lec23.ipynb

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      "\u001b[32m\u001b[1m    Updating\u001b[22m\u001b[39m registry at `~/.julia/registries/General.toml`\n",
      "\u001b[32m\u001b[1m   Resolving\u001b[22m\u001b[39m package versions...\n",
      "\u001b[32m\u001b[1m  No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.7/Project.toml`\n",
      "\u001b[32m\u001b[1m  No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.7/Manifest.toml`\n"
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       "LogLevel(1)"
      ]
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   ],
   "source": [
    "import Pkg; Pkg.add([\"Plots\",\"Interact\",\"SparseArrays\",\"FFTW\"]);\n",
    "import Logging; Logging.disable_logging(Logging.Info)"
   ]
  },
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   "source": [
    "const STATIC_LECTURE = false;"
   ]
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       "    WebIO Jupyter integration documentation\n",
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       "for more information.\n",
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       "WebIO._IJuliaInit()"
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   ],
   "source": [
    "using LinearAlgebra;\n",
    "using Plots; gr(); default(l=5, size=(900,550), label=false);\n",
    "using Printf;\n",
    "using Interact;\n",
    "using SparseArrays;\n",
    "\n",
    "BLAS.set_num_threads(1)\n",
    "\n",
    "if STATIC_LECTURE\n",
    "    default(fmt=:png)\n",
    "    macro manipulate(v)\n",
    "        return :($v; plot!())\n",
    "    end\n",
    "end\n",
    "\n",
    "prettyprint(a) = map(x -> @sprintf(\"%+.2f\", x), a)\n",
    "sciprint(a) = map(x -> @sprintf(\"%+.2e\", x), a)\n",
    "\n",
    "function imshow(x; kwargs...)\n",
    "    return heatmap(x\n",
    "        , aspect_ratio=:equal\n",
    "        , framestyle=:none\n",
    "        , yflip=true\n",
    "        , c=cgrad(:diverging_gwv_55_95_c39_n256, rev=true)\n",
    "        ; kwargs...\n",
    "    )\n",
    "end\n",
    "function imshowlog(x; kwargs...)\n",
    "    return heatmap(max.(log.(abs.(x)), log(eps()))\n",
    "        , aspect_ratio=:equal\n",
    "        , framestyle=:none\n",
    "        , yflip=true\n",
    "        , c=cgrad(:linear_worb_100_25_c53_n256)\n",
    "        ; kwargs...\n",
    "    )\n",
    "end\n",
    "bigspy(x; kwargs...) = spy(x, m=4, c=:diverging_gwv_55_95_c39_n256; kwargs...)\n",
    "p = plot!([0,1],[0,1]);"
   ]
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  {
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   "metadata": {
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   "source": [
    "### Warmup\n",
    "\n",
    "Consider the **High performance Linpack benchmark:** timing LU factorization with partial pivoting:\n",
    "\n",
    "- Current best: [Fugaku supercomputer](https://top500.org/system/179807/) sustains $4 \\times 10^{17}$ flops/s on this problem...\n",
    "\n",
    "- ...for a problem of size $m = 2 \\times 10^7$\n",
    "\n",
    "How long did it take Fugaku to compute the LU factorization?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "040a7b44",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Introduction to Iterative Methods"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e836600e",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Beyond Dense Linear Algebra"
   ]
  },
  {
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   "id": "1aa44a88",
   "metadata": {
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     "slide_type": "subslide"
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   "source": [
    "### All semester we have analyzed algorithms for factorizations\n",
    "\n",
    "| name | complexity |\n",
    "| :-----: | :-----: |\n",
    "| Householder $QR$ | ${\\sim 2mn^2 - \\frac{2}{3}n^3}$ |\n",
    "| $LU$             | ${\\sim \\frac{2}{3} m^3}$ |\n",
    "| Cholesky $LL^*$ | ${\\sim \\frac{1}{3} m^3}$ |\n",
    "| Eigenvalue (hermitian) $Q \\Lambda Q^*$ | ${\\sim \\frac{4}{3}m^3 + O(m^3)}$ |\n",
    "| Schur $Q T Q^*$ | ${\\sim \\frac{10}{3}m^3 + O(m^3)}$ |\n",
    "| SVD $U \\Sigma V^*$ | ${\\sim 4mn^2 - \\frac{4}{3}n^3 + O(n^3)}$ |"
   ]
  },
  {
   "cell_type": "markdown",
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    }
   },
   "source": [
    "- All of these algorithms assume the matrix is stored as a dense array"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9a73daaf",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- If $n = O(m)$ all of these factorizations are $O(m^3)$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e6fbb554",
   "metadata": {
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     "slide_type": "subslide"
    }
   },
   "source": [
    "### $O(m^3)$ is prohibitive for large problems\n",
    "\n",
    "As the warmup shows, the worlds fastest computer can solve $Ax = b$ by LU factorization for $m = 2 \\times 10^7$ in $O(\\text{hours})$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4793de2c",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Baker, Falgout, Kolev, Yang **(2012)**. _Scaling hypre's multigrid solvers to 100,000 cores_. [doi:10.1007/978-1-4471-2437-5_13](https://doi.org/10.1007/978-1-4471-2437-5_13).  Solving $Ax = b$ where $A$ is a discretized PDE operator for $m = 10^{12}$ in $O(\\text{minutes})$."
