docs/src/notebooks/Krylov Solver and Preconditioners.ipynb

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    "## PETSc KSP and PETSc PC\n",
    "In this tutorial we will have an in-depth look in to the way we can use a PETSc `KSP` object to solve a linear system arising from the discretization of a linear partial differential equation. In particular, we will focus our attention the usual Poisson problem in variational form, i.e. find $u_h\\in P^k(\\mathcal{T}_h)$ such for any $v_h \\in P^k(\\mathcal{T}_h)$ the following equation is verified,\n",
    "$$(\\nabla u_h,\\nabla v_h) = (f,v_h),\\;\\textrm{ for a certain data }f\\in [P^k(\\mathcal{T}_h)]^*.$$"
   ]
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   "id": "e1fe36f9",
   "metadata": {},
   "outputs": [
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     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Starting 1 engines with <class 'ipyparallel.cluster.launcher.LocalEngineSetLauncher'>\n"
     ]
    },
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       "  0%|          | 0/1 [00:00<?, ?engine/s]"
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   "source": [
    "from ipyparallel import Cluster\n",
    "c = await Cluster().start_and_connect(n=1, activate=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4ba8f95c",
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   "source": [
    "Let's test if the cluster has been initialized correctly by checking the size of the `COMM_WORLD`-"
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       "\u001b[0;31mOut[0:1]: \u001b[0m1"
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      "execute_input": "from mpi4py.MPI import COMM_WORLD\nCOMM_WORLD.Get_size()\n",
      "execute_result": {
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      "submitted": "2023-09-23T16:35:43.576214Z"
     },
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   "source": [
    "%%px\n",
    "from mpi4py.MPI import COMM_WORLD\n",
    "COMM_WORLD.Get_size()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "77b29016",
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   "source": [
    "\n",
    "First we need to construct the distributed mesh that will be interested in dealing with, then define the `BilinearForm` corresponding to the Poisson problem and the load vector `f` we are interested in solving for, which is a NGSolve `LinearForm` object."
   ]
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   "source": [
    "%%px\n",
    "from ngsolve import Mesh, BilinearForm, LinearForm, H1\n",
    "from ngsolve import x, y, dx, grad\n",
    "from netgen.geom2d import unit_square\n",
    "\n",
    "if COMM_WORLD.rank == 0:\n",
    "    mesh = Mesh(unit_square.GenerateMesh(maxh=0.2).Distribute(COMM_WORLD))\n",
    "else:\n",
    "    mesh = Mesh(ngm.Mesh.Receive(COMM_WORLD))\n",
    "fes = H1(mesh, order=3, dirichlet=\"left|right|top|bottom\")\n",
    "u,v = fes.TnT()\n",
    "a = BilinearForm(grad(u)*grad(v)*dx).Assemble()\n",
    "f = LinearForm(fes)\n",
    "f += 32 * (y*(1-y)+x*(1-x)) * v * dx"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "61dcb959",
   "metadata": {},
   "source": [
    "We now initialize a new `KrylovSolver` which wraps a PETSc `KSP` object and can be used to solve the linear system $A\\vec{u}_h = \\vec{f}_h$, obtained from the above discretization. In particular, it is possible to control the setup of the PETSc `KSP` solver using the `solverParameters` dictionary which is equivalent to passing PETSc command line options. More detail on available options and solvers can be found in the [PETSc KSP Manual](https://petsc.org/release/manual/ksp/#ch-ksp). Once the solver has been initialized it is possible to obtain a solution of the linear system in terms of `GridFunction` using the `solve` method."
