Tutorial¶

Using plans and generating log files¶

PEXSI is written in C++, and the subroutines cannot directly interface with other programming languages such as C or FORTRAN. To solve this problem, the PEXSI internal data structure is handled using a datatype PPEXSIPlan. The idea and the usage of PPEXSIPlan is similar to fftw_plan in the FFTW (http://www.fftw.org/~fftw/fftw3_doc/Using-Plans.html) package.

In PEXSI, a matrix (or more accurately, its inverse) is generally referred to as a “pole”. The factorization and selected inversion procedure for a pole is computed in parallel using numProcRow * numProcCol processors.

When only selected inversion (PSelInv) is used, it is recommended to set the mpisize of the communicator comm to be just numProcRow * numProcCol.

When PEXSI is used to evaluate a large number of inverse matrices such as in the electronic structure calculation, it is best to set mpisize to be numPole*numProcRow*numProcCol, where numPole is the number of poles can be processed in parallel.

Starting from v1.0, when PPEXSIDFTDriver2 is used, it is best set mpisize to be numPoint*numPole*numProcRow*numProcCol, where numPoint is the number of PEXSI evaluations that can be performed in parallel. When mpisize < numPoint*numPole*numProcRow*numProcCol, PPEXSIDFTDriver2 will first parallel over the numProcRow*numProcCol and numPoint.

The output information is controlled by the outputFileIndex variable, which is a local variable for each processor. For instance, if this index is 1, then the corresponding processor will output to the file logPEXSI1. If outputFileIndex is negative, then this processor does NOT output logPEXSI files.

Note

• Each processor must output to a different file if outputFileIndex is non-negative.
• When many processors are used, it is not recommended for all processors to output the log files. This is because the IO takes time and can be the bottleneck on many architecture. A good practice is to let the master processor output information (generating logPEXSI0) or to let the master processor of each pole to output the information.

Parallel selected inversion for a real symmetric matrix¶

The parallel selected inversion routine for a real symmetric matrix can be used as follows. This assumes that the mpisize of MPI_COMM_WORLD is nprow * npcol.

#include  "c_pexsi_interface.h"
...
{
/* Setup the input matrix in distributed compressed sparse column (CSC) format */
...;
/* Initialize PEXSI.
* PPEXSIPlan is a handle communicating with the C++ internal data structure */
PPEXSIPlan   plan;

plan = PPEXSIPlanInitialize(
MPI_COMM_WORLD,
nprow,
npcol,
mpirank,
&info );

/* Tuning parameters of PEXSI. The default options is reasonable to
* start, and the parameters in options can be changed.  */
PPEXSIOptions  options;
PPEXSISetDefaultOptions( &options );

/* Load the matrix into the internal data structure */
plan,
options,
nrows,
nnz,
nnzLocal,
numColLocal,
colptrLocal,
rowindLocal,
AnzvalLocal,
1,     // S is an identity matrix here
NULL,  // S is an identity matrix here
&info );

/* Factorize the matrix symbolically */
PPEXSISymbolicFactorizeRealSymmetricMatrix(
plan,
options,
&info );

/* Main routine for computing selected elements and save into AinvnzvalLocal */
PPEXSISelInvRealSymmetricMatrix (
plan,
options,
AnzvalLocal,
AinvnzvalLocal,
&info );

...;
/* Post processing AinvnzvalLocal */
...;

PPEXSIPlanFinalize(
plan,
&info );
}


This routine computes the selected elements of the matrix $$A^{-1}=(H - z S)^{-1}$$ in parallel. The input matrix $$H$$ follows the Distribute CSC format, defined through the variables colptrLocal,rowindLocal, HnzvalLocal. The input matrix $$S$$ can be omitted if it is an identity matrix and by setting isSIdentity=1. If $$S$$ is not an identity matrix, the nonzero sparsity pattern is assumed to be the same as the nonzero sparsity pattern of $$H$$. Both HnzvalLocal and SnzvalLocal are double precision arrays.

An example is given in examples/driver_pselinv_real.c, which evaluates the selected elements of the inverse of the matrix saved in examples/lap2dr.matrix. See also PEXSI Real Symmetric Matrix for detailed information of its usage.

Parallel selected inversion for a complex symmetric matrix¶

The parallel selected inversion routine for a complex symmetric matrix is very similar to the real symmetric case. An example is given in examples/driver_pselinv_complex.c. See also PEXSI Real Symmetric Matrix for detailed information of its usage.