   ]
  },
  {
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   "id": "7e82c514",
   "metadata": {
    "slideshow": {
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    }
   },
   "source": [
    "### [Edelman's law](https://doi.org/10.1177/109434209300700203)\n",
    "\n",
    "> [N]ature is not so perverse as to throw $n^2$ numbers at us haphazardly. Therefore, when faced with large $n$-by-$n$ matrices, we ought to try our hardest to take advantage of the structure that is surely there."
   ]
  },
  {
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   "id": "e2da8862",
   "metadata": {
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    }
   },
   "source": [
    "### Simplest improvement: sparse direct linear algebra\n",
    "\n",
    "Do the same numerical operations as dense linear algebra, but avoid storing and computing zeros.\n",
    "\n",
    "(Except sometimes orderings are chosen to promote sparsity)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "977895fa",
   "metadata": {
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    }
   },
   "source": [
    "### Sparse Linear Algebra Libraries\n",
    "\n",
    "(There is not an agreed upon interface to the same extent there is for dense linear algebra)\n",
    "\n",
    "- [SuiteSparse](https://people.engr.tamu.edu/davis/suitesparse.html)\n",
    "\n",
    "- [SuperLU](https://portal.nersc.gov/project/sparse/superlu/)\n",
    "\n",
    "- [MUMPS](http://mumps.enseeiht.fr/)\n",
    "\n",
    "- [Pastix](https://ramet.github.io/project/pastix6/)\n",
    "\n",
    "- [Pardiso](https://www.pardiso-project.org/)\n",
    "\n",
    "- [cuSPARSE](https://docs.nvidia.com/cuda/cusparse/index.html)\n",
    "\n",
    "- [Kokkos kernels](https://github.com/kokkos/kokkos-kernels)"
   ]
  },
  {
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   "id": "5ed7500a",
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   "source": [
    "### Example: a discrete Laplacian $L_d$\n",
    "\n",
    "Represents the PDE $u_{xx} = f$ (1D), $u_{xx} + u_{yy} = f$ (2D), or $u_{xx} + u_{yy} + u_{zz} = f$ (3D)"
   ]
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  {
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     "execution_count": 4,
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   ],
   "source": [
    "m = 4^3\n",
    "\n",
    "L₁ = SymTridiagonal(2 * ones(m), -1 * ones(m-1))\n",
    "\n",
    "p = bigspy(L₁, m=4, clim=(-2,2), colorbar=false, framestyle=:none, title=\"1D Laplacian\")\n",
    "\n",
    "m = 8\n",
    "\n",
    "L₁ = SymTridiagonal(2 * ones(m), -1 * ones(m-1))\n",
    "I₁ = Diagonal(ones(m))\n",
    "L₂ = kron(L₁,I₁) + kron(I₁,L₁)\n",
    "q = bigspy(L₂,clim=(-4,4), colorbar=false, framestyle=:none, title=\"2D Laplacian\")\n",
    "\n",
    "m = 4\n",
    "\n",
    "L₁ = SymTridiagonal(2 * ones(m), -1 * ones(m-1))\n",
    "I₁ = Diagonal(ones(m))\n",
    "I₂ = Diagonal(ones(m*m))\n",
    "L₂ = kron(L₁,I₁) + kron(I₁,L₁)\n",
    "L₃ = kron(L₂,I₁) + kron(I₂,L₁)\n",
    "r = bigspy(L₃,clim=(-4,4), colorbar=false, framestyle=:none, title=\"3D Laplacian\")\n",
    "#L₂ = kron(L₁,L₁)\n",
    "#L₃ = kron(L₂,L₁)\n",
    "plot(p,q,r,layout=(1,3))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c97b8d4b",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- symmetric positive definite\n",
    "- $L_d$ has at most $2d + 1$ nonzeros per row / column"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "e6b3554e",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "function laplacian(d, m)\n",
    "    if d == 1\n",
    "        return SparseMatrixCSC(SymTridiagonal(2 * ones(m), -1 * ones(m-1)))\n",
    "    elseif d > 1\n",
    "        L₁ = laplacian(1, m)\n",
    "        L = laplacian(d-1,m)\n",
    "        Id = SparseMatrixCSC(Diagonal(ones(m^(d-1))))\n",
    "        I₁ = SparseMatrixCSC(Diagonal(ones(m)))\n",
