/**
* This example program sets up and solves the Laplace
* equation \f$-\nabla^2 u=0\f$. See the
* document GettingStarted.pdf for more information.
*/
int main(int argc, char** argv)
{
try
{
/* command-line options */
std::string meshFile="plateWithHole3D-1";
std::string solverFile = "aztec-ml.xml";
Sundance::setOption("meshFile", meshFile, "mesh file");
Sundance::setOption("solver", solverFile,
"name of XML file for solver");
/* Initialize */
Sundance::init(&argc, &argv);
/* --- Specify vector representation to be used --- */
VectorType<double> vecType = new EpetraVectorType();
/* --- Read mesh --- */
MeshType meshType = new BasicSimplicialMeshType();
MeshSource meshSrc
= new ExodusMeshReader(meshFile, meshType);
Mesh mesh = meshSrc.getMesh();
/* --- Specification of geometric regions --- */
/* Region "interior" consists of all maximal-dimension cells */
CellFilter interior = new MaximalCellFilter();
/* Identify boundary regions via labels in mesh */
CellFilter edges = new DimensionalCellFilter(2);
CellFilter south = edges.labeledSubset(1);
CellFilter east = edges.labeledSubset(2);
CellFilter north = edges.labeledSubset(3);
CellFilter west = edges.labeledSubset(4);
CellFilter hole = edges.labeledSubset(5);
CellFilter down = edges.labeledSubset(6);
CellFilter up = edges.labeledSubset(7);
/* --- Symbolic equation definition --- */
/* Test and unknown function */
BasisFamily basis = new Lagrange(1);
Expr u = new UnknownFunction(basis, "u");
Expr v = new TestFunction(basis, "v");
/* Gradient operator */
Expr dx = new Derivative(0);
Expr dy = new Derivative(1);
Expr dz = new Derivative(2);
Expr grad = List(dx, dy, dz);
/* We need a quadrature rule for doing the integrations */
QuadratureFamily quad1 = new GaussianQuadrature(1);
QuadratureFamily quad2 = new GaussianQuadrature(2);
/** Write the weak form */
Expr eqn
= Integral(interior, (grad*u)*(grad*v), quad1)
+ Integral(east, v, quad1);
/* Write the essential boundary conditions */
Expr h = new CellDiameterExpr();
Expr bc = EssentialBC(west, v*u/h, quad2);
/* Set up linear problem */
LinearProblem prob(mesh, eqn, bc, v, u, vecType);
/* --- solve the problem --- */
/* Create the solver as specified by parameters in
* an XML file */
LinearSolver<double> solver
= LinearSolverBuilder::createSolver(solverFile);
/* Solve! The solution is returned as an Expr containing a
* DiscreteFunction */
Expr soln = prob.solve(solver);
/* --- Postprocessing --- */
/* Project the derivative onto the P1 basis */
DiscreteSpace discSpace(mesh, List(basis, basis, basis), vecType);
L2Projector proj(discSpace, grad*soln);
Expr gradU = proj.project();
/* Write the solution and its projected gradient to a VTK file */
FieldWriter w = new VTKWriter("LaplaceDemo3D");
w.addMesh(mesh);
w.addField("soln", new ExprFieldWrapper(soln[0]));
w.addField("du_dx", new ExprFieldWrapper(gradU[0]));
w.addField("du_dy", new ExprFieldWrapper(gradU[1]));
w.addField("du_dz", new ExprFieldWrapper(gradU[2]));
w.write();
/* Check flux balance */
Expr n = CellNormalExpr(3, "n");
CellFilter wholeBdry = east+west+north+south+up+down+hole;
Expr fluxExpr
= Integral(wholeBdry, (n*grad)*soln, quad1);
double flux = evaluateIntegral(mesh, fluxExpr);
Out::root() << "numerical flux = " << flux << std::endl;
/* --- Let's compute a few other quantities, such as the centroid of
* the mesh:*/
/* Coordinate functions let us build up functions of position */
Expr x = new CoordExpr(0);
Expr y = new CoordExpr(1);
Expr z = new CoordExpr(2);
Expr xCMExpr = Integral(interior, x, quad1);
Expr yCMExpr = Integral(interior, y, quad1);
Expr zCMExpr = Integral(interior, z, quad1);
Expr volExpr = Integral(interior, 1.0, quad1);
double vol = evaluateIntegral(mesh, volExpr);
double xCM = evaluateIntegral(mesh, xCMExpr)/vol;
double yCM = evaluateIntegral(mesh, yCMExpr)/vol;
double zCM = evaluateIntegral(mesh, zCMExpr)/vol;
Out::root() << "centroid = (" << xCM << ", " << yCM
<< ", " << zCM << ")" << std::endl;
/* Next, compute the first Fourier sine coefficient of the solution on the
* surface of the hole.*/
Expr r = sqrt(x*x + y*y);
Expr sinPhi = y/r;
/* Use a higher-order quadrature rule for these integrals */
QuadratureFamily quad4 = new GaussianQuadrature(4);
Expr fourierSin1Expr = Integral(hole, sinPhi*soln, quad4);
Expr fourierDenomExpr = Integral(hole, sinPhi*sinPhi, quad2);
double fourierSin1 = evaluateIntegral(mesh, fourierSin1Expr);
double fourierDenom = evaluateIntegral(mesh, fourierDenomExpr);
Out::root() << "fourier sin m=1 = " << fourierSin1/fourierDenom << std::endl;
/* Compute the L2 norm of the solution */
Expr L2NormExpr = Integral(interior, soln*soln, quad2);
double l2Norm_method1 = sqrt(evaluateIntegral(mesh, L2NormExpr));
Out::os() << "method #1: ||soln|| = " << l2Norm_method1 << endl;
/* Use the L2Norm() function to do the same calculation */
double l2Norm_method2 = L2Norm(mesh, interior, soln, quad2);
Out::os() << "method #2: ||soln|| = " << l2Norm_method2 << endl;
/*
* Check that the flux is acceptably close to zero. The flux calculation
* is only O(h) so keep the tolerance loose. This
* is just a sanity check to ensure the code doesn't get completely
* broken after a change to the library.