   ]
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      "text/plain": [
       "[stdout:0] KSP Object: 1 MPI process\n",
       "  type: cg\n",
       "  maximum iterations=10000, initial guess is zero\n",
       "  tolerances:  relative=1e-05, absolute=1e-50, divergence=10000.\n",
       "  left preconditioning\n",
       "  using PRECONDITIONED norm type for convergence test\n",
       "PC Object: 1 MPI process\n",
       "  type: lu\n",
       "    out-of-place factorization\n",
       "    tolerance for zero pivot 2.22045e-14\n",
       "    matrix ordering: external\n",
       "    factor fill ratio given 0., needed 0.      Factored matrix follows:\n",
       "        Mat Object: 1 MPI process\n",
       "          type: mumps\n",
       "          rows=205, cols=205\n",
       "          package used to perform factorization: mumps\n",
       "          total: nonzeros=7603, allocated nonzeros=7603\n",
       "            MUMPS run parameters:\n",
       "              Use -ksp_view ::ascii_info_detail to display information for all processes\n",
       "              RINFOG(1) (global estimated flops for the elimination after analysis): 153913.\n",
       "              RINFOG(2) (global estimated flops for the assembly after factorization): 1916.\n",
       "              RINFOG(3) (global estimated flops for the elimination after factorization): 153913.\n",
       "              (RINFOG(12) RINFOG(13))*2^INFOG(34) (determinant): (0.,0.)*(2^0)              INFOG(3) (estimated real workspace for factors on all processors after analysis): 7603\n",
       "              INFOG(4) (estimated integer workspace for factors on all processors after analysis): 998\n",
       "              INFOG(5) (estimated maximum front size in the complete tree): 35\n",
       "              INFOG(6) (number of nodes in the complete tree): 14\n",
       "              INFOG(7) (ordering option effectively used after analysis): 2\n",
       "              INFOG(8) (structural symmetry in percent of the permuted matrix after analysis): 100\n",
       "              INFOG(9) (total real/complex workspace to store the matrix factors after factorization): 7603\n",
       "              INFOG(10) (total integer space store the matrix factors after factorization): 998\n",
       "              INFOG(11) (order of largest frontal matrix after factorization): 35\n",
       "              INFOG(12) (number of off-diagonal pivots): 0\n",
       "              INFOG(13) (number of delayed pivots after factorization): 0\n",
       "              INFOG(14) (number of memory compress after factorization): 0\n",
       "              INFOG(15) (number of steps of iterative refinement after solution): 0\n",
       "              INFOG(16) (estimated size (in MB) of all MUMPS internal data for factorization after analysis: value on the most memory consuming processor): 0\n",
       "              INFOG(17) (estimated size of all MUMPS internal data for factorization after analysis: sum over all processors): 0\n",
       "              INFOG(18) (size of all MUMPS internal data allocated during factorization: value on the most memory consuming processor): 0\n",
       "              INFOG(19) (size of all MUMPS internal data allocated during factorization: sum over all processors): 0\n",
       "              INFOG(20) (estimated number of entries in the factors): 7603\n",
       "              INFOG(21) (size in MB of memory effectively used during factorization - value on the most memory consuming processor): 0\n",
       "              INFOG(22) (size in MB of memory effectively used during factorization - sum over all processors): 0\n",
       "              INFOG(23) (after analysis: value of ICNTL(6) effectively used): 0\n",
       "              INFOG(24) (after analysis: value of ICNTL(12) effectively used): 1\n",
       "              INFOG(25) (after factorization: number of pivots modified by static pivoting): 0\n",
       "              INFOG(28) (after factorization: number of null pivots encountered): 0\n",
       "              INFOG(29) (after factorization: effective number of entries in the factors (sum over all processors)): 7603\n",
       "              INFOG(30, 31) (after solution: size in Mbytes of memory used during solution phase): 0, 0\n",
       "              INFOG(32) (after analysis: type of analysis done): 1\n",
       "              INFOG(33) (value used for ICNTL(8)): 7\n",
       "              INFOG(34) (exponent of the determinant if determinant is requested): 0\n",
       "              INFOG(35) (after factorization: number of entries taking into account BLR factor compression - sum over all processors): 7603\n",
       "              INFOG(36) (after analysis: estimated size of all MUMPS internal data for running BLR in-core - value on the most memory consuming processor): 0\n",
       "              INFOG(37) (after analysis: estimated size of all MUMPS internal data for running BLR in-core - sum over all processors): 0\n",
       "              INFOG(38) (after analysis: estimated size of all MUMPS internal data for running BLR out-of-core - value on the most memory consuming processor): 0\n",
       "              INFOG(39) (after analysis: estimated size of all MUMPS internal data for running BLR out-of-core - sum over all processors): 0\n",
       "  linear system matrix = precond matrix:\n",
       "  Mat Object: 1 MPI process\n",
       "    type: seqaij\n",
       "    rows=205, cols=205\n",
       "    total: nonzeros=2863, allocated nonzeros=2863\n",
       "    total number of mallocs used during MatSetValues calls=0\n",
       "      using I-node routines: found 137 nodes, limit used is 5\n"
      ]
     },
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   ],
   "source": [
    "%%px\n",
    "from ngsPETSc import KrylovSolver\n",
    "from ngsolve.webgui import  Draw\n",
    "solver = KrylovSolver(a,fes, solverParameters={'ksp_type': 'cg', 'pc_type': 'lu',\n",
    "                                'pc_factor_mat_solver_type': 'mumps'})\n",
    "gfu = solver.solve(f)\n",
    "Draw(gfu,mesh, \"solution\")\n",
    "solver.ksp.view()"
   ]
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   "source": [
    "In the previous example we have solved using a CG method and a LU factorization preconditioned computed using MUMPS. We can access all this information from the using the `view` of the `solver.ksp` object. We can also solve using more exotic methods and preconditioners, such as a biconjugate gradient method using HYPRE for preconditioning."