    "        return SparseMatrixCSC(kron(L,I₁)) + SparseMatrixCSC(kron(Id, L₁))\n",
    "    end\n",
    "end;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "b448c7d1",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false,
    "slideshow": {
     "slide_type": "subslide"
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   },
   "outputs": [],
   "source": [
    "ms = Int64.(round.(10 .^ LinRange(1, 4, 20)))\n",
    "ts = Float64[]\n",
    "for m in ms\n",
    "    L₁ = Matrix(laplacian(1, m)) # dense\n",
    "    b = randn(m)\n",
    "    push!(ts, @elapsed L₁ \\ b)\n",
    "end"
   ]
  },
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   "execution_count": 7,
   "id": "7ecee00a",
   "metadata": {
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      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = plot(ms, ts\n",
    "    , xlabel = \"m\"\n",
    "    , ylabel = \"time (seconds)\"\n",
    "    , title = \"Dense LU solve time, 1D Laplacian\"\n",
    "    , scale = :log10\n",
    "    , m = :o\n",
    "    , legend = :bottomright\n",
    ")\n",
    "plot!(p, [ms[1], ms[end]], [ts[end] .* (ms[1] / ms[end]) .^ 3 , ts[end]]\n",
    "    , l = (:dash, :black)\n",
    "    , label=\"O(m³)\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "8f3bfda4",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "ms_2 = Int64.(round.(10 .^ LinRange(1, 7, 20)))\n",
    "ts_2 = Float64[]\n",
    "for m in ms_2\n",
    "    L₁ = laplacian(1, m) # sparse\n",
    "    b = randn(m)\n",
    "    push!(ts_2, @elapsed L₁ \\ b)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "fb6bb42a",
   "metadata": {
    "hideCode": true,
    "hidePrompt": false
   },
   "outputs": [
    {
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     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = plot([ms ms_2], [ts ts_2]\n",
    "    , xlabel = \"m\"\n",
    "    , ylabel = \"time (seconds)\"\n",
    "    , title = \"LU solve time, 1D Laplacian\"\n",
    "    , scale = :log10\n",
    "    , m = :o\n",
    "    , legend = :bottomright\n",
    ")\n",
    "plot!(p, [ms[1], ms[end]], [ts[end] .* (ms[1] / ms[end]) .^ 3 , ts[end]]\n",
    "    , l = (:dash, :black)\n",
    "    , label=\"O(m³)\")\n",
    "plot!(p, [ms_2[1], ms_2[end]], [ts_2[end] .* (ms_2[1] / ms_2[end]) .^ 1 , ts_2[end]]\n",
    "    , l = (:dot, :black)\n",
    "    , label=\"O(m¹)\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "8efb7ecf",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "ms_3 = Int64.(round.(10 .^ LinRange(1, 2, 20)))\n",
    "Ms_3 = Float64[]\n",
    "ts_3 = Float64[]\n",
    "for m in ms_3\n",
    "    L₂ = Matrix(laplacian(2, m)) # dense\n",
    "    M = m * m\n",
    "    push!(Ms_3, M)\n",
    "    b = randn(M)\n",
    "    push!(ts_3, @elapsed L₂ \\ b)\n",
    "end"
   ]
  },
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   "execution_count": 11,
   "id": "bc721c17",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
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     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = plot([ms Ms_3], [ts ts_3]\n",
    "    , xlabel = \"m = mₓ²\"\n",
    "    , ylabel = \"time (seconds)\"\n",
    "    , title = \"Dense LU solve time, 1D & 2D Laplacians\"\n",
    "    , scale = :log10\n",
    "    , m = :o\n",
    "    , label = [\"1D\" \"2D\"]\n",
    "    , legend = :bottomright\n",
    ")\n",
    "plot!(p, [Ms_3[1], Ms_3[end]], [ts_3[end] .* (Ms_3[1] / Ms_3[end]) .^ 3 , ts_3[end]]\n",
    "    , l = (:dash, :black)\n",
    "    , label=\"O(M³)\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "5e618c4a",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "ms_4 = Int64.(round.(10 .^ LinRange(1, 3.3, 20)))\n",
    "Ms_4 = Float64[]\n",
    "ts_4 = Float64[]\n",
    "for m in ms_4\n",
    "    L₂ = laplacian(2, m) # sparse\n",
    "    M = m * m\n",
    "    push!(Ms_4, M)\n",
    "    b = randn(M)\n",
