# Euclid¶

The Euclid library is a scalable implementation of the Parallel ILU algorithm that was presented at SC99 [HyPo1999], and published in expanded form in the SIAM Journal on Scientific Computing [HyPo2001]. By scalable we mean that the factorization (setup) and application (triangular solve) timings remain nearly constant when the global problem size is scaled in proportion to the number of processors. As with all ILU preconditioning methods, the number of iterations is expected to increase with global problem size.

Experimental results have shown that PILU preconditioning is in general more effective than Block Jacobi preconditioning for minimizing total solution time. For scaled problems, the relative advantage appears to increase as the number of processors is scaled upwards. Euclid may also be used to good advantage as a smoother within multigrid methods.

## Overview¶

Euclid is best thought of as an “extensible ILU preconditioning framework.” Extensible means that Euclid can (and eventually will, time and contributing agencies permitting) support many variants of ILU($$k$$) and ILUT preconditioning. (The current release includes Block Jacobi ILU($$k$$) and Parallel ILU($$k$$) methods.) Due to this extensibility, and also because Euclid was developed independently of the hypre project, the methods by which one passes runtime parameters to Euclid preconditioners differ in some respects from the hypre norm. While users can directly set options within their code, options can also be passed to Euclid preconditioners via command line switches and/or small text-based configuration files. The latter strategies have the advantage that users will not need to alter their codes as Euclid’s capabilities are extended.

The following fragment illustrates the minimum coding required to invoke Euclid preconditioning within hypre application contexts. The next subsection provides examples of the various ways in which Euclid’s options can be set. The final subsection lists the options, and provides guidance as to the settings that (in our experience) will likely prove effective for minimizing execution time.

#include "HYPRE_parcsr_ls.h"

HYPRE_Solver eu;
HYPRE_Solver pcg_solver;
HYPRE_ParVector b, x;
HYPRE_ParCSRMatrix A;

//Instantiate the preconditioner.
HYPRE_EuclidCreate(comm, &eu);

//Optionally use the following three methods to set runtime options.
// 1. pass options from command line or string array.
HYPRE_EuclidSetParams(eu, argc, argv);

// 2. pass options from a configuration file.
HYPRE_EuclidSetParamsFromFile(eu, "filename");

// 3. pass options using interface functions.
HYPRE_EuclidSetLevel(eu, 3);
...

//Set Euclid as the preconditioning method for some
//other solver, using the function calls HYPRE_EuclidSetup
//and HYPRE_EuclidSolve.  We assume that the pcg_solver
//has been properly initialized.
HYPRE_PCGSetPrecond(pcg_solver,
(HYPRE_PtrToSolverFcn) HYPRE_EuclidSolve,
(HYPRE_PtrToSolverFcn) HYPRE_EuclidSetup,
eu);

//Solve the system by calling the Setup and Solve methods for,
//in this case, the HYPRE_PCG solver.  We assume that A, b, and x
//have been properly initialized.
HYPRE_PCGSetup(pcg_solver, (HYPRE_Matrix)A, (HYPRE_Vector)b, (HYPRE_Vector)x);
HYPRE_PCGSolve(pcg_solver, (HYPRE_Matrix)parcsr_A, (HYPRE_Vector)b, (HYPRE_Vector)x);

//Destroy the Euclid preconditioning object.
HYPRE_EuclidDestroy(eu);


## Setting Options: Examples¶

For expositional purposes, assume you wish to set the ILU($$k$$) factorization level to the value $$k = 3$$. There are several methods of accomplishing this. Internal to Euclid, options are stored in a simple database that contains (name, value) pairs. Various of Euclid’s internal (private) functions query this database to determine, at runtime, what action the user has requested. If you enter the option -eu_stats 1, a report will be printed when Euclid’s destructor is called; this report lists (among other statistics) the options that were in effect during the factorization phase.

Method 1. By default, Euclid always looks for a file titled database in the working directory. If it finds such a file, it opens it and attempts to parse it as a configuration file. Configuration files should be formatted as follows.