Parallel selected inversion for a real unsymmetric matrix¶

The parallel selected inversion routine for a real unsymmetric matrix can be used as follows. This assumes that the size of MPI_COMM_WORLD is nprow * npcol.

#include  "c_pexsi_interface.h"
...
{
/* Setup the input matrix in distributed compressed sparse column (CSC) format */
...;
/* Initialize PEXSI.
* PPEXSIPlan is a handle communicating with the C++ internal data structure */
PPEXSIPlan   plan;

plan = PPEXSIPlanInitialize(
MPI_COMM_WORLD,
nprow,
npcol,
mpirank,
&info );

/* Tuning parameters of PEXSI. The default options is reasonable to
* start, and the parameters in options can be changed.  */
PPEXSIOptions  options;
PPEXSISetDefaultOptions( &options );

/* Load the matrix into the internal data structure */
plan,
options,
nrows,
nnz,
nnzLocal,
numColLocal,
colptrLocal,
rowindLocal,
AnzvalLocal,
1,     // S is an identity matrix here
NULL,  // S is an identity matrix here
&info );

/* Factorize the matrix symbolically */
PPEXSISymbolicFactorizeRealUnsymmetricMatrix(
plan,
options,
&info );

/* Main routine for computing selected elements and save into AinvnzvalLocal */
PPEXSISelInvRealUnsymmetricMatrix (
plan,
options,
AnzvalLocal,
AinvnzvalLocal,
&info );

...;
/* Post processing AinvnzvalLocal */
...;

PPEXSIPlanFinalize(
plan,
&info );
}


This routine computes the selected elements of the matrix $$A^{-1}=(H - z S)^{-1}$$ in parallel. The input matrix $$H$$ follows the Distribute CSC format, defined through the variables colptrLocal, rowindLocal, HnzvalLocal. The input matrix $$S$$ can be omitted if it is an identity matrix and by setting isSIdentity=1. If $$S$$ is not an identity matrix, the nonzero sparsity pattern is assumed to be the same as the nonzero sparsity pattern of $$H$$. Both HnzvalLocal and SnzvalLocal are double precision arrays.

Note

As discussed in selected elements, for general non-symmetric matrices, the selected elements are the elements such that $$\{A_{j,i}\ne 0\}$$. This means that the matrix elements computed corresponding to the sparsity pattern of $$A^T$$. However, storing the matrix elements $$\{A^{-1}_{i,j}\vert A_{j,i}\ne 0\}$$ is practically cumbersome, especially in the context of distributed computing. Hence we choose to store the selected elements for $$A^{-T}$$, i.e. $$\{A^{-T}_{i,j}\vert A_{i,j}\ne 0\}$$. These are the values obtained from the non-symmetric version of PSelInv.

An example is given in examples/driver_pselinv_real_unsym.c, which evaluates the selected elements of the inverse of the matrix saved in examples/big.unsym.matrix. See also PPEXSISelInvRealUnsymmetricMatrix for detailed information of its usage.

Parallel selected inversion for a complex unsymmetric matrix¶

The parallel selected inversion routine for a complex unsymmetric matrix is very similar to the real unsymmetric case. An example is given in examples/driver_pselinv_complex_unsym.c. See also PPEXSISelInvComplexUnsymmetricMatrix for detailed information of its usage.

Similar to the case of real unsymmetric matrices, the values $$\{A^{-T}_{i,j}\vert A_{i,j}\ne 0\}$$ are the values obtained from the non-symmetric version of PSelInv.

Solving Kohn-Sham density functional theory: I¶

The simplest way to use PEXSI to solve Kohn-Sham density functional theory is to use the PPEXSIDFTDriver2 routine. This routine uses built-in heuristics to obtain values of some parameters in PEXSI and provides a relatively small set of adjustable parameters for users to tune. This routine estimates the chemical potential self-consistently using a combined approach of inertia counting procedure and Newton’s iteration through PEXSI. Some heuristic approach is also implemented in this routine for dynamic adjustment of the chemical potential and some stopping criterion.

An example routine is given in examples/driver_ksdft.c, which solves a fake DFT problem by taking a Hamiltonian matrix from examples/lap2dr.matrix.

Here is the structure of the code using the simple driver routine.