    "    push!(ts_4, @elapsed L₂ \\ b)\n",
    "end"
   ]
  },
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   "execution_count": 13,
   "id": "bef144c3",
   "metadata": {
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    "hidePrompt": false
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     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = plot([Ms_3 Ms_4], [ts_3 ts_4]\n",
    "    , xlabel = \"m\"\n",
    "    , ylabel = \"time (seconds)\"\n",
    "    , title = \"LU solve time, 2D Laplacian\"\n",
    "    , scale = :log10\n",
    "    , m = :o\n",
    "    , legend = :bottomright\n",
    ")\n",
    "plot!(p, [Ms_3[1], Ms_3[end]], [ts_3[end] .* (Ms_3[1] / Ms_3[end]) .^ 3 , ts_3[end]]\n",
    "    , l = (:dash, :black)\n",
    "    , label=\"O(M³)\")\n",
    "plot!(p, [Ms_4[1], Ms_4[end]], [ts_4[end] .* (Ms_4[1] / Ms_4[end]) .^ 1.5 , ts_4[end]]\n",
    "    , l = (:dot, :black)\n",
    "    , label=\"O(M¹.⁵)\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "de4b2784",
   "metadata": {
    "cell_style": "split",
    "hideCode": false,
    "hidePrompt": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "ms_5 = collect(3:22)\n",
    "Ms_5 = Float64[]\n",
    "ts_5 = Float64[]\n",
    "for m in ms_5\n",
    "    L₃ = Matrix(laplacian(3, m)) # dense\n",
    "    M = m * m * m\n",
    "    push!(Ms_5, M)\n",
    "    b = randn(M)\n",
    "    push!(ts_5, @elapsed L₃ \\ b)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "27c2fd38",
   "metadata": {
    "cell_style": "split",
    "hideCode": false,
    "hidePrompt": false
   },
   "outputs": [],
   "source": [
    "ms_6 = collect(3:2:41)\n",
    "Ms_6 = Float64[]\n",
    "ts_6 = Float64[]\n",
    "for m in ms_6\n",
    "    L₃ = laplacian(3, m) # sparse\n",
    "    M = m * m * m\n",
    "    push!(Ms_6, M)\n",
    "    b = randn(M)\n",
    "    push!(ts_6, @elapsed L₃ \\ b)\n",
    "end"
   ]
  },
  {
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   "execution_count": 16,
   "id": "c3a746fa",
   "metadata": {
    "hideCode": true,
    "hidePrompt": false
   },
   "outputs": [
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     "execution_count": 16,
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   ],
   "source": [
    "p = plot([Ms_5 Ms_6], [ts_5 ts_6]\n",
    "    , xlabel = \"m = mₓ³\"\n",
    "    , ylabel = \"time (seconds)\"\n",
    "    , title = \"LU solve time, 3D Laplacian\"\n",
    "    , scale = :log10\n",
    "    , m = :o\n",
    "    , legend = :bottomright\n",
    ")\n",
    "plot!(p, [Ms_5[1], Ms_5[end]], [ts_5[end] .* (Ms_5[1] / Ms_5[end]) .^ 3 , ts_5[end]]\n",
    "    , l = (:dash, :black)\n",
    "    , label=\"O(m³)\")\n",
    "plot!(p, [Ms_6[1], Ms_6[end]], [ts_6[end] .* (Ms_6[1] / Ms_6[end]) .^ 2 , ts_6[end]]\n",
    "    , l = (:dot, :black)\n",
    "    , label=\"O(m²)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0c801b5d",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Conclusion\n",
    "\n",
    "- Sparse direct performance depends not just on the sparsity of $A$ but on the sparsity of the computed factors"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "735c2c10",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### More complicated improvement: application specific algorithms\n",
    "\n",
    "Applications that generate matrices can develop algorithms that depend on more structure than just sparsity\n",
    "\n",
    "**Example:** there are $O(m[\\log m])$ solvers for the Laplacian and related discretized PDEs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "0807754c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "inverse_laplacian (generic function with 1 method)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import FFTW\n",
    "\n",
    "function inverse_laplacian(d, b)\n",
    "    y = copy(b)\n",
    "    M = length(b)\n",
    "    m = Int64(round(M^(1/d)))\n",