>cat database
#this is an optional comment
-level 3


Any line in a configuration file that contains a “#” character in the first column is ignored. All other lines should begin with an option name, followed by one or more blanks, followed by the option value. Note that option names always begin with a - character. If you include an option name that is not recognized by Euclid, no harm should ensue.

Method 2. To pass options on the command line, call

HYPRE_EuclidSetParams(HYPRE_Solver solver, int argc, char *argv[]);


where argc and argv carry the usual connotation: main(int argc, char *argv[]). If your hypre application is called phoo, you can then pass options on the command line per the following example.

mpirun -np 2 phoo -level 3


Since Euclid looks for the database file when HYPRE_EuclidCreate is called, and parses the command line when HYPRE_EuclidSetParams is called, option values passed on the command line will override any similar settings that may be contained in the database file. Also, if same option name appears more than once on the command line, the final appearance determines the setting.

Some options, such as -bj (see next subsection) are boolean. Euclid always treats these options as the value 1 (true) or 0 (false). When passing boolean options from the command line the value may be committed, in which case it assumed to be 1. Note, however, that when boolean options are contained in a configuration file, either the 1 or 0 must stated explicitly.

Method 3. There are two ways in which you can read in options from a file whose name is other than database. First, you can call HYPRE_EuclidSetParamsFromFile to specify a configuration filename. Second, if you have passed the command line arguments as described above in Method 2, you can then specify the configuration filename on the command line using the -db_filename filename option, e.g.,

mpirun -np 2 phoo -db_filename ../myConfigFile


Method 4. One can also set parameters via interface functions, e.g

int HYPRE_EuclidSetLevel(HYPRE_Solver solver, int level);


For a full set of functions, see the reference manual.

## Options Summary¶

• -level $$\langle int \rangle$$ Factorization level for ILU($$k$$). Default: 1. Guidance: for 2D convection-diffusion and similar problems, fastest solution time is typically obtained with levels 4 through 8. For 3D problems fastest solution time is typically obtained with level 1.

• -bj Use Block Jacobi ILU preconditioning instead of PILU. Default: 0 (false). Guidance: if subdomains contain relatively few nodes (less than 1,000), or the problem is not well partitioned, Block Jacobi ILU may give faster solution time than PILU.

• -eu_stats When Euclid’s destructor is called a summary of runtime settings and timing information is printed to stdout. Default: 0 (false). The timing marks in the report are the maximum over all processors in the MPI communicator.

• -eu_mem When Euclid’s destructor is called a summary of Euclid’s memory usage is printed to stdout. Default: 0 (false). The statistics are for the processor whose rank in MPI_COMM_WORLD is 0.

• -printTestData This option is used in our autotest procedures, and should not normally be invoked by users.

• -sparseA $$\langle float \rangle$$ Drop-tolerance for ILU($$k$$) factorization. Default: 0 (no dropping). Entries are treated as zero if their absolute value is less than sparseA * max, where max is the largest absolute value of any entry in the row. Guidance: try this in conjunction with -rowScale. CAUTION: If the coefficient matrix $$A$$ is symmetric, this setting is likely to cause the filled matrix, $$F = L+U-I$$, to be unsymmetric. This setting has no effect when ILUT factorization is selected.

• -rowScale Scale values prior to factorization such that the largest value in any row is +1 or -1. Default: 0 (false). CAUTION: If the coefficient matrix $$A$$ is symmetric, this setting is likely to cause the filled matrix, $$F = L+U-I$$, to be unsymmetric. Guidance: if the matrix is poorly scaled, turning on row scaling may help convergence.

• -ilut $$\langle float \rangle$$ Use ILUT factorization instead of the default, ILU($$k$$). Here, $$\langle float \rangle$$ is the drop tolerance, which is relative to the largest absolute value of any entry in the row being factored. CAUTION: If the coefficient matrix $$A$$ is symmetric, this setting is likely to cause the filled matrix, $$F = L+U-I$$, to be unsymmetric. NOTE: this option can only be used sequentially!