*/
Sundance::passFailTest(fabs(flux), 1.0e-2);
}
catch(std::exception& e)
{
Sundance::handleException(e);
}
Sundance::finalize();
return Sundance::testStatus();
}
int main(int argc, char** argv)
{
try
{
int nx = 32;
double convTol = 1.0e-8;
double lambda = 0.5;
Sundance::setOption("nx", nx, "Number of elements");
Sundance::setOption("tol", convTol, "Convergence tolerance");
Sundance::setOption("lambda", lambda, "Lambda (parameter in Bratu's equation)");
Sundance::init(&argc, &argv);
Out::root() << "Bratu problem (lambda=" << lambda << ")" << endl;
Out::root() << "Newton's method, linearized by hand" << endl << endl;
VectorType<double> vecType = new EpetraVectorType();
MeshType meshType = new BasicSimplicialMeshType();
MeshSource mesher = new PartitionedLineMesher(0.0, 1.0, nx, meshType);
Mesh mesh = mesher.getMesh();
CellFilter interior = new MaximalCellFilter();
CellFilter sides = new DimensionalCellFilter(mesh.spatialDim()-1);
CellFilter left = sides.subset(new CoordinateValueCellPredicate(0, 0.0));
CellFilter right = sides.subset(new CoordinateValueCellPredicate(0, 1.0));
BasisFamily basis = new Lagrange(1);
Expr w = new UnknownFunction(basis, "w");
Expr v = new TestFunction(basis, "v");
Expr grad = gradient(1);
Expr x = new CoordExpr(0);
const double pi = 4.0*atan(1.0);
Expr uExact = sin(pi*x);
Expr R = pi*pi*uExact - lambda*exp(uExact);
QuadratureFamily quad4 = new GaussianQuadrature(4);
QuadratureFamily quad2 = new GaussianQuadrature(2);
DiscreteSpace discSpace(mesh, basis, vecType);
Expr uPrev = new DiscreteFunction(discSpace, 0.5);
Expr stepVal = copyDiscreteFunction(uPrev);
Expr eqn
= Integral(interior, (grad*v)*(grad*w) + (grad*v)*(grad*uPrev)
- v*lambda*exp(uPrev)*(1.0+w) - v*R, quad4);
Expr h = new CellDiameterExpr();
Expr bc = EssentialBC(left+right, v*(uPrev+w)/h, quad2);
LinearProblem prob(mesh, eqn, bc, v, w, vecType);
LinearSolver<double> linSolver
= LinearSolverBuilder::createSolver("amesos.xml");
Out::root() << "Newton iteration" << endl;
int maxIters = 20;
Expr soln ;
bool converged = false;
for (int i=0; i<maxIters; i++)
{
/* solve for the next u */
prob.solve(linSolver, stepVal);
Vector<double> stepVec = getDiscreteFunctionVector(stepVal);
double deltaU = stepVec.norm2();
Out::root() << "Iter=" << setw(3) << i << " ||Delta u||=" << setw(20)
<< deltaU << endl;
addVecToDiscreteFunction(uPrev, stepVec);
if (deltaU < convTol)
{
soln = uPrev;
converged = true;
break;
}
}
TEUCHOS_TEST_FOR_EXCEPTION(!converged, std::runtime_error,
"Newton iteration did not converge after "
<< maxIters << " iterations");
FieldWriter writer = new DSVWriter("HandCodedBratu.dat");
writer.addMesh(mesh);
writer.addField("soln", new ExprFieldWrapper(soln[0]));
writer.write();
Out::root() << "Converged!" << endl << endl;
double L2Err = L2Norm(mesh, interior, soln-uExact, quad4);
Out::root() << "L2 Norm of error: " << L2Err << endl;
Sundance::passFailTest(L2Err, 1.5/((double) nx*nx));
}
catch(std::exception& e)
{
Sundance::handleException(e);
}
Sundance::finalize();
}
int main(int argc, char** argv)
{
try
{
Sundance::init(&argc, &argv);
int np = MPIComm::world().getNProc();
int nx = 64;
const double pi = 4.0*atan(1.0);
MeshType meshType = new BasicSimplicialMeshType();
MeshSource mesher = new PartitionedLineMesher(0.0, pi, nx, meshType);
Mesh mesh = mesher.getMesh();
CellFilter interior = new MaximalCellFilter();
CellFilter bdry = new BoundaryCellFilter();
/* Create a vector space factory, used to
* specify the low-level linear algebra representation */
VectorType<double> vecType = new EpetraVectorType();
/* create symbolic coordinate functions */
Expr x = new CoordExpr(0);
/* create target function */
double R = 0.01;
Expr sx = sin(x);
Expr cx = cos(x);
Expr ssx = sin(sx);
Expr sx2 = sx*sx;
Expr cx2 = cx*cx;
Expr f = sx2 - sx - ssx;
Expr uStar = 2.0*R*(sx2-cx2) + R*sx2*ssx + sx;
/* Form exact solution */
Expr uEx = sx;
Expr lambdaEx = R*sx2;
Expr alphaEx = -lambdaEx/R;
/* create a discrete space on the mesh */
BasisFamily bas = new Lagrange(1);
DiscreteSpace discreteSpace(mesh, bas, vecType);
/* initialize the design, state, and multiplier vectors to constants */
Expr alpha0 = new DiscreteFunction(discreteSpace, 0.25, "alpha0");
Expr u0 = new DiscreteFunction(discreteSpace, 0.5, "u0");
Expr lambda0 = new DiscreteFunction(discreteSpace, 0.25, "lambda0");
/* create symbolic objects for test and unknown functions */
Expr u = new UnknownFunction(bas, "u");
Expr lambda = new UnknownFunction(bas, "lambda");
Expr alpha = new UnknownFunction(bas, "alpha");
/* create symbolic differential operators */
Expr dx = new Derivative(0);
Expr grad = dx;
/* create quadrature rules of different orders */
QuadratureFamily q1 = new GaussianQuadrature(1);
QuadratureFamily q2 = new GaussianQuadrature(2);
QuadratureFamily q4 = new GaussianQuadrature(4);
/* Form objective function */
Expr reg = Integral(interior, 0.5 * R * alpha*alpha, q2);
Expr fit = Integral(interior, 0.5 * pow(u-uStar, 2.0), q4);
Expr constraintEqn = Integral(interior,
(grad*lambda)*(grad*u) + lambda*(alpha + sin(u) + f), q4);
Expr L = reg + fit + constraintEqn;
Expr constraintBC = EssentialBC(bdry, lambda*u, q2);
Functional Lagrangian(mesh, L, constraintBC, vecType);
LinearSolver<double> adjSolver
= LinearSolverBuilder::createSolver("amesos.xml");
ParameterXMLFileReader reader("nox-amesos.xml");
ParameterList noxParams = reader.getParameters();
NOXSolver nonlinSolver(noxParams);
RCP<PDEConstrainedObjBase> obj = rcp(new NonlinearPDEConstrainedObj(
Lagrangian, u, u0, lambda, lambda0, alpha, alpha0,
nonlinSolver, adjSolver));
Vector<double> xInit = obj->getInit();
bool doFDCheck = true;
if (doFDCheck)
{
Out::root() << "Doing FD check of gradient..." << endl;
bool fdOK = obj->fdCheck(xInit, 1.0e-6, 0);
if (fdOK)
{
Out::root() << "FD check OK" << endl;
}
else
{
Out::root() << "FD check FAILED" << endl;
TEUCHOS_TEST_FOR_EXCEPT(!fdOK);
}
}
RCP<UnconstrainedOptimizerBase> opt
= OptBuilder::createOptimizer("basicLMBFGS.xml");
opt->setVerb(3);
OptState state = opt->run(obj, xInit);
if (state.status() != Opt_Converged)
{
Out::root()<< "optimization failed: " << state.status() << endl;
TEUCHOS_TEST_FOR_EXCEPT(state.status() != Opt_Converged);
}
else
{
Out::root() << "opt converged: " << state.iter() << " iterations"
<< endl;
}
FieldWriter w = new MatlabWriter("NonlinControl1D");
w.addMesh(mesh);
w.addField("u", new ExprFieldWrapper(u0));
w.addField("alpha", new ExprFieldWrapper(alpha0));
w.addField("lambda", new ExprFieldWrapper(lambda0));
w.write();
double uErr = L2Norm(mesh, interior, u0-uEx, q4);
double lamErr = L2Norm(mesh, interior, lambda0-lambdaEx, q4);
double aErr = L2Norm(mesh, interior, alpha0-alphaEx, q4);
Out::root() << "error in u = " << uErr << endl;
Out::root() << "error in lambda = " << lamErr << endl;
Out::root() << "error in alpha = " << aErr << endl;
double tol = 0.05;
Sundance::passFailTest(uErr + lamErr + aErr, tol);
}
catch(exception& e)
{
cerr << "main() caught exception: " << e.what() << endl;
}
Sundance::finalize();
return Sundance::testStatus();
}
int main(int argc, char** argv)
{
try
{
const double pi = 4.0*atan(1.0);
double lambda = 1.25*pi*pi;
int nx = 32;
int nt = 10;
double tFinal = 1.0/lambda;
Sundance::setOption("nx", nx, "Number of elements");
Sundance::setOption("nt", nt, "Number of timesteps");
Sundance::setOption("tFinal", tFinal, "Final time");
Sundance::init(&argc, &argv);
/* Creation of vector type */
VectorType<double> vecType = new EpetraVectorType();
/* Set up mesh */
MeshType meshType = new BasicSimplicialMeshType();
MeshSource meshSrc = new PartitionedRectangleMesher(
0.0, 1.0, nx,
0.0, 1.0, nx,
meshType);
Mesh mesh = meshSrc.getMesh();
/*
* Specification of cell filters
*/
CellFilter interior = new MaximalCellFilter();
CellFilter edges = new DimensionalCellFilter(1);