   ]
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     "data": {
      "text/plain": [
       "[stdout:0] KSP Object: 1 MPI process\n",
       "  type: bcgs\n",
       "  maximum iterations=10000, initial guess is zero\n",
       "  tolerances:  relative=1e-05, absolute=1e-50, divergence=10000.\n",
       "  left preconditioning\n",
       "  using PRECONDITIONED norm type for convergence test\n",
       "PC Object: 1 MPI process\n",
       "  type: hypre\n",
       "    HYPRE BoomerAMG preconditioning\n",
       "      Cycle type V\n",
       "      Maximum number of levels 25\n",
       "      Maximum number of iterations PER hypre call 1\n",
       "      Convergence tolerance PER hypre call 0.\n",
       "      Threshold for strong coupling 0.25\n",
       "      Interpolation truncation factor 0.\n",
       "      Interpolation: max elements per row 0\n",
       "      Number of levels of aggressive coarsening 0\n",
       "      Number of paths for aggressive coarsening 1\n",
       "      Maximum row sums 0.9\n",
       "      Sweeps down         1\n",
       "      Sweeps up           1\n",
       "      Sweeps on coarse    1\n",
       "      Relax down          symmetric-SOR/Jacobi\n",
       "      Relax up            symmetric-SOR/Jacobi\n",
       "      Relax on coarse     Gaussian-elimination\n",
       "      Relax weight  (all)      1.\n",
       "      Outer relax weight (all) 1.\n",
       "      Maximum size of coarsest grid 9\n",
       "      Minimum size of coarsest grid 1\n",
       "      Using CF-relaxation\n",
       "      Not using more complex smoothers.\n",
       "      Measure type        local\n",
       "      Coarsen type        Falgout\n",
       "      Interpolation type  classical\n",
       "      SpGEMM type         hypre\n",
       "  linear system matrix = precond matrix:\n",
       "  Mat Object: 1 MPI process\n",
       "    type: seqaij\n",
       "    rows=205, cols=205\n",
       "    total: nonzeros=2863, allocated nonzeros=2863\n",
       "    total number of mallocs used during MatSetValues calls=0\n",
       "      using I-node routines: found 137 nodes, limit used is 5\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%px\n",
    "solver = KrylovSolver(a,fes, solverParameters={'ksp_type': 'bcgs', 'pc_type': 'hypre'})\n",
    "gfu = solver.solve(f)\n",
    "Draw(gfu,mesh, \"solution\")\n",
    "solver.ksp.view()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f65d64aa",
   "metadata": {},
   "source": [
    "It is also possible to import any PETSc `PC` object and used it NGSolve using the `PETScPC` NGSolve `Preconditioner`. This is very useful if one is interested in using NGSolve linear algebra functionality with \"foreign\" preconditioner. "
   ]
  },
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   "execution_count": 6,
   "id": "71305a02",
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      "text/plain": [
       "[stdout:0] \u001b[2KCG iteration 1, residual = 2.3515313068754367     \n",
       "\u001b[2KCG iteration 2, residual = 0.11703951230766897     \n",
       "\u001b[2KCG iteration 3, residual = 0.02532238998559266     \n",
       "\u001b[2KCG iteration 4, residual = 0.002762280521164809     \n",
       "\u001b[2KCG iteration 5, residual = 0.00036992984473413113     \n",
       "\u001b[2KCG iteration 6, residual = 5.0193813921945675e-05     \n",