    "    dims = repeat([m],d)\n",
    "    y = reshape(y, dims...)\n",
    "    kinds = repeat([FFTW.RODFT00],d)\n",
    "    ϕ = π / (m + 1)\n",
    "    λs = [2 * (1 - cos(k * ϕ)) for k in 1:m]\n",
    "    inds = [reshape(repeat(λs, inner=m^(k-1), outer=m^(d-k)), dims...) for k in 1:d]\n",
    "    factor = sum(inds)\n",
    "    sqs = collect(1:m)\n",
    "    sqs .*= sqs\n",
    "    sqs = kron(sqs, ones(m))\n",
    "    FFTW.r2r!(y, kinds)\n",
    "    y ./= factor * (2 * (m+1))^d\n",
    "    FFTW.r2r!(y, kinds)\n",
    "    return reshape(y, M)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "59c9ffd4",
   "metadata": {
    "cell_style": "split",
    "hideCode": false,
    "hidePrompt": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "ms_7 = Int64.(round.(3 .^ LinRange(1, 4,20)))\n",
    "Ms_7 = Float64[]\n",
    "ts_7 = Float64[]\n",
    "for m in ms_7\n",
    "    L₃ = laplacian(3, m) # sparse\n",
    "    M = m * m * m\n",
    "    push!(Ms_7, M)\n",
    "    b = randn(M)\n",
    "    push!(ts_7, @elapsed inverse_laplacian(3, b))\n",
    "end"
   ]
  },
  {
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   "execution_count": 41,
   "id": "5def3598",
   "metadata": {
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      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = plot([Ms_5 Ms_6 Ms_7], [ts_5 ts_6 ts_7]\n",
    "    , xlabel = \"M = m³\"\n",
    "    , ylabel = \"time (seconds)\"\n",
    "    , title = \"Solve time, 3D Laplacian\"\n",
    "    , scale = :log10\n",
    "    , m = :o\n",
    "    , legend = :bottomright\n",
    ")\n",
    "plot!(p, [Ms_5[1], Ms_5[end]], [ts_5[end] .* (Ms_5[1] / Ms_5[end]) .^ 3 , ts_5[end]]\n",
    "    , l = (:dash, :black)\n",
    "    , label=\"O(m³)\")\n",
    "plot!(p, [Ms_6[1], Ms_6[end]], [ts_6[end] .* (Ms_6[1] / Ms_6[end]) .^ 2 , ts_6[end]]\n",
    "    , l = (:dot, :black)\n",
    "    , label=\"O(m²)\")\n",
    "plot!(p, [Ms_7[1], Ms_7[end]], [ts_7[end] .* (Ms_7[1] / Ms_7[end]) .^ 1 , ts_7[end]]\n",
    "    , l = (:dashdot, :black)\n",
    "    , label=\"O(m¹)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1ebacfc1",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Between dense, sparse, and application specific\n",
    "\n",
    "**Aim:** methods for solving\n",
    "\n",
    "$$\\large Ax = b$$\n",
    "\n",
    "that are better than dense / sparse direct methods for many problems, but without using too many application-specific details"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bb8470f2",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "**Observation:** Many applications, because of their structure, have **fast matrix-vector products:**\n",
    "\n",
    "$$\\large T_{\\text{matvec}}(A) = o(m^2)$$\n",
    "\n",
    "and for many applications,\n",
    "\n",
    "$$\\large T_{\\text{matvec}}(A) = O(m)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3862d1f1",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Fast matvec example: discrete Laplacian\n",
    "\n",
    "$$\\large T_{\\text{matvec}}(L_d) = O(d(m_x)^d) = O(d m) = o(m^2)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ce22b830",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Example: general sparse matrix\n",
    "\n",
    "If $A$ has $n_{\\text{nnz}}$ nonzeros, then\n",
    "\n",
    "$$\\large T_{\\text{matvec}}(A) = O(n_{\\text{nnz}})$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8a9ad911",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Linear operator AKA \"black-box\" methods\n",
    "\n",
    "If the only computational kernel available to us about a ~~matrix~~ linear operator is its action on a vector,\n",
    "\n",
    "$$\\large A: x \\mapsto Ax,$$\n",
    "\n",
    "then we need a **black box algorithm** that can solve $Ax = b$ relying only on this operation"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7bddbcac",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Exercise: simplest bound\n",