CellFilter west = edges.coordSubset(0, 0.0);
CellFilter east = edges.coordSubset(0, 1.0);
CellFilter south = edges.coordSubset(1, 0.0);
CellFilter north = edges.coordSubset(1, 1.0);
/* set up test and unknown functions */
BasisFamily basis = new Lagrange(1);
Expr u = new UnknownFunction(basis, "u");
Expr v = new TestFunction(basis, "v");
/* set up differential operators */
Expr grad = gradient(2);
Expr x = new CoordExpr(0);
Expr y = new CoordExpr(1);
Expr t = new Sundance::Parameter(0.0);
Expr tPrev = new Sundance::Parameter(0.0);
DiscreteSpace discSpace(mesh, basis, vecType);
Expr uExact = cos(0.5*pi*y)*sin(pi*x)*exp(-lambda*t);
L2Projector proj(discSpace, uExact);
Expr uPrev = proj.project();
/*
* We need a quadrature rule for doing the integrations
*/
QuadratureFamily quad = new GaussianQuadrature(2);
double deltaT = tFinal/nt;
Expr gWest = -pi*exp(-lambda*t)*cos(0.5*pi*y);
Expr gWestPrev = -pi*exp(-lambda*tPrev)*cos(0.5*pi*y);
/* Create the weak form */
Expr eqn = Integral(interior, v*(u-uPrev)/deltaT
+ 0.5*(grad*v)*(grad*u + grad*uPrev), quad)
+ Integral(west, -0.5*v*(gWest+gWestPrev), quad);
Expr bc = EssentialBC(east + north, v*u, quad);
LinearProblem prob(mesh, eqn, bc, v, u, vecType);
LinearSolver<double> solver
= LinearSolverBuilder::createSolver("amesos.xml");
FieldWriter w0 = new VTKWriter("TransientHeat2D-0");
w0.addMesh(mesh);
w0.addField("T", new ExprFieldWrapper(uPrev[0]));
w0.write();
for (int i=0; i<nt; i++)
{
t.setParameterValue((i+1)*deltaT);
tPrev.setParameterValue(i*deltaT);
Out::root() << "t=" << (i+1)*deltaT << endl;
Expr uNext = prob.solve(solver);
ostringstream oss;
oss << "TransientHeat2D-" << i+1;
FieldWriter w = new VTKWriter(oss.str());
w.addMesh(mesh);
w.addField("T", new ExprFieldWrapper(uNext[0]));
w.write();
updateDiscreteFunction(uNext, uPrev);
}
double err = L2Norm(mesh, interior, uExact-uPrev, quad);
Out::root() << "error norm=" << err << endl;
double h = 1.0/(nx-1.0);
double tol = 0.1*(pow(h,2.0) + pow(lambda*deltaT, 2.0));
Out::root() << "tol=" << tol << endl;
Sundance::passFailTest(err, tol);
}
catch(std::exception& e)
{
Sundance::handleException(e);
}
Sundance::finalize();
return Sundance::testStatus();
}
int main(int argc, char** argv)
{
try
{
int depth = 0;
bool useCCode = false;
Sundance::ElementIntegral::alwaysUseCofacets() = true;
Sundance::clp().setOption("depth", &depth, "expression depth");
Sundance::clp().setOption("C", "symb", &useCCode, "Code type (C or symbolic)");
Sundance::init(&argc, &argv);
/* We will do our linear algebra using Epetra */
VectorType<double> vecType = new EpetraVectorType();
/* Read the mesh */
MeshType meshType = new BasicSimplicialMeshType();
MeshSource mesher
= new ExodusMeshReader("cube-0.1", meshType);
Mesh mesh = mesher.getMesh();
/* Create a cell filter that will identify the maximal cells
* in the interior of the domain */
CellFilter interior = new MaximalCellFilter();
CellFilter faces = new DimensionalCellFilter(2);
CellFilter side1 = faces.labeledSubset(1);
CellFilter side2 = faces.labeledSubset(2);
CellFilter side3 = faces.labeledSubset(3);
CellFilter side4 = faces.labeledSubset(4);
CellFilter side5 = faces.labeledSubset(5);
CellFilter side6 = faces.labeledSubset(6);
/* Create unknown and test functions, discretized using second-order
* Lagrange interpolants */
Expr u = new UnknownFunction(new Lagrange(1), "u");
Expr v = new TestFunction(new Lagrange(1), "v");
/* Create differential operator and coordinate functions */
Expr dx = new Derivative(0);
Expr dy = new Derivative(1);
Expr dz = new Derivative(2);
Expr grad = List(dx, dy, dz);
Expr x = new CoordExpr(0);
Expr y = new CoordExpr(1);
Expr z = new CoordExpr(2);
/* We need a quadrature rule for doing the integrations */
QuadratureFamily quad2 = new GaussianQuadrature(2);
QuadratureFamily quad4 = new GaussianQuadrature(4);
/* Define the weak form */
//Expr eqn = Integral(interior, (grad*v)*(grad*u) + v, quad);
Expr coeff = 1.0;
#ifdef FOR_TIMING
if (useCCode)
{
coeff = Poly(depth, x);
}
else
{
for (int i=0; i<depth; i++)
{
Expr t = 1.0;
for (int j=0; j<depth; j++) t = t*x;
coeff = coeff + 2.0*t - t - t;
}
}
#endif
Expr eqn = Integral(interior, coeff*(grad*v)*(grad*u) /*+ 2.0*v*/, quad2);
/* Define the Dirichlet BC */
Expr exactSoln = x;//(x + 1.0)*x - 1.0/4.0;
Expr h = new CellDiameterExpr();
WatchFlag watchBC("watch BCs");
watchBC.setParam("integration setup", 6);
watchBC.setParam("integration", 6);
watchBC.setParam("fill", 6);
watchBC.setParam("evaluation", 6);
watchBC.deactivate();
Expr bc = EssentialBC(side4, v*(u-exactSoln), quad4)
+ EssentialBC(side6, v*(u-exactSoln), quad4, watchBC);
/* We can now set up the linear problem! */
LinearProblem prob(mesh, eqn, bc, v, u, vecType);
#ifdef HAVE_CONFIG_H
ParameterXMLFileReader reader(searchForFile("SolverParameters/aztec-ml.xml"));
#else
ParameterXMLFileReader reader("aztec-ml.xml");
#endif
ParameterList solverParams = reader.getParameters();
std::cerr << "params = " << solverParams << std::endl;
LinearSolver<double> solver
= LinearSolverBuilder::createSolver(solverParams);
Expr soln = prob.solve(solver);
#ifndef FOR_TIMING
DiscreteSpace discSpace(mesh, new Lagrange(1), vecType);
L2Projector proj1(discSpace, exactSoln);
L2Projector proj2(discSpace, soln-exactSoln);
L2Projector proj3(discSpace, pow(soln-exactSoln, 2.0));
Expr exactDisc = proj1.project();
Expr errorDisc = proj2.project();
// Expr errorSqDisc = proj3.project();
std::cerr << "writing fields" << std::endl;
/* Write the field in VTK format */
FieldWriter w = new VTKWriter("Poisson3d");
w.addMesh(mesh);
w.addField("soln", new ExprFieldWrapper(soln[0]));
w.addField("exact soln", new ExprFieldWrapper(exactDisc));
w.addField("error", new ExprFieldWrapper(errorDisc));
// w.addField("errorSq", new ExprFieldWrapper(errorSqDisc));
w.write();
std::cerr << "computing error" << std::endl;
Expr errExpr = Integral(interior,
pow(soln-exactSoln, 2.0),
new GaussianQuadrature(4));
double errorSq = evaluateIntegral(mesh, errExpr);
std::cerr << "error norm = " << sqrt(errorSq) << std::endl << std::endl;
#else
double errorSq = 1.0;
#endif
double tol = 1.0e-10;
Sundance::passFailTest(sqrt(errorSq), tol);
}
catch(std::exception& e)
{
Sundance::handleException(e);
}
Sundance::finalize(); return Sundance::testStatus();
}
int main(int argc, char** argv)
{
try
{
/*
* Initialization code
*/
std::string meshFile="plateWithHole2D-1";
std::string solverFile = "nox-aztec.xml";
Sundance::setOption("meshFile", meshFile, "mesh file");
Sundance::setOption("solver", solverFile,
"name of XML file for solver");
Sundance::init(&argc, &argv);
// This next line is just a hack to deal with some
// transitional code in the
// element integration logic.
Sundance::ElementIntegral::alwaysUseCofacets() = false;
/*
* Creation of vector type
*/
VectorType<double> vecType = new EpetraVectorType();
/*
* Creation of mesh
*/
MeshType meshType = new BasicSimplicialMeshType();
MeshSource meshSrc
= new ExodusMeshReader(meshFile, meshType);
Mesh mesh = meshSrc.getMesh();
/*
* Specification of cell filters
*/
CellFilter interior = new MaximalCellFilter();
CellFilter edges = new DimensionalCellFilter(1);
CellFilter south = edges.labeledSubset(1);
CellFilter east = edges.labeledSubset(2);
CellFilter north = edges.labeledSubset(3);
CellFilter west = edges.labeledSubset(4);
/*
* <Header level="subsubsection" name="symb_setup">
* Setup of symbolic problem description
* </Header>
*
* Create unknown and test functions discretized on the space
* first-order Lagrange polynomials.
*/
BasisFamily basis = new Lagrange(2);
Expr u = new UnknownFunction(basis, "u");
Expr v = new TestFunction(basis, "v");
/*
* Create differential operators and coordinate functions. Directions
* are indexed starting from zero. The \verb+List()+ function can
* collect expressions into a vector.
