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BlockKrylovSchur/BlockKrylovSchurEpetraExGenAztecOO.cpp

This is an example of how to use the Anasazi::BlockKrylovSchurSolMgr solver manager to solve a generalized eigenvalue problem, using Epetra data stuctures and the AztecOO solver package.

// This example computes the eigenvalues of smallest magnitude of a
// generalized eigenvalue problem $K x = \lambda M x$, using Anasazi's
// implementation of the block Krylov-Schur method.  It implements
// inverse iteration using an AztecOO iterative linear solver with an
// Ifpack preconditioner.
//
// Anasazi computes the smallest-magnitude eigenvalues using a
// shift-and-invert strategy.  For simplicity, the code below uses a
// shift of zero.  It illustrates the general pattern for using
// Anasazi for this problem:
//
//   1. Construct an "operator" A such that $Az = K^{-1} M z$.
//   2. Use Anasazi to solve $Az = \sigma z$, which is a spectral
//      transformation of the original problem $K x = \lambda M x$.
//   3. The eigenvalues $\lambda$ of the original problem are the
//      inverses of the eigenvalues $\sigma$ of the transformed
//      problem.
//
// The "operator A such that $A z = K^{-1} M z$" is a subclass of
// Epetra_Operator.  The Apply method of that operator takes the
// vector b, and computes $x = K^{-1} M b$.  It does so first by
// applying the matrix M, and then by solving the linear system $K x = // M b$ for x.  Trilinos implements many different ways to solve
// linear systems.  The example below uses an iterative linear solver
// provided by the AztecOO package, with an Ifpack preconditioner, to
// solve linear systems.

// Include header for block Krylov-Schur solver
#include "AnasaziBlockKrylovSchurSolMgr.hpp"
// Include header to define basic eigenproblem Ax = \lambda*Bx
#include "AnasaziBasicEigenproblem.hpp"
// plan to use Tpetra objects instead of Epetra objects, include
// Thyra objects instead of Epetra objects.
// Include header for Epetra sparse matrix, Map (representation of
// parallel distributions), and linear problem.  AztecOO uses the
// latter to encapsulate linear systems to solve.
#include "Epetra_Map.h"
#include "Epetra_CrsMatrix.h"
#include "Epetra_LinearProblem.h"
// Include header for AztecOO iterative linear solver, and
// AztecOO_Operator.  The latter wraps an AztecOO solver in an
// Epetra_Operator.
#include "AztecOO.h"
#include "AztecOO_Operator.h"
// Include header for Ifpack preconditioner factory
#include "Ifpack.h"
// Include header for Teuchos serial dense matrix, which we use to
// compute the eigenvectors.
#include "Teuchos_SerialDenseMatrix.hpp"
// Include header for the problem definition
#include "ModeLaplace2DQ2.h"

// Include selected communicator class and map required by Epetra objects
#ifdef EPETRA_MPI
#  include "Epetra_MpiComm.h"
#else
#  include "Epetra_SerialComm.h"
#endif // EPETRA_MPI

// ****************************************************************************
// BEGIN MAIN ROUTINE
// ****************************************************************************

int
main (int argc, char *argv[])
{
using Teuchos::RCP;
using Teuchos::rcp;
using std::cerr;
using std::cout;
using std::endl;
// Anasazi solvers have the following template parameters:
//
//   - Scalar: The type of dot product results.
//   - MV: The type of (multi)vectors.
//   - OP: The type of operators (functions from multivector to
//     multivector).  A matrix (like Epetra_CrsMatrix) is an example
//     of an operator; an Ifpack preconditioner is another example.
//
// Here, Scalar is double, MV is Epetra_MultiVector, and OP is
// Epetra_Operator.
typedef Epetra_MultiVector MV;
typedef Epetra_Operator OP;
typedef Anasazi::MultiVecTraits<double, MV> MVT;

#ifdef EPETRA_MPI
// Initialize MPI
MPI_Init (&argc, &argv);
Epetra_MpiComm Comm (MPI_COMM_WORLD);
#else
Epetra_SerialComm Comm;
#endif // EPETRA_MPI

const int MyPID = Comm.MyPID ();

//
// Set up the test problem
//

// Dimensionality of the spatial domain to discretize
const int space_dim = 2;

// Size of each of the dimensions of the (discrete) domain
std::vector<double> brick_dim (space_dim);
brick_dim[0] = 1.0;
brick_dim[1] = 1.0;

// Number of elements in each of the dimensions of the domain
std::vector<int> elements (space_dim);
elements[0] = 10;
elements[1] = 10;

// Create the test problem
RCP<ModalProblem> testCase =
rcp (new ModeLaplace2DQ2 (Comm, brick_dim[0], elements[0],
brick_dim[1], elements[1]));