    "\n",
    "**Claim:** $m T_{\\text{matvec}}(A) + O(m^3)$ is an upper bound on the complexity of solving $Ax = b.$\n",
    "\n",
    "_Proof._"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5cade39f",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Example: Richardson Iteration"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a6b4f5b3",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "$$\\begin{aligned}\n",
    "x^{(0)} &\\gets 0 \\\\\n",
    "r^{(k)} &\\gets b - A x^{(k)} \\\\\n",
    "x^{(k+1)} &\\gets x^{(k)} + \\alpha r^{(k)}\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7703af7d",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- $x^{(k)}$: the approximate solution after $k$ steps"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fe5e4c39",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- $r^{(k)}$: the **residual** after $k$ steps"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "db1b6bcc",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Relate $r^{(k+1)}$ to $r^{(k)}$: the iteration matrix"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6a71d4d8",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "Unroll a little more:\n",
    "\n",
    "$$\\begin{aligned}\n",
    "x^{(0)} &\\gets 0 \\\\\n",
    "r^{(k)} &\\gets b - A x^{(k)} \\\\\n",
    "x^{(k+1)} &\\gets x^{(k)} + \\alpha r^{(k)} \\\\\n",
    "r^{(k+1)} &\\gets b - A x^{(k+1)} \\\\\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "58e774df",
   "metadata": {
    "cell_style": "split",
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Substite the third line in the fourth:\n",
    "\n",
    "$$\\begin{aligned}\n",
    "x^{(0)} &\\gets 0 \\\\\n",
    "r^{(k)} &\\gets b - A x^{(k)} \\\\\n",
    "x^{(k+1)} &\\gets x^{(k)} + \\alpha r^{(k)} \\\\\n",
    "r^{(k+1)} &\\gets b - A (x^{(k)} + \\alpha r^{(k)}) \\\\\n",
    "&\\phantom{\\gets} = \\underbrace{b - A x^{(k)}}_{r^{(k)}} - \\alpha A r^{(k)}\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f11b1701",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$\\large r^{(k+1)} = \\underbrace{(I - \\alpha A)}_{\\text{iteration matrix}} r^{(k)} = (I - \\alpha A)^{k+1} \\underbrace{r^{(0)}}_{b}.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "19e4a1b5",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Sufficient condition for convergence\n",
    "\n",
    "**Claim.** If $\\rho = \\max_{\\lambda \\in \\Lambda(I - \\alpha A)} |\\lambda| < 1$, then $r^{(k)} \\to 0$, and\n",
    "\n",
    "$$\\Large \\|r^{(k)}\\| = O(\\rho^k \\|b\\|)$$"
   ]
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   "id": "9ce3b939",
   "metadata": {
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   },
   "source": [
    "### Example: convergence of Richardson for the Laplacian"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "34fb12f9",
   "metadata": {
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    {
     "data": {
      "text/plain": [
       "richardson! (generic function with 1 method)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@views function richardson!(x, A, b, α, n_iter)\n",
    "    r = copy(b)\n",
    "    axpby!(α, r, 0.0, x) \n",
    "    for k in 1:(n_iter-1)\n",
    "        axpby!(1.0, b, 0.0, r)\n",
    "        mul!(r,A,x,-1.0,1.0)\n",
    "        axpy!(α, r, x)\n",
    "    end\n",
    "    return nothing\n",
    "end"
   ]
  },
  {
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   "execution_count": null,
   "id": "dee69737",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "d = 3\n",
    "m = 10\n",
    "L₃ = laplacian(d, m)\n",
    "b = randn(m^3)\n",
    "M = Matrix(L₃)\n",
    "x_dense = copy(b)\n",
    "t_dense = @elapsed M \\ b\n",
    "r_dense = norm(b - L₃ * (M \\ b))\n",
    "t_sparse = @elapsed L₃ \\ b\n",
