_exact.t, sol_exact.y; label=\"exact solution\", ls=:auto)\n",
    "title!(\"On the Earth\"; xlabel=\"t\", legend=:bottomleft)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e815fae5-421d-4c73-9a0f-d230514d251f",
   "metadata": {},
   "source": [
    "Since the time step `dt = 0.1` is not enough small, the error of the Euler method is rather large."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d1cb8a2c-825e-484f-b7cb-1ab1741e6c60",
   "metadata": {},
   "source": [
    "### Example 2: Change the algorithm parameter `dt` smaller"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "8a9091d9-9604-47a1-aca5-5a073113819d",
   "metadata": {},
   "outputs": [
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       "  1804.16,622.587 1806.72,625.638 1809.28,628.698 1811.85,631.769 1814.41,634.85 1816.97,637.941 1819.54,641.042 1822.1,644.153 1824.66,647.274 1827.23,650.405 \n",
       "  1829.79,653.546 1832.36,656.698 1834.92,659.859 1837.48,663.03 1840.05,666.211 1842.61,669.403 1845.17,672.604 1847.74,675.815 1850.3,679.037 1852.86,682.268 \n",
       "  1855.43,685.509 1857.99,688.761 1860.55,692.022 1863.12,695.294 1865.68,698.576 1868.25,701.867 1870.81,705.169 1873.37,708.48 1875.94,711.802 1878.5,715.134 \n",
       "  1881.06,718.476 1883.63,721.828 1886.19,725.189 1888.75,728.561 1891.32,731.943 1893.88,735.335 1896.44,738.737 1899.01,742.149 1901.57,745.571 1904.13,749.003 \n",
       "  1906.7,752.445 1909.26,755.897 1911.83,759.36 1914.39,762.832 1916.95,766.314 1919.52,769.806 1922.08,773.309 1924.64,776.821 1927.21,780.343 1929.77,783.876 \n",
       "  1932.33,787.418 1934.9,790.971 1937.46,794.533 1940.02,798.106 1942.59,801.688 1945.15,805.281 1947.72,808.883 1950.28,812.496 1952.84,816.119 1955.41,819.752 \n",
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       "  1983.6,860.374 1986.17,864.127 1988.73,867.89 1991.3,871.663 1993.86,875.446 1996.42,879.24 1998.99,883.043 2001.55,886.856 2004.11,890.68 2006.68,894.513 \n",
       "  2009.24,898.356 2011.8,902.21 2014.37,906.073 2016.93,909.947 2019.49,913.83 2022.06,917.724 2024.62,921.628 2027.19,925.541 2029.75,929.465 2032.31,933.399 \n",
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       "  2086.15,1018.32 2088.71,1022.48 2091.27,1026.64 2093.84,1030.82 2096.4,1035 2098.96,1039.2 2101.53,1043.4 2104.09,1047.62 2106.66,1051.84 2109.22,1056.08 \n",
       "  2111.78,1060.32 2114.35,1064.57 2116.91,1068.84 2119.47,1073.11 2122.04,1077.4 2124.6,1081.69 2127.16,1086 2129.73,1090.31 2132.29,1094.64 2134.85,1098.97 \n",
       "  2137.42,1103.32 2139.98,1107.67 2142.54,1112.04 2145.11,1116.41 2147.67,1120.8 2150.24,1125.19 2152.8,1129.6 2155.36,1134.02 2157.93,1138.44 2160.49,1142.88 \n",
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      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "earth = FreeFall.Problem()\n",
    "sol_euler = FreeFall.solve(earth, FreeFall.EulerMethod(dt = 0.01))\n",
    "sol_exact = FreeFall.solve(earth, FreeFall.ExactFormula())\n",
    "\n",
    "plot(sol_euler.t, sol_euler.y; label=\"Euler's method (dt = $(sol_euler.alg.dt))\", ls=:auto)\n",
    "plot!(sol_exact.t, sol_exact.y; label=\"exact solution\", ls=:auto)\n",
    "title!(\"On the Earth\"; xlabel=\"t\", legend=:bottomleft)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1c998b2d-d0e8-4dde-b847-b1c6b333b59c",
   "metadata": {},
   "source": [
    "### Example 3: Change the problem parameter `g` to the gravitational acceleration on the moon"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "780da4d3-9cfb-4a88-a25a-459d4dac5060",
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "moon = FreeFall.Problem(g = 1.62, tspan = (0.0, 40.0))\n",
    "sol_euler = FreeFall.solve(moon)\n",
    "sol_exact = FreeFall.solve(moon, FreeFall.ExactFormula(dt = sol_euler.alg.dt))\n",
    "\n",
    "plot(sol_euler.t, sol_euler.y; label=\"Euler's method (dt = $(sol_euler.alg.dt))\", ls=:auto)\n",
    "plot!(sol_exact.t, sol_exact.y; label=\"exact solution\", ls=:auto)\n",