*/
Expr dx = new Derivative(0);
Expr dy = new Derivative(1);
Expr grad = List(dx, dy);
Expr x = new CoordExpr(0);
Expr y = new CoordExpr(1);
/*
* We need a quadrature rule for doing the integrations
*/
QuadratureFamily quad2 = new GaussianQuadrature(2);
QuadratureFamily quad4 = new GaussianQuadrature(4);
/*
* Create the weak form and the BCs
*/
Expr source=exp(u);
Expr eqn
= Integral(interior, (grad*u)*(grad*v)+v*source, quad4);
Expr h = new CellDiameterExpr();
Expr bc = EssentialBC(west+east, v*(u-1.0)/h, quad2);
/*
* <Header level="subsubsection" name="lin_prob">
* Creation of initial guess
* </Header>
*
* So far the setup has been almost identical to that for the linear
* problem, the only difference being the nonlinear term in the
* equation set.
*/
DiscreteSpace discSpace(mesh, basis, vecType);
L2Projector proj(discSpace, 1.0);
Expr u0 = proj.project();
/*
* <Header level="subsubsection" name="lin_prob">
* Creation of nonlinear problem
* </Header>
*
* Similar to the setup of a \verb+LinearProblem+, the equation, BCs,
* and mesh are put into a \verb+NonlinearProblem+ object which
* controls the construction of the \verb+Assembler+ and its use
* in building Jacobians and residuals during a nonlinear solve.
*/
NonlinearProblem prob(mesh, eqn, bc, v, u, u0, vecType);
/*
*
*/
ParameterXMLFileReader reader(solverFile);
ParameterList solverParams = reader.getParameters();
NOXSolver solver(solverParams);
prob.solve(solver);
/*
* Visualization output
*/
FieldWriter w = new VTKWriter("PoissonBoltzmannDemo2D");
w.addMesh(mesh);
w.addField("soln", new ExprFieldWrapper(u0));
w.write();
/*
* <Header level="subsubsection" name="postproc">
* Postprocessing
* </Header>
*
* Postprocessing can be done using the same symbolic language
* as was used for the problem specification. Here, we define
* an integral giving the flux, then evaluate it on the mesh.
*/
Expr n = CellNormalExpr(2, "n");
Expr fluxExpr
= Integral(east + west, (n*grad)*u0, quad2);
double flux = evaluateIntegral(mesh, fluxExpr);
Out::os() << "numerical flux = " << flux << std::endl;
Expr sourceExpr
= Integral(interior, exp(u0), quad4);
double src = evaluateIntegral(mesh, sourceExpr);
Out::os() << "numerical integrated source = " << src << std::endl;
/*
* Check that the flux is acceptably close to zero. This
* is just a sanity check to ensure the code doesn't get completely
* broken after a change to the library.
*/
Sundance::passFailTest(fabs(flux-src), 1.0e-3);
/*
* <Header level="subsubsection" name="finalize">
* Finalization boilerplate
* </Header>
* Finally, we have boilerplate code for exception handling
* and finalization.
*/
}
catch(std::exception& e)
{
Sundance::handleException(e);
}
Sundance::finalize(); return Sundance::testStatus();
return Sundance::testStatus();
}
void LinearPDEConstrainedObj
::solveStateAndAdjoint(const Vector<double>& x) const
{
Tabs tab(0);
PLAYA_MSG2(verb(), tab << "solving state and adjoint");
PLAYA_MSG3(verb(), tab << "|x|=" << x.norm2());
PLAYA_MSG5(verb(), tab << "x=" << endl << tab << x.norm2());
Tabs tab1;
setDiscreteFunctionVector(designVarVal(), x);
PLAYA_MSG3(verb(), tab1 << "solving state eqns");
/* solve the state equations in order */
for (int i=0; i<stateProbs_.size(); i++)
{
SolverState<double> status
= stateProbs_[i].solve(solvers_[i], stateVarVals(i));
/* if the solve failed, write out the design var and known state
* variables */
if (status.finalState() != SolveConverged)
{
FieldWriter w = new VTKWriter("badSolve");
w.addMesh(Lagrangian().mesh());
w.addField("designVar", new ExprFieldWrapper(designVarVal()));
for (int j=0; j<i; j++)
{
Expr tmp = stateVarVals(j).flatten();
for (int k=0; k<tmp.size(); k++)
{
w.addField("stateVar-"+Teuchos::toString(j)+"-"+Teuchos::toString(k),
new ExprFieldWrapper(tmp[k]));
}
}
w.write();
}
TEUCHOS_TEST_FOR_EXCEPTION(status.finalState() != SolveConverged,
std::runtime_error,
"state equation " << i
<< " could not be solved: status="
<< status.stateDescription());
}
PLAYA_MSG3(verb(), tab1 << "done solving state eqns");
/* do postprocessing */
statePostprocCallback();
PLAYA_MSG3(verb(), tab1 << "solving adjoint eqns");
/* solve the adjoint equations in reverse order */
for (int i=adjointProbs_.size()-1; i>=0; i--)
{
SolverState<double> status
= adjointProbs_[i].solve(solvers_[i], adjointVarVals(i));
/* if the solve failed, write out the design var and known state
* and adjoint variables */
if (status.finalState() != SolveConverged)
{
FieldWriter w = new VTKWriter("badSolve");
w.addMesh(Lagrangian().mesh());
w.addField("designVar", new ExprFieldWrapper(designVarVal()));
for (int j=0; j<stateProbs_.size(); j++)
{
Expr tmp = stateVarVals(j).flatten();
for (int k=0; k<tmp.size(); k++)
{
w.addField("stateVar-"+Teuchos::toString(j)+"-"+Teuchos::toString(k),
new ExprFieldWrapper(tmp[k]));
}
}
for (int j=adjointProbs_.size()-1; j>i; j--)
{
Expr tmp = adjointVarVals(j).flatten();
for (int k=0; k<tmp.size(); k++)
{
w.addField("adjointVar-"+Teuchos::toString(j)+"-"+Teuchos::toString(k),
new ExprFieldWrapper(tmp[k]));
}
}
w.write();
}
TEUCHOS_TEST_FOR_EXCEPTION(status.finalState() != SolveConverged,
std::runtime_error,
"adjoint equation " << i
<< " could not be solved: status="
<< status.stateDescription());
}
PLAYA_MSG3(verb(), tab1 << "done solving adjoint eqns");
PLAYA_MSG2(verb(), tab1 << "done solving state and adjoint eqns");
}
bool BlockStochPoissonTest1D()
{
/* We will do our linear algebra using Epetra */
VectorType<double> vecType = new EpetraVectorType();
/* Read a mesh */
MeshType meshType = new BasicSimplicialMeshType();
int nx = 32;
MeshSource mesher = new PartitionedLineMesher(0.0, 1.0, nx,
meshType);
Mesh mesh = mesher.getMesh();
/* Create a cell filter that will identify the maximal cells
* in the interior of the domain */
CellFilter interior = new MaximalCellFilter();
CellFilter pts = new DimensionalCellFilter(0);
CellFilter left = pts.subset(new CoordinateValueCellPredicate(0,0.0));
CellFilter right = pts.subset(new CoordinateValueCellPredicate(0,1.0));
Expr x = new CoordExpr(0);
/* Create the stochastic coefficients */
int nDim = 1;
int order = 6;
#ifdef HAVE_SUNDANCE_STOKHOS
Out::root() << "using Stokhos hermite basis" << std::endl;
SpectralBasis pcBasis = new Stokhos::HermiteBasis<int,double>(order);
#else
Out::root() << "using George's hermite basis" << std::endl;
SpectralBasis pcBasis = new HermiteSpectralBasis(nDim, order);
#endif
Array<Expr> q(pcBasis.nterms());
Array<Expr> kappa(pcBasis.nterms());
Array<Expr> uEx(pcBasis.nterms());
double a = 0.1;
q[0] = -2 + pow(a,2)*(4 - 9*x)*x - 2*pow(a,3)*(-1 + x)*(1 + 3*x*(-3 + 4*x));
q[1] = -(a*(-3 + 10*x + 2*a*(-1 + x*(8 - 9*x +
a*(-4 + 3*(5 - 4*x)*x + 12*a*(-1 + x)*(1 + 5*(-1 + x)*x))))));
q[2] = a*(-4 + 6*x + a*(1 - x*(2 + 3*x) + a*(4 - 28*x + 30*pow(x,2))));
q[3] = -(pow(a,2)*(-3 + x*(20 - 21*x +
a*(-4 + 3*(5 - 4*x)*x + 24*a*(-1 + x)*(1 + 5*(-1 + x)*x)))));
q[4] = pow(a,3)*(1 + x*(-6 + x*(3 + 4*x)));
q[5] = -4*pow(a,4)*(-1 + x)*x*(1 + 5*(-1 + x)*x);
q[6] = 0.0;
uEx[0] = -((-1 + x)*x);
uEx[1] = -(a*(-1 + x)*pow(x,2));
uEx[2] = a*pow(-1 + x,2)*x;
uEx[3] = pow(a,2)*pow(-1 + x,2)*pow(x,2);
uEx[4] = 0.0;
uEx[5] = 0.0;
uEx[6] = 0.0;
kappa[0] = 1.0;
kappa[1] = a*x;
kappa[2] = -(pow(a,2)*(-1 + x)*x);
kappa[3] = 1.0; // unused
kappa[4] = 1.0; // unused
kappa[5] = 1.0; // unused
kappa[6] = 1.0; // unused
Array<Expr> uBC(pcBasis.nterms());
for (int i=0; i<pcBasis.nterms(); i++) uBC[i] = 0.0;
int L = nDim+2;
int P = pcBasis.nterms();
Out::os() << "L = " << L << std::endl;
Out::os() << "P = " << P << std::endl;
/* Create the unknown and test functions. Do NOT use the spectral
* basis here */
Expr u = new UnknownFunction(new Lagrange(4), "u");
Expr v = new TestFunction(new Lagrange(4), "v");
/* Create differential operator and coordinate function */
Expr dx = new Derivative(0);
Expr grad = dx;
/* We need a quadrature rule for doing the integrations */
QuadratureFamily quad = new GaussianQuadrature(12);
/* Now we create problem objects to build each $K_j$ and $f_j$.