    "r_sparse = norm(b - L₃ * (L₃ \\ b))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "2562ded6",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "d = 3\n",
    "m = 10\n",
    "t_fft = @elapsed inverse_laplacian(d, b)\n",
    "r_fft = norm(b - L₃ * inverse_laplacian(d, b));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "e1b35a78",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "t_rich = Float64[]\n",
    "r_rich = Float64[]\n",
    "for n_iter in 1:2000\n",
    "    x = zeros(m^3)\n",
    "    push!(t_rich, @elapsed richardson!(x, L₃, b, 0.5 / d, n_iter))\n",
    "    push!(r_rich, norm(b - L₃ * x))\n",
    "end\n",
    "t_rich = accumulate(min, t_rich[end:-1:1])[end:-1:1];"
   ]
  },
  {
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   "execution_count": 43,
   "id": "ec074c1c",
   "metadata": {
    "hideCode": true,
    "slideshow": {
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      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = plot(t_rich, r_rich, yscale=:log10, label=\"richardson\", xlabel=\"time\", ylabel=\"||r||\", title = \"Solver residuals, Laplacian in 3D, mₓ = $m\")\n",
    "plot!(p, [0, t_dense, t_dense, 2 * t_dense], [norm(b), norm(b), r_dense ,r_dense], label=\"dense\")\n",
    "plot!(p, [0, t_sparse, t_sparse, 2 * t_dense], [norm(b), norm(b), r_sparse ,r_sparse], label=\"sparse\")\n",
    "plot!(p, [0, t_fft, t_fft, 2 * t_dense], [norm(b), norm(b), r_fft ,r_fft], label=\"fft\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "a6dae3b3",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "d = 3\n",
    "m = 20\n",
    "L₃ = laplacian(d, m)\n",
    "b = randn(m^3)\n",
    "M = Matrix(L₃)\n",
    "t_fft = @elapsed inverse_laplacian(d, b)\n",
    "r_fft = norm(b - L₃ * inverse_laplacian(d, b));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "2b82b3dd",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "t_rich = Float64[]\n",
    "r_rich = Float64[]\n",
    "for n_iter in 1:2000\n",
    "    x = zeros(m^3)\n",
    "    push!(t_rich, @elapsed richardson!(x, L₃, b, 0.5 / d, n_iter))\n",
    "    push!(r_rich, norm(b - L₃ * x))\n",
    "end\n",
    "t_rich = accumulate(min, t_rich[end:-1:1])[end:-1:1];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "00db47c5",
   "metadata": {
    "hideCode": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
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   "source": [
    "r_rich = r_rich[1:200]\n",
    "t_rich = t_rich[1:200]\n",
    "p = plot(t_rich, r_rich, yscale=:log10, label=\"richardson\", xlabel=\"time\", ylabel=\"||r||\", title = \"Solver residuals, Laplacian in 3D, mₓ = $m\")\n",
    "plot!(p, [0, t_fft, t_fft, 2 * t_fft], [norm(b), norm(b), r_fft ,r_fft], label=\"fft\", c = 4)"
   ]
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   "id": "9c65447f",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
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   },
   "source": [
    "### Observations about Richardson for the Laplacian (about all iterative methods)\n",
    "\n",
    "- Whether Richardson is better than other methods depends on how small you want the norm of the residual to be\n",
    "\n",
    "- How well Richardson works depends on $m_x$: why?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "75f9d471",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Convergence behavior and eigenvalues\n",
    "\n",
    "Suppose $A$ is diagonalizable and expand $b$ in the eigenbasis,\n",
    "\n",
    "$$\\large b = \\sum_j v_j$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c9d2f882",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
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   },
   "source": [
    "Then $r^{(k)} = (I - \\alpha A)^k b$ implies\n",
    "\n",
    "$$\\large r^{(k)} = \\sum_{j} (1 - \\alpha \\lambda_j)^k v_j.$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "aca0199f",
   "metadata": {