// Get the stiffness and mass matrices.
//
// rcp (T*, false) returns a nonowning (doesn't deallocate) RCP.
RCP<Epetra_CrsMatrix> K =
rcp (const_cast<Epetra_CrsMatrix* > (testCase->getStiffness ()), false);
RCP<Epetra_CrsMatrix> M =
rcp (const_cast<Epetra_CrsMatrix*> (testCase->getMass ()), false);

//
// Create linear solver for linear systems with K
//
// Anasazi uses shift and invert, with a "shift" of zero, to find
// the eigenvalues of least magnitude.  In this example, we
// implement the "invert" part of shift and invert by using an
// AztecOO iterative linear solver with an Ifpack preconditioner.
//

//
// Construct Ifpack preconditioner
//

// An Ifpack "factory" knows how to create Ifpack preconditioners.
Ifpack Factory;

// Use the Factory to create the preconditioner.  Factory.Create()
// returns a raw Ifpack2_Preconditioner pointer.  The caller (that's
// us!) is responsible for deallocation, so we use an owning RCP to
// deallocate it automatically.
//
// The Factory needs a string name of the preconditioner, and the
// overlap level.  (Almost) all Ifpack preconditioners are local to
// each MPI process.  Ifpack automatically does an additive Schwarz
// decomposition across processes.  The default overlap level for
// additive Schwarz is zero, but you can use a different overlap
// level here if you want.  The overlap level must be nonnegative,
// and only matters if running with more than one MPI process.
std::string PrecType = "ICT"; // incomplete Cholesky
const int OverlapLevel = 0;
// Create the preconditioner.
RCP<Ifpack_Preconditioner> Prec =
rcp (Factory.Create (PrecType, &*K, OverlapLevel));
if (Prec.is_null ()) {
throw std::logic_error ("Ifpack's factory returned a NULL preconditioner!");
}

//
// Set Ifpack preconditioner parameters.
//
// Set parameters after creating the preconditioner.  Please refer
// to Ifpack's documentation for a list of valid parameters.
//
Teuchos::ParameterList ifpackList;
ifpackList.set ("fact: drop tolerance", 1e-4);
ifpackList.set ("fact: ict level-of-fill", 0.0);
// The "combine mode" describes how to combine contributions from
// other MPI processes.  It only matters if the overlap level is
// nonzero.  See Epetra_CombineMode.h for documentation of of the
// various combine modes.

// Set the parameters.
IFPACK_CHK_ERR(Prec->SetParameters(ifpackList));

// Initialize the preconditioner.  This only looks at the structure
// of the matrix, not the values.  Nevertheless, the matrix must
// generally be fill complete at this point.  Compare Initialize()
// to a symbolic factorization, and Compute() (see below) to a
// numeric factorization.
IFPACK_CHK_ERR(Prec->Initialize());

// Compute the preconditioner.  This looks at both the structure and
// the values of the matrix.  Compare Initialize() (see above) to a
// symbolic factorization, and Compute() (see below) to a numeric
// factorization.
IFPACK_CHK_ERR(Prec->Compute());

//
// Set up AztecOO iterative solver for solving linear systems with K.
//

// Create Epetra linear problem class for solving linear systems
// with K.  This implements the inverse operator for shift and
// invert.
Epetra_LinearProblem precProblem;
// Tell the linear problem about the matrix K.  Epetra_LinearProblem
// doesn't know about RCP, so we have to give it a raw pointer.
precProblem.SetOperator (K.getRawPtr ());

// Create AztecOO iterative solver for solving linear systems with K.
AztecOO precSolver (precProblem);
// Tell the solver to use the Ifpack preconditioner we created above.
precSolver.SetPrecOperator (Prec.get ());
// Set AztecOO solver options.
precSolver.SetAztecOption (AZ_output, AZ_none); // Don't print output
precSolver.SetAztecOption (AZ_solver, AZ_cg); // Use CG

// Use the above AztecOO solver to create the AztecOO_Operator.
// This is the place where we tell the AztecOO solver the maximum
// number of iterations (here, we use the matrix dimension; in
// practice, you'll want a smaller number) and the convergence
// tolerance (here, 1e-12).
RCP<AztecOO_Operator> precOperator =
rcp (new AztecOO_Operator (&precSolver, K->NumGlobalRows (), 1e-12));

// Create an Operator that computes y = K^{-1} M x.
//
// This Operator object is the operator we give to Anasazi.  Thus,
// Anasazi just sees an operator that computes y = K^{-1} M x.  The
// matrix K got absorbed into precOperator via precProblem (the
// Epetra_LinearProblem object).  Later, when we set up the Anasazi
// eigensolver, we will need to tell it about M, so that it can
// orthogonalize basis vectors with respect to the inner product
// defined by M (since it is symmetric positive definite).
RCP<Anasazi::EpetraGenOp> Aop = rcp (new Anasazi::EpetraGenOp (precOperator, M));