* There will be L matrix-vector pairs */
Array<Expr> eqn(P);
Array<Expr> bc(P);
Array<LinearProblem> prob(P);
Array<LinearOperator<double> > KBlock(L);
Array<Vector<double> > fBlock(P);
Array<Vector<double> > solnBlock;
for (int j=0; j<P; j++)
{
eqn[j] = Integral(interior, kappa[j]*(grad*v)*(grad*u) + v*q[j], quad);
bc[j] = EssentialBC(left+right, v*(u-uBC[j]), quad);
prob[j] = LinearProblem(mesh, eqn[j], bc[j], v, u, vecType);
if (j<L) KBlock[j] = prob[j].getOperator();
fBlock[j] = -1.0*prob[j].getSingleRHS();
}
/* Read the solver to be used on the diagonal blocks */
LinearSolver<double> diagSolver
= LinearSolverBuilder::createSolver("amesos.xml");
double convTol = 1.0e-12;
int maxIters = 30;
int verb = 1;
StochBlockJacobiSolver solver(diagSolver, pcBasis,
convTol, maxIters, verb);
solver.solve(KBlock, fBlock, solnBlock);
/* write the solution */
FieldWriter w = new MatlabWriter("Stoch1D");
w.addMesh(mesh);
DiscreteSpace discSpace(mesh, new Lagrange(4), vecType);
for (int i=0; i<P; i++)
{
L2Projector proj(discSpace, uEx[i]);
Expr ue_i = proj.project();
Expr df = new DiscreteFunction(discSpace, solnBlock[i]);
w.addField("u["+ Teuchos::toString(i)+"]",
new ExprFieldWrapper(df));
w.addField("uEx["+ Teuchos::toString(i)+"]",
new ExprFieldWrapper(ue_i));
}
w.write();
double totalErr2 = 0.0;
DiscreteSpace discSpace4(mesh, new Lagrange(4), vecType);
for (int i=0; i<P; i++)
{
Expr df = new DiscreteFunction(discSpace4, solnBlock[i]);
Expr errExpr = Integral(interior, pow(uEx[i]-df, 2.0), quad);
Expr scaleExpr = Integral(interior, pow(uEx[i], 2.0), quad);
double errSq = evaluateIntegral(mesh, errExpr);
double scale = evaluateIntegral(mesh, scaleExpr);
if (scale > 0.0)
Out::os() << "mode i=" << i << " error=" << sqrt(errSq/scale) << std::endl;
else
Out::os() << "mode i=" << i << " error=" << sqrt(errSq) << std::endl;
}
double tol = 1.0e-12;
return SundanceGlobal::checkTest(sqrt(totalErr2), tol);
}
bool DuffingFloquet()
{
int np = MPIComm::world().getNProc();
TEUCHOS_TEST_FOR_EXCEPT(np != 1);
const double pi = 4.0*atan(1.0);
/* We will do our linear algebra using Epetra */
VectorType<double> vecType = new EpetraVectorType();
/* Create a periodic mesh */
int nx = 128;
MeshType meshType = new PeriodicMeshType1D();
MeshSource mesher = new PeriodicLineMesher(0.0, 2.0*pi, nx, meshType);
Mesh mesh = mesher.getMesh();
/* Create a cell filter that will identify the maximal cells
* in the interior of the domain */
CellFilter interior = new MaximalCellFilter();
CellFilter pts = new DimensionalCellFilter(0);
CellFilter left = pts.subset(new CoordinateValueCellPredicate(0,0.0));
CellFilter right = pts.subset(new CoordinateValueCellPredicate(0,2.0*pi));
/* Create unknown and test functions, discretized using first-order
* Lagrange interpolants */
Expr u1 = new UnknownFunction(new Lagrange(1), "u1");
Expr u2 = new UnknownFunction(new Lagrange(1), "u2");
Expr v1 = new TestFunction(new Lagrange(1), "v1");
Expr v2 = new TestFunction(new Lagrange(1), "v2");
/* Create differential operator and coordinate function */
Expr dx = new Derivative(0);
Expr x = new CoordExpr(0);
/* We need a quadrature rule for doing the integrations */
QuadratureFamily quad = new GaussianQuadrature(4);
double F0 = 0.5;
double gamma = 2.0/3.0;
double a0 = 1.0;
double w0 = 1.0;
double eps = 0.5;
Expr u1Guess = -0.75*cos(x) + 0.237*sin(x);
Expr u2Guess = 0.237*cos(x) + 0.75*sin(x);
DiscreteSpace discSpace(mesh,
List(new Lagrange(1), new Lagrange(1)),
vecType);
L2Projector proj(discSpace, List(u1Guess, u2Guess));
Expr u0 = proj.project();
Expr rhs1 = u2;
Expr rhs2 = -w0*w0*u1 - gamma*u2 - eps*w0*w0*pow(u1,3.0)/a0/a0
+ F0*w0*w0*sin(x);
/* Define the weak form */
Expr eqn = Integral(interior,
v1*(dx*u1 - rhs1) + v2*(dx*u2 - rhs2),
quad);
Expr dummyBC ;
NonlinearProblem prob(mesh, eqn, dummyBC, List(v1,v2), List(u1,u2),
u0, vecType);
ParameterXMLFileReader reader("nox.xml");
ParameterList solverParams = reader.getParameters();
Out::root() << "finding periodic solution" << endl;
NOXSolver solver(solverParams);
prob.solve(solver);
/* unfold the solution onto a non-periodic mesh */
Expr uP = unfoldPeriodicDiscreteFunction(u0, "u_p");
Out::root() << "uP=" << uP << endl;
Mesh unfoldedMesh = DiscreteFunction::discFunc(uP)->mesh();
DiscreteSpace unfDiscSpace = DiscreteFunction::discFunc(uP)->discreteSpace();
FieldWriter writer = new MatlabWriter("Floquet.dat");
writer.addMesh(unfoldedMesh);
writer.addField("u_p[0]", new ExprFieldWrapper(uP[0]));
writer.addField("u_p[1]", new ExprFieldWrapper(uP[1]));
Array<Expr> a(2);
a[0] = new Sundance::Parameter(0.0, "a1");
a[1] = new Sundance::Parameter(0.0, "a2");
Expr bc = EssentialBC(left, v1*(u1-uP[0]-a[0]) + v2*(u2-uP[1]-a[1]), quad);
NonlinearProblem unfProb(unfoldedMesh, eqn, bc,
List(v1,v2), List(u1,u2), uP, vecType);
unfProb.setEvalPoint(uP);
LinearOperator<double> J = unfProb.allocateJacobian();
Vector<double> b = J.domain().createMember();
LinearSolver<double> linSolver
= LinearSolverBuilder::createSolver("amesos.xml");
SerialDenseMatrix<int, double> F(a.size(), a.size());
for (int i=0; i<a.size(); i++)
{
Out::root() << "doing perturbed orbit #" << i << endl;
for (int j=0; j<a.size(); j++)
{
if (i==j) a[j].setParameterValue(1.0);
else a[j].setParameterValue(0.0);
}
unfProb.computeJacobianAndFunction(J, b);
Vector<double> w = b.copy();
linSolver.solve(J, b, w);
Expr w_i = new DiscreteFunction(unfDiscSpace, w);
for (int j=0; j<a.size(); j++)
{
Out::root() << "postprocessing" << i << endl;
writer.addField("w[" + Teuchos::toString(i)
+ ", " + Teuchos::toString(j) + "]", new ExprFieldWrapper(w_i[j]));
Expr g = Integral(right, w_i[j], quad);
F(j,i) = evaluateIntegral(unfoldedMesh, g);
}
}
writer.write();
Out::root() << "Floquet matrix = " << endl
<< F << endl;
Out::root() << "doing eigenvalue analysis" << endl;
Array<double> ew_r(a.size());
Array<double> ew_i(a.size());
int lWork = 6*a.size();
Array<double> work(lWork);
int info = 0;
LAPACK<int, double> lapack;
lapack.GEEV('N','N', a.size(), F.values(),
a.size(), &(ew_r[0]), &(ew_i[0]), 0, 1, 0, 1, &(work[0]), lWork,
&info);
TEUCHOS_TEST_FOR_EXCEPTION(info != 0,
std::runtime_error,
"LAPACK GEEV returned error code =" << info);
Array<double> ew(a.size());
for (int i=0; i<a.size(); i++)
{
ew[i] = sqrt(ew_r[i]*ew_r[i]+ew_i[i]*ew_i[i]);
Out::root() << setw(5) << i
<< setw(16) << ew_r[i]
<< setw(16) << ew_i[i]
<< setw(16) << ew[i]
<< endl;
}
double err = ::fabs(ew[0] - 0.123);
return SundanceGlobal::checkTest(err, 0.001);
}
int main( int argc , char **argv )
{
try {