    "hideCode": true,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "α = 1/6\n",
    "ρs = [eigmax(Matrix(I - α * laplacian(3, m))) for m in ms_5];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "59618134",
   "metadata": {
    "hideCode": true
   },
   "outputs": [
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     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = plot(ms_5, ρs, xlabel=\"mₓ\", ylabel=\"ρ\")\n",
    "plot!(p, ms_5[[1,end]], [1,1], l=(:dash, :black))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b34fcc76",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### The iteration polynomial for Richardson\n",
    "\n",
    "Let\n",
    "\n",
    "$$\\large p_k(x) = (1 - \\alpha x)^k.$$\n",
    "\n",
    "Then\n",
    "\n",
    "$$\\large r^{(k)} = \\sum_{j} p_k(\\lambda_j) v_j.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "92476e91",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "We can see the iteration polynomial without expanding $b$ in the eigenbasis:\n",
    "\n",
    "$$\\large r^{(k)} = p_k(A) b.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7e3d2508",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Polynomials and black box iterations\n",
    "\n",
    "When designing iterative algorithms that only use the matvec kernel, the goal is to make\n",
    "\n",
    "$$\\|\\large p_k(A) b\\|$$\n",
    "\n",
    "as small as possible (where $p_k$ is the iteration polynomial of the algorithm)."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3f3ee327",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Krylov subspaces"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e7af5429",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "The **Krylov subspace** $K_n(A, b)$ is\n",
    "\n",
    "$$\\large K_n(A, b) = \\{b, Ab, A^2 b, \\dots, A^{n-1} b\\}$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7a7630ba",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "A **Krylov subspace method** seeks an approximate solution $\\tilde{x}$  to $Ax = b$ in $K_n(A,b)$ for some $n$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0953f242",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Exercise\n",
    "\n",
    "Let $A \\in \\mathbb{C}^{m \\times m}$ and $b \\in \\mathbb{C}^m$ be arbitrary.  Show that any $x \\in K_n$ is equal to $p(A)b$ for some polynomial $b$ of degree $\\leq n - 1$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fdf51c18",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Because Krylov subspaces are nested, and because only one matvec is needed to grow a basis for $K_n(A,b)$ into a basis for $K_{n+1}(A,b)$, Krylov subspace methods are a useful class of black-box iterative algoritms."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bba4fdad",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### The conditioning of the literal Krylov basis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "fbc0cdf6",
   "metadata": {
    "cell_style": "center"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "K (generic function with 1 method)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function K(n, A, b)\n",
    "    if n == 1\n",
    "        return [copy(b);;]\n",
    "    else\n",
    "        Ks = K(n-1, A, b)\n",
    "        return [Ks A*Ks[:,end]]\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "05201c3e",
   "metadata": {
    "cell_style": "center",
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   "outputs": [
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     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d = 3\n",
    "m = 10\n",
    "n = 20\n",
    "A = laplacian(d, m)\n",
    "b = randn(m^d)\n",
    "conds = [cond(K(k, A, b)) for k in 1:20];\n",
    "plot(1:20,conds,xlabel=\"n\", ylabel=\"cond(Kₙ)\", yscale=:log10, st=:scatter)"
   ]
  }
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