       "\u001b[2KCG iteration 7, residual = 8.048185966060108e-06     \n",
       "\u001b[2KCG iteration 8, residual = 1.1206592184815855e-06     \n",
       "\u001b[2KCG iteration 9, residual = 1.3930271656429977e-07     \n",
       "\u001b[2KCG iteration 10, residual = 2.358932103568244e-08     \n",
       "\u001b[2KCG iteration 11, residual = 3.2889334523253205e-09     \n",
       "\u001b[2KCG iteration 12, residual = 4.688724686311323e-10     \n",
       "\u001b[2KCG iteration 13, residual = 6.51694553700458e-11     \n",
       "\u001b[2KCG iteration 14, residual = 9.11213172981437e-12     \n",
       "\u001b[2KCG iteration 15, residual = 7.541816903801115e-13     \n"
      ]
     },
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     "data": {
      "text/plain": [
       "[stderr:0] WARNING: kwarg 'pc_type' is an undocumented flags option for class <class 'ngsolve.comp.Preconditioner'>, maybe there is a typo?\n"
      ]
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      "error": null,
      "execute_input": "from ngsPETSc import pc\nfrom ngsolve import Preconditioner, GridFunction\nfrom ngsolve.solvers import CG\npre = Preconditioner(a, \"PETScPC\", pc_type=\"hypre\")\ngfu = GridFunction(fes)\ngfu.vec.data = CG(a.mat, rhs=f.vec, pre=pre, printrates=mesh.comm.rank==0)\nDraw(gfu,mesh, \"solution\")\n",
      "execute_result": {
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        "text/plain": "BaseWebGuiScene"
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       "execution_count": 5,
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      "status": "ok",
      "stderr": "WARNING: kwarg 'pc_type' is an undocumented flags option for class <class 'ngsolve.comp.Preconditioner'>, maybe there is a typo?\n",
      "stdout": "\u001b[2KCG iteration 1, residual = 2.3515313068754367     \n\u001b[2KCG iteration 2, residual = 0.11703951230766897     \n\u001b[2KCG iteration 3, residual = 0.02532238998559266     \n\u001b[2KCG iteration 4, residual = 0.002762280521164809     \n\u001b[2KCG iteration 5, residual = 0.00036992984473413113     \n\u001b[2KCG iteration 6, residual = 5.0193813921945675e-05     \n\u001b[2KCG iteration 7, residual = 8.048185966060108e-06     \n\u001b[2KCG iteration 8, residual = 1.1206592184815855e-06     \n\u001b[2KCG iteration 9, residual = 1.3930271656429977e-07     \n\u001b[2KCG iteration 10, residual = 2.358932103568244e-08     \n\u001b[2KCG iteration 11, residual = 3.2889334523253205e-09     \n\u001b[2KCG iteration 12, residual = 4.688724686311323e-10     \n\u001b[2KCG iteration 13, residual = 6.51694553700458e-11     \n\u001b[2KCG iteration 14, residual = 9.11213172981437e-12     \n\u001b[2KCG iteration 15, residual = 7.541816903801115e-13     \n",
      "submitted": "2023-09-23T16:35:45.056262Z"
     },
     "output_type": "display_data"
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   ],
   "source": [
    "%%px\n",
    "from ngsPETSc import pc\n",
    "from ngsolve import Preconditioner, GridFunction\n",
    "from ngsolve.solvers import CG\n",
    "pre = Preconditioner(a, \"PETScPC\", pc_type=\"hypre\")\n",
    "gfu = GridFunction(fes)\n",
    "gfu.vec.data = CG(a.mat, rhs=f.vec, pre=pre, printrates=mesh.comm.rank==0)\n",
    "Draw(gfu,mesh, \"solution\")"
   ]
  }
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