#include  "c_pexsi_interface.h"
...
{
/* Setup the input matrix in distributed compressed sparse column (CSC) format */
...;
/* Initialize PEXSI.
* PPEXSIPlan is a handle communicating with the C++ internal data structure */
PPEXSIPlan   plan;

/* Set the outputFileIndex to be the pole index */
/* The first processor for each pole outputs information */
if( mpirank % (nprow * npcol) == 0 ){
outputFileIndex = mpirank / (nprow * npcol);
}
else{
outputFileIndex = -1;
}

plan = PPEXSIPlanInitialize(
MPI_COMM_WORLD,
nprow,
npcol,
outputFileIndex,
&info );

/* Tuning parameters of PEXSI. See PPEXSIOption for explanation of the
* parameters */
PPEXSIOptions  options;
PPEXSISetDefaultOptions( &options );

options.temperature  = 0.019; // 3000K
options.muPEXSISafeGuard  = 0.2;
options.numElectronPEXSITolerance = 0.001;
/* method = 1: Contour integral ; method = 2: Moussa optimized poles; default is 2*/
options.method = 2;
/* typically 20-30 poles when using method = 2; 40-80 poles when method = 1 */
options.numPole  = 20;
/* 2 points parallelization is set as default. */
options.nPoints = 2;

/* Load the matrix into the internal data structure */
plan,
options,
nrows,
nnz,
nnzLocal,
numColLocal,
colptrLocal,
rowindLocal,
HnzvalLocal,
isSIdentity,
SnzvalLocal,
&info );

/* Call the simple DFT driver2 using PEXSI */
PPEXSIDFTDriver2(
plan,
options,
numElectronExact,
&muPEXSI,
&numElectronPEXSI,
&numTotalInertiaIter,
&info );

/* Retrieve the density matrix and other quantities from the plan */
if(mpirank < nprow * npcol ) {

PPEXSIRetrieveRealDM(
plan,
DMnzvalLocal,
&totalEnergyH,
&info );

PPEXSIRetrieveRealEDM(
plan,
options,
EDMnzvalLocal,
&totalEnergyS,
&info );
}

/* Clean up */
PPEXSIPlanFinalize(
plan,
&info );
}


Solving Kohn-Sham density functional theory: II¶

In a DFT calculation, the information of the symbolic factorization can be reused for different $$(H,S)$$ matrix pencil if the sparsity pattern does not change. An example routine is given in examples/driver2_ksdft.c, which solves a fake DFT problem by taking a Hamiltonian matrix from examples/lap2dr.matrix.

Here is the structure of the code using the simple driver routine.

#include  "c_pexsi_interface.h"
...
{
/* Perform DFT calculation as in the previous note */

/* Update and obtain another set of H and S */

/* Solve the problem once again without symbolic factorization */
plan,
options,
nrows,
nnz,
nnzLocal,
numColLocal,
colptrLocal,
rowindLocal,
HnzvalLocal,
isSIdentity,
SnzvalLocal,
&info );

// No need to perform symbolic factorization
options.isSymbolicFactorize = 0;
// Given a good guess of the chemical potential, no need to perform
// inertia counting.
options.isInertiaCount = 0;
// Optional update mu0, muMin0, muMax0 in PPEXSIOptions

PPEXSIDFTDriver2(
plan,
options,
numElectronExact,
&muPEXSI,
&numElectronPEXSI,
&numTotalInertiaIter,
&info );

}


Note

The built-in heuristics in PPEXSIDFTDriver2 may not be optimal. It handles only one $$(H,S)$$ pair at a time, and does not accept multiple matrix pairs $$\{(H_l,S_l)\}$$ as in the case of spin-orbit polarized calculations. For expert users and developers, it should be relatively easy to dig into the driver routine, and only use PEXSI::PPEXSIData::SymbolicFactorizeRealSymmetricMatrix (for symbolic factorization), PEXSI::PPEXSIData::CalculateNegativeInertiaReal (for inertia counting), and PEXSI::PPEXSIData::CalculateFermiOperatorReal (for one-shot PEXSI calculation) to improve heuristics and extend the functionalities.

Parallel computation of the Fermi operator for complex Hermitian matrices¶

The PPEXSIDFTDriver routine and PPEXSIDFTDriver2 routines are standalone routines for solving the density matrix with the capability of finding the chemical potential effectively. This can be used for $$\Gamma$$ point calculation. For electronic structure calculations with k-points, multiple Hamiltonian operators may be needed to compute the number of electrons. The PEXSI package provides expert level routines for such purpose. See driver_fermi_complex.c for an example of the components.