int nx = 128;
double C = 4.0;
std::string solverFile = "aztec-ml.xml";
Sundance::setOption("nx", nx, "number of elements in x");
Sundance::setOption("C", C, "Nitsche penalty");
Sundance::setOption("solver", solverFile, "name of XML file for solver");
Sundance::init( &argc , &argv );
int np = MPIComm::world().getNProc();
int npx = -1;
int npy = -1;
balanceXY(np, &npx, &npy);
TEST_FOR_EXCEPT(npx < 1);
TEST_FOR_EXCEPT(npy < 1);
TEST_FOR_EXCEPT(npx * npy != np);
VectorType<double> vecType = new EpetraVectorType();
const int k = 1;
const int splitBC = 1;
MeshType meshType = new BasicSimplicialMeshType();
MeshSource mesher = new PartitionedRectangleMesher( 0.0 , 1.0 , nx , npx ,
0.0 , 1.0 , nx , npy,
meshType );
Mesh mesh = mesher.getMesh();
BasisFamily L = new Lagrange( k );
Expr u = new UnknownFunction( L , "u" );
Expr v = new TestFunction( L , "v" );
QuadratureFamily quad = new GaussianQuadrature( 2 * k );
Expr h = new CellDiameterExpr();
Expr alpha = C / h;
Expr eqn = poissonEquationNitsche( splitBC, u , v , alpha , quad );
Expr bc;
LinearProblem prob( mesh , eqn , bc , v , u , vecType);
#ifdef HAVE_CONFIG_H
ParameterXMLFileReader reader(searchForFile("SolverParameters/" + solverFile));
#else
ParameterXMLFileReader reader(solverFile);
#endif
ParameterList solverParams = reader.getParameters();
LinearSolver<double> solver
= LinearSolverBuilder::createSolver(solverParams);
Expr soln = prob.solve( solver );
FieldWriter w = new VTKWriter( "NitschePoisson2D" );
w.addMesh( mesh );
w.addField( "u" , new ExprFieldWrapper( soln ) );
w.write();
Expr x = new CoordExpr(0);
Expr y = new CoordExpr(1);
QuadratureFamily quad4 = new GaussianQuadrature(4);
CellFilter interior = new MaximalCellFilter();
const double pi = 4.0*atan(1.0);
Expr exactSoln = sin(pi*x)*sin(pi*y);
Expr err = exactSoln - soln;
Expr errExpr = Integral(interior,
err*err,
quad4);
FunctionalEvaluator errInt(mesh, errExpr);
double errorSq = errInt.evaluate();
cout << "error norm = " << sqrt(errorSq) << std::endl << std::endl;
Sundance::passFailTest(sqrt(errorSq), 1.0e-4);
}
catch (std::exception &e)
{
Sundance::handleException(e);
}
Sundance::finalize();
return Sundance::testStatus();
}
int main(int argc, char** argv)
{
try
{
int nx = 32;
double convTol = 1.0e-8;
double lambda = 0.5;
Sundance::setOption("nx", nx, "Number of elements");
Sundance::setOption("tol", convTol, "Convergence tolerance");
Sundance::setOption("lambda", lambda, "Lambda (parameter in Bratu's equation)");
Sundance::init(&argc, &argv);
Out::root() << "Bratu problem (lambda=" << lambda << ")" << endl;
Out::root() << "Fixed-point iteration" << endl << endl;
VectorType<double> vecType = new EpetraVectorType();
MeshType meshType = new BasicSimplicialMeshType();
MeshSource mesher = new PartitionedLineMesher(0.0, 1.0, nx, meshType);
Mesh mesh = mesher.getMesh();
CellFilter interior = new MaximalCellFilter();
CellFilter sides = new DimensionalCellFilter(mesh.spatialDim()-1);
CellFilter left = sides.subset(new CoordinateValueCellPredicate(0, 0.0));
CellFilter right = sides.subset(new CoordinateValueCellPredicate(0, 1.0));
BasisFamily basis = new Lagrange(1);
Expr u = new UnknownFunction(basis, "u");
Expr v = new TestFunction(basis, "v");
Expr grad = gradient(1);
Expr x = new CoordExpr(0);
const double pi = 4.0*atan(1.0);
Expr uExact = sin(pi*x);
Expr R = pi*pi*uExact - lambda*exp(uExact);
QuadratureFamily quad4 = new GaussianQuadrature(4);
QuadratureFamily quad2 = new GaussianQuadrature(2);
DiscreteSpace discSpace(mesh, basis, vecType);
Expr uPrev = new DiscreteFunction(discSpace, 0.5);
Expr uCur = copyDiscreteFunction(uPrev);
Expr eqn
= Integral(interior, (grad*u)*(grad*v) - v*lambda*exp(uPrev) - v*R, quad4);
Expr h = new CellDiameterExpr();
Expr bc = EssentialBC(left+right, v*u/h, quad4);
LinearProblem prob(mesh, eqn, bc, v, u, vecType);
Expr normSqExpr = Integral(interior, pow(u-uPrev, 2.0), quad2);
Functional normSqFunc(mesh, normSqExpr, vecType);
FunctionalEvaluator normSqEval = normSqFunc.evaluator(u, uCur);
LinearSolver<double> linSolver
= LinearSolverBuilder::createSolver("amesos.xml");
Out::root() << "Fixed-point iteration" << endl;
int maxIters = 20;
Expr soln ;
bool converged = false;
for (int i=0; i<maxIters; i++)
{
/* solve for the next u */
prob.solve(linSolver, uCur);
/* evaluate the norm of (uCur-uPrev) using
* the FunctionalEvaluator defined above */
double deltaU = sqrt(normSqEval.evaluate());
Out::root() << "Iter=" << setw(3) << i << " ||Delta u||=" << setw(20)
<< deltaU << endl;
/* check for convergence */
if (deltaU < convTol)
{
soln = uCur;
converged = true;
break;
}
/* get the vector from the current discrete function */
Vector<double> uVec = getDiscreteFunctionVector(uCur);
/* copy the vector into the previous discrete function */
setDiscreteFunctionVector(uPrev, uVec);
}
TEUCHOS_TEST_FOR_EXCEPTION(!converged, std::runtime_error,
"Fixed point iteration did not converge after "
<< maxIters << " iterations");
FieldWriter writer = new DSVWriter("FixedPointBratu.dat");
writer.addMesh(mesh);
writer.addField("soln", new ExprFieldWrapper(soln[0]));
writer.write();
Out::root() << "Converged!" << endl << endl;
double L2Err = L2Norm(mesh, interior, soln-uExact, quad4);
Out::root() << "L2 Norm of error: " << L2Err << endl;
Sundance::passFailTest(L2Err, 1.5/((double) nx*nx));
}
catch(exception& e)
{
Sundance::handleException(e);
}
Sundance::finalize();
}
int main(int argc, char** argv)
{
try
{
int nx = 32;
double convTol = 1.0e-8;
double lambda = 0.5;
Sundance::setOption("nx", nx, "Number of elements");
Sundance::setOption("tol", convTol, "Convergence tolerance");
Sundance::setOption("lambda", lambda, "Lambda (parameter in Bratu's equation)");
Sundance::init(&argc, &argv);
Out::root() << "Bratu problem (lambda=" << lambda << ")" << endl;
Out::root() << "Newton's method with automated linearization"
<< endl << endl;
VectorType<double> vecType = new EpetraVectorType();
MeshType meshType = new BasicSimplicialMeshType();
MeshSource mesher = new PartitionedLineMesher(0.0, 1.0, nx, meshType);
Mesh mesh = mesher.getMesh();
CellFilter interior = new MaximalCellFilter();
CellFilter sides = new DimensionalCellFilter(mesh.spatialDim()-1);
CellFilter left = sides.subset(new CoordinateValueCellPredicate(0, 0.0));
CellFilter right = sides.subset(new CoordinateValueCellPredicate(0, 1.0));
BasisFamily basis = new Lagrange(1);
Expr u = new UnknownFunction(basis, "w");
Expr v = new TestFunction(basis, "v");
Expr grad = gradient(1);
Expr x = new CoordExpr(0);
const double pi = 4.0*atan(1.0);