    "title!(\"On the Moon\"; xlabel=\"t\", legend=:bottomleft)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e01c24c5-9cfa-4d66-800d-027b9952c630",
   "metadata": {},
   "source": [
    "## Extensibility of the Julia ecosystem\n",
    "\n",
    "A different author of the `FreeFall` module can define , in the `SomeExtension` module, a new algorithm type and a new method of the `FreeFall.solve` function.  Both a new type and a new method!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "7f257c50-bfa4-4593-a409-868891984563",
   "metadata": {},
   "outputs": [
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     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "module SomeExtension\n",
    "\n",
    "using Parameters\n",
    "using ..FreeFall\n",
    "using ..FreeFall: Problem, Solution\n",
    "\n",
    "Base.@kwdef struct Symplectic2ndOrder{T}  dt::T = 0.1 end\n",
    "\n",
    "function FreeFall.solve(prob::Problem, alg::Symplectic2ndOrder)\n",
    "    # (; g, y0, v0, tspan) = prob\n",
    "    # (; dt) = alg\n",
    "    @unpack g, y0, v0, tspan = prob\n",
    "    @unpack dt = alg\n",
    "    t0, t1 = tspan\n",
    "    t = range(t0, t1 + dt/2; step = dt)    \n",
    "    n = length(t)    \n",
    "    y = Vector{typeof(y0)}(undef, n)\n",
    "    v = Vector{typeof(v0)}(undef, n)\n",
    "    y[1] = y0\n",
    "    v[1] = v0\n",
    "    for i in 1:n-1\n",
    "        ytmp = y[i] + v[i]*dt/2\n",
    "        v[i+1] = v[i] - g*dt\n",
    "        y[i+1] = ytmp + v[i+1]*dt/2\n",
    "    end\n",
    "    Solution(y, v, t, prob, alg)\n",
    "end\n",
    "\n",
    "end\n",
    "\n",
    "earth = FreeFall.Problem()\n",
    "sol_sympl = FreeFall.solve(earth, SomeExtension.Symplectic2ndOrder(dt = 2.0))\n",
    "sol_exact = FreeFall.solve(earth, FreeFall.ExactFormula())\n",
    "\n",
    "plot(sol_sympl.t, sol_sympl.y; label=\"2nd order symplectic (dt = $(sol_sympl.alg.dt))\", ls=:auto)\n",
    "plot!(sol_exact.t, sol_exact.y; label=\"exact solution\", ls=:auto)\n",
    "title!(\"On the Earth\"; xlabel=\"t\", legend=:bottomleft)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "876cf29b-a732-40de-b47f-dfb03ff4a0db",
   "metadata": {},
   "source": [
    "The second-order symplectic method gives exact discretizations of the free-fall motions under uniform gravity potentials."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "90696808-a7a9-4567-b345-2cde048c7566",
   "metadata": {},
   "source": [
    "## Composability of the Julia ecosystem\n",
    "\n",
    "By combining the MonteCarloMeasurements.jl package with the modules given above, we can plot the behavior of the system for perturbations of the initial value.  We did not intend for them to be used in this way!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "e6a8adb3-60aa-40f1-903c-fa4365d7c6c7",
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using MonteCarloMeasurements\n",
    "\n",
    "earth = FreeFall.Problem(y0 = 0.0 ± 0.0, v0 = 30.0 ± 1.0)\n",
    "sol_euler = FreeFall.solve(earth)\n",
    "sol_sympl = FreeFall.solve(earth, SomeExtension.Symplectic2ndOrder(dt = 2.0))\n",
    "sol_exact = FreeFall.solve(earth, FreeFall.ExactFormula())\n",
    "\n",
    "ylim = (-100, 60)\n",
    "P = plot(sol_euler.t, sol_euler.y; label=\"Euler's method (dt = $(sol_euler.alg.dt))\", ls=:auto)\n",
    "title!(\"On the Earth\"; xlabel=\"t\", legend=:bottomleft, ylim)\n",
    "Q = plot(sol_sympl.t, sol_sympl.y; label=\"2nd order symplectic (dt = $(sol_sympl.alg.dt))\", ls=:auto)\n",
    "title!(\"On the Earth\"; xlabel=\"t\", legend=:bottomleft, ylim)\n",
    "R = plot(sol_exact.t, sol_exact.y; label=\"exact solution\", ls=:auto)\n",
    "title!(\"On the Earth\"; xlabel=\"t\", legend=:bottomleft, ylim)\n",
    "plot(P, Q, R; size=(720, 600))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "faf19208-7bba-48e8-9d64-f5a9816a657d",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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