//
// Set parameters for the block Krylov-Schur eigensolver
//

double tol = 1.0e-8;
int nev = 10;
int blockSize = 3;
int numBlocks = 3 * nev / blockSize;
int maxRestarts = 10;

// We're looking for the largest-magnitude eigenvalues of the
// _inverse_ operator, thus, the smallest-magnitude eigenvalues of
// the original operator.
std::string which = "LM";
int verbosity = Anasazi::Errors + Anasazi::Warnings + Anasazi::FinalSummary;

// Create ParameterList to pass into eigensolver
Teuchos::ParameterList MyPL;
MyPL.set ("Verbosity", verbosity);
MyPL.set ("Which", which);
MyPL.set ("Block Size", blockSize);
MyPL.set ("Num Blocks", numBlocks);
MyPL.set ("Maximum Restarts", maxRestarts);
MyPL.set ("Convergence Tolerance", tol);

// Create an initial set of vectors to start the eigensolver.  Note:
// This needs to have the same number of columns as the block size.
RCP<MV> ivec = rcp (new MV (K->Map (), blockSize));
MVT::MvRandom (*ivec);

// This object holds all the stuff about your problem that Anasazi
// will see.
//
// Anasazi only needs M so that it can orthogonalize basis vectors
// with respect to the M inner product.  Wouldn't it be nice if
// Anasazi didn't require M in two different places?  Alas, this is
// not currently the case.
RCP<Anasazi::BasicEigenproblem<double,MV,OP> > MyProblem =
rcp (new Anasazi::BasicEigenproblem<double,MV,OP> (Aop, M, ivec));

// Tell the eigenproblem that the matrix pencil (K,M) is symmetric.
MyProblem->setHermitian (true);

// Set the number of eigenvalues requested
MyProblem->setNEV (nev);

// Tell the eigenproblem that you are finished passing it information.
bool boolret = MyProblem->setProblem();
if (boolret != true) {
if (MyPID == 0) {
cout << "Anasazi::BasicEigenproblem::setProblem() returned with error." << endl;
}
#ifdef EPETRA_MPI
MPI_Finalize ();
#endif // EPETRA_MPI
return -1;
}

// Create the Block Krylov-Schur eigensolver.
Anasazi::BlockKrylovSchurSolMgr<double, MV, OP> MySolverMgr (MyProblem, MyPL);

// Solve the eigenvalue problem.
//
// Note that creating the eigensolver is separate from solving it.
// After creating the eigensolver, you may call solve() multiple
// times with different parameters or initial vectors.  This lets
// you reuse intermediate state, like allocated basis vectors.
Anasazi::ReturnType returnCode = MySolverMgr.solve ();
if (returnCode != Anasazi::Converged && MyPID == 0) {
cout << "Anasazi::EigensolverMgr::solve() returned unconverged." << endl;
}

// Get the eigenvalues and eigenvectors from the eigenproblem.
Anasazi::Eigensolution<double,MV> sol = MyProblem->getSolution ();
// Anasazi returns eigenvalues as Anasazi::Value, so that if
// Anasazi's Scalar type is real-valued (as it is in this case), but
// some eigenvalues are complex, you can still access the
// eigenvalues correctly.  In this case, there are no complex
// eigenvalues, since the matrix pencil is symmetric.
std::vector<Anasazi::Value<double> > evals = sol.Evals;
Teuchos::RCP<MV> evecs = sol.Evecs;
int numev = sol.numVecs;

if (numev > 0) {
// Reconstruct the eigenvalues.  The ones that Anasazi gave back
// are the inverses of the original eigenvalues.  Reconstruct the
// eigenvectors too.
Teuchos::SerialDenseMatrix<int,double> dmatr(numev,numev);
MV tempvec (K->Map (), MVT::GetNumberVecs (*evecs));
K->Apply (*evecs, tempvec);
MVT::MvTransMv (1.0, tempvec, *evecs, dmatr);

if (MyPID == 0) {
double compeval = 0.0;
cout << "Actual Eigenvalues (obtained by Rayleigh quotient) : " << endl;
cout << "------------------------------------------------------" << endl;
cout << std::setw(16) << "Real Part"
<< std::setw(16) << "Rayleigh Error" << endl;
cout << "------------------------------------------------------" << endl;
for (int i = 0; i < numev; ++i) {
compeval = dmatr(i,i);
cout << std::setw(16) << compeval
<< std::setw(16)
<< std::fabs (compeval - 1.0/evals[i].realpart)
<< endl;
}
cout << "------------------------------------------------------" << endl;
}

}

#ifdef EPETRA_MPI
MPI_Finalize ();
#endif // EPETRA_MPI

return 0;
}