Expr uExact = sin(pi*x);
Expr R = pi*pi*uExact - lambda*exp(uExact);
QuadratureFamily quad4 = new GaussianQuadrature(4);
QuadratureFamily quad2 = new GaussianQuadrature(2);
DiscreteSpace discSpace(mesh, basis, vecType);
Expr uPrev = new DiscreteFunction(discSpace, 0.5);
Expr eqn
= Integral(interior, (grad*v)*(grad*u) - v*lambda*exp(u) - v*R, quad4);
Expr h = new CellDiameterExpr();
Expr bc = EssentialBC(left+right, v*u/h, quad2);
NonlinearProblem prob(mesh, eqn, bc, v, u, uPrev, vecType);
NonlinearSolver<double> solver
= NonlinearSolverBuilder::createSolver("playa-newton-amesos.xml");
Out::root() << "Newton solve" << endl;
SolverState<double> state = prob.solve(solver);
TEUCHOS_TEST_FOR_EXCEPTION(state.finalState() != SolveConverged,
std::runtime_error,
"Nonlinear solve failed to converge: message=" << state.finalMsg());
Expr soln = uPrev;
FieldWriter writer = new DSVWriter("AutoLinearizedBratu.dat");
writer.addMesh(mesh);
writer.addField("soln", new ExprFieldWrapper(soln[0]));
writer.write();
Out::root() << "Converged!" << endl << endl;
double L2Err = L2Norm(mesh, interior, soln-uExact, quad4);
Out::root() << "L2 Norm of error: " << L2Err << endl;
Sundance::passFailTest(L2Err, 1.5/((double) nx*nx));
}
catch(std::exception& e)
{
Sundance::handleException(e);
}
Sundance::finalize();
return Sundance::testStatus();
}
bool AToCDensitySample()
{
/* We will do our linear algebra using Epetra */
VectorType<double> vecType = new EpetraVectorType();
/* Create a mesh. It will be of type BasisSimplicialMesh, and will
* be built using a PartitionedLineMesher. */
MeshType meshType = new BasicSimplicialMeshType();
MeshSource mesher = new PartitionedRectangleMesher(-1.0, 1.0, 32, 1,
-1.0, 1.0, 32, 1,
meshType);
Mesh mesh = mesher.getMesh();
/* Create a cell filter that will identify the maximal cells
* in the interior of the domain */
CellFilter interior = new MaximalCellFilter();
Expr x = new CoordExpr(0);
Expr y = new CoordExpr(1);
BasisFamily L1 = new Lagrange(1);
DiscreteSpace discSpace(mesh, List(L1, L1), vecType);
/* Discretize some expression for the force. We'll pick a linear function
* so that it can be interpolated exactly, letting us check the
* validity of our interpolations. */
L2Projector proj(discSpace, List(x, y));
Expr F = proj.project();
/* create a sampler */
cout << "making grid" << std::endl;
AToCPointLocator locator(mesh, interior, createVector(tuple(200, 200)));
AToCDensitySampler sampler(locator, vecType);
CToAInterpolator forceInterpolator(locator, F);
cout << "making points" << std::endl;
/* create a bunch of particles */
int nCells = mesh.numCells(2);
int nPts = 15000;
Array<double> pos(2*nPts);
Array<double> f(F.size() * nPts);
Array<Point> physPts;
/* We'll generate random sample points in a way that lets us make an exact check
* of the density recovery. We pick random cells, then random local coordinates
* within each cell. This way, we can compute the density exactly as we
* go, giving us something to check the recovered density against. */
Array<int> counts(nCells);
for (int i=0; i<nPts; i++)
{
/* pick a random cell */
int cell = (int) floor(nCells * drand48());
counts[cell]++;
/* generate a point in local coordinates */
double s = drand48();
double t = drand48() * (1.0-s);
Point refPt(s, t);
/* map to physical coordinates */
mesh.pushForward(2, tuple(cell), tuple(refPt), physPts);
Point X = physPts[0];
pos[2*i] = X[0];
pos[2*i+1] = X[1];
}
cout << "sampling..." << std::endl;
Expr density = sampler.sample(createVector(pos), 1.0);
cout << "computing forces..." << std::endl;
forceInterpolator.interpolate(pos, f);
double maxForceErr = 0.0;
for (int i=0; i<nPts; i++)
{
double x0 = pos[2*i];
double y0 = pos[2*i+1];
double fx = x0;
double fy = y0;
double df = ::fabs(fx - f[2*i]) + ::fabs(fy - f[2*i+1]);
maxForceErr = max(maxForceErr, df);
}
cout << "max force error = " << maxForceErr << std::endl;
cout << "writing..." << std::endl;
/* Write the field in VTK format */
FieldWriter w = new VTKWriter("Density2d");
w.addMesh(mesh);
w.addField("rho", new ExprFieldWrapper(density));
w.write();
double errorSq = 0.0;
double tol = 1.0e-6;
return SundanceGlobal::passFailTest(::sqrt(errorSq), tol);
}
bool PoissonOnDisk()
{
#ifdef HAVE_SUNDANCE_EXODUS
/* We will do our linear algebra using Epetra */
VectorType<double> vecType = new EpetraVectorType();
/* Get a mesh */
MeshType meshType = new BasicSimplicialMeshType();
MeshSource meshReader = new ExodusNetCDFMeshReader("disk.ncdf", meshType);
Mesh mesh = meshReader.getMesh();
/* Create a cell filter that will identify the maximal cells
* in the interior of the domain */
CellFilter interior = new MaximalCellFilter();
CellFilter bdry = new BoundaryCellFilter();
/* Create unknown and test functions, discretized using first-order
* Lagrange interpolants */
Expr u = new UnknownFunction(new Lagrange(1), "u");
Expr v = new TestFunction(new Lagrange(1), "v");
/* Create differential operator and coordinate functions */
Expr dx = new Derivative(0);
Expr dy = new Derivative(1);
Expr grad = List(dx, dy);
Expr x = new CoordExpr(0);
Expr y = new CoordExpr(1);
/* We need a quadrature rule for doing the integrations */
QuadratureFamily quad2 = new GaussianQuadrature(2);
QuadratureFamily quad4 = new GaussianQuadrature(4);
/* Define the weak form */
Expr eqn = Integral(interior, (grad*v)*(grad*u) + v, quad2);
/* Define the Dirichlet BC */
Expr bc = EssentialBC(bdry, v*u, quad4);
/* We can now set up the linear problem! */
LinearProblem prob(mesh, eqn, bc, v, u, vecType);
LinearSolver<double> solver
= LinearSolverBuilder::createSolver("amesos.xml");
Expr soln = prob.solve(solver);
double R = 1.0;
Expr exactSoln = 0.25*(x*x + y*y - R*R);
DiscreteSpace discSpace(mesh, new Lagrange(1), vecType);
Expr du = L2Projector(discSpace, exactSoln-soln).project();
/* Write the field in VTK format */
FieldWriter w = new VTKWriter("PoissonOnDisk");
w.addMesh(mesh);
w.addField("soln", new ExprFieldWrapper(soln[0]));
w.addField("error", new ExprFieldWrapper(du));
w.write();
Expr errExpr = Integral(interior,
pow(soln-exactSoln, 2),
new GaussianQuadrature(4));
double errorSq = evaluateIntegral(mesh, errExpr);
std::cerr << "soln error norm = " << sqrt(errorSq) << std::endl << std::endl;
/* Check error in automatically-computed cell normals */
/* Create a cell normal expression. Note that this is NOT a constructor
* call, hence no "new" before the CellNormalExpr() function. The
* argument "2" is the spatial dimension (mandatory), and
* the "n" is the name of the expression (optional).
*/
Expr n = CellNormalExpr(2, "n");
Expr nExact = List(x, y)/sqrt(x*x + y*y);
Expr nErrExpr = Integral(bdry, pow(n-nExact, 2.0), new GaussianQuadrature(1));
double nErrorSq = evaluateIntegral(mesh, nErrExpr);
Out::root() << "normalVector error norm = "
<< sqrt(nErrorSq) << std::endl << std::endl;
double tol = 1.0e-4;
return SundanceGlobal::checkTest(sqrt(errorSq + nErrorSq), tol);
#else
std::cout << "dummy PoissonOnDisk PASSED. Enable Exodus to run the actual test" << std::endl;
return true;
#endif
}
int main(int argc, char** argv)
{
try
{
Sundance::init(&argc, &argv);
/* We will do our linear algebra using Epetra */
VectorType<double> vecType = new EpetraVectorType();
/* Create a mesh. It will be of type BasisSimplicialMesh, and will
* be built using a PartitionedLineMesher. */
MeshType meshType = new BasicSimplicialMeshType();
int nx = 32;
MeshSource mesher = new PartitionedRectangleMesher(
0.0, 1.0, nx,
0.0, 1.0, nx,
meshType);
Mesh mesh = mesher.getMesh();
/* Create a cell filter that will identify the maximal cells
* in the interior of the domain */
CellFilter interior = new MaximalCellFilter();
/* Make cell filters for the east and west boundaries */
CellFilter edges = new DimensionalCellFilter(1);
CellFilter west = edges.coordSubset(0, 0.0);
CellFilter east = edges.coordSubset(0, 1.0);
/* Create unknown and test functions */
Expr u = new UnknownFunction(new Lagrange(1), "u");
Expr v = new TestFunction(new Lagrange(1), "v");
/* Create differential operator and coordinate function */
Expr x = new CoordExpr(0);
Expr grad = gradient(1);
/* We need a quadrature rule for doing the integrations */
QuadratureFamily quad = new GaussianQuadrature(4);
/* Define the parameter */
Expr xi = new Sundance::Parameter(0.0);
/* Construct a forcing term to provide an exact solution for
* validation purposes. This will involve the parameter. */
Expr uEx = x*(1.0-x)*(1.0+xi*exp(x));
Expr f = -(-20 - exp(x)*xi*(1 + 32*x + 10*x*x +
exp(x)*(-1 + 2*x*(2 + x))*xi))/10.0;
/* Define the weak form, using the parameter expression. This weak form
* can be used for all parameter values. */
Expr eqn = Integral(interior,
(1.0 + 0.1*xi*exp(x))*(grad*v)*(grad*u) - f*v, quad);
/* Define the Dirichlet BC */
Expr h = new CellDiameterExpr();
Expr bc = EssentialBC(east + west, v*u/h, quad);
/* We can now set up the linear problem. This can be reused
* for different parameter values. */
LinearProblem prob(mesh, eqn, bc, v, u, vecType);
/* make a projector for the exact solution. Just like the
* problem, this can be reused for different parameter values. */
DiscreteSpace ds(mesh, new Lagrange(1), vecType);
L2Projector proj(ds, uEx);
/* Get the solver and declare variables for the results */
LinearSolver<double> solver = LinearSolverBuilder::createSolver("aztec-ml.xml");
Expr soln;
SolverState<double> state;
/* Set up the sweep from xi=0 to xi=xiMax in nSteps steps. */
int nSteps = 10;
double xiMax = 2.0;
/* Make an array in which to keep the observed errors */
Array<double> err(nSteps);
/* Do the sweep */
for (int n=0; n<nSteps; n++)
{
/* Update the parameter value */
double xiVal = xiMax*n/(nSteps - 1.0);
xi.setParameterValue(xiVal);
Out::root() << "step n=" << n << " of " << nSteps << " xi=" << xiVal;
/* Solve the problem. The updated parameter value is automatically used. */
state = prob.solve(solver, soln);
TEUCHOS_TEST_FOR_EXCEPTION(state.finalState() != SolveConverged,
std::runtime_error,
"solve failed!");
/* Project the exact solution onto a discrrete space for viz. The updated
* parameter value is automatically used. */
Expr uEx0 = proj.project();
/* Write the approximate and exact solutions for viz */
FieldWriter w = new VTKWriter("ParameterSweep-" + Teuchos::toString(n));
w.addMesh(mesh);
w.addField("u", new ExprFieldWrapper(soln[0]));
w.addField("uEx", new ExprFieldWrapper(uEx0[0]));
w.write();
/* Compute the L2 norm of the error */
err[n] = L2Norm(mesh, interior, soln-uEx, quad);
Out::root() << " L2 error = " << err[n] << endl;
}
/* The errors are O(h^2), so use that to set a tolerance */
double hVal = 1.0/(nx-1.0);
double fudge = 2.0;
double tol = fudge*hVal*hVal;
/* Find the max error over all parameter values */
double maxErr = *std::max_element(err.begin(), err.end());
/* Check the error */
Sundance::passFailTest(maxErr, tol);
}
catch(std::exception& e)
{
Sundance::handleException(e);
}
Sundance::finalize();
return Sundance::testStatus();
}
int main(int argc, char** argv)
{
try
{
Sundance::init(&argc, &argv);
int np = MPIComm::world().getNProc();
int nx = 48;
int ny = 48;
int npx = -1;
int npy = -1;
PartitionedRectangleMesher::balanceXY(np, &npx, &npy);
TEUCHOS_TEST_FOR_EXCEPT(npx < 1);
TEUCHOS_TEST_FOR_EXCEPT(npy < 1);
TEUCHOS_TEST_FOR_EXCEPT(npx * npy != np);
MeshType meshType = new BasicSimplicialMeshType();
MeshSource mesher = new PartitionedRectangleMesher(0.0, 1.0, nx, npx,
0.0, 1.0, ny, npy, meshType);
Mesh mesh = mesher.getMesh();
CellFilter interior = new MaximalCellFilter();
CellFilter bdry = new BoundaryCellFilter();
/* Create a vector space factory, used to
* specify the low-level linear algebra representation */
VectorType<double> vecType = new EpetraVectorType();
/* create a discrete space on the mesh */
DiscreteSpace discreteSpace(mesh, new Lagrange(1), vecType);
/* initialize the design, state, and multiplier vectors */
Expr alpha0 = new DiscreteFunction(discreteSpace, 1.0, "alpha0");
Expr u0 = new DiscreteFunction(discreteSpace, 1.0, "u0");
Expr lambda0 = new DiscreteFunction(discreteSpace, 1.0, "lambda0");
/* create symbolic objects for test and unknown functions */
Expr u = new UnknownFunction(new Lagrange(1), "u");
Expr lambda = new UnknownFunction(new Lagrange(1), "lambda");
Expr alpha = new UnknownFunction(new Lagrange(1), "alpha");
/* create symbolic differential operators */
Expr dx = new Derivative(0);
Expr dy = new Derivative(1);
Expr grad = List(dx, dy);
/* create symbolic coordinate functions */
Expr x = new CoordExpr(0);
Expr y = new CoordExpr(1);
/* create target function */
const double pi = 4.0*atan(1.0);
Expr uStar = sin(pi*x)*sin(pi*y);
/* create quadrature rules of different orders */
QuadratureFamily q1 = new GaussianQuadrature(1);
QuadratureFamily q2 = new GaussianQuadrature(2);
QuadratureFamily q4 = new GaussianQuadrature(4);
/* Regularization weight */
double R = 0.001;
double U0 = 1.0/(1.0 + 4.0*pow(pi,4.0)*R);
double A0 = -2.0*pi*pi*U0;
/* Form objective function */
Expr reg = Integral(interior, 0.5 * R * alpha*alpha, q2);
Expr fit = Integral(interior, 0.5 * pow(u-uStar, 2.0), q4);
Expr constraintEqn = Integral(interior,
(grad*lambda)*(grad*u) + lambda*alpha, q2);
Expr L = reg + fit + constraintEqn;
Expr constraintBC = EssentialBC(bdry, lambda*u, q2);
Functional Lagrangian(mesh, L, constraintBC, vecType);
LinearSolver<double> solver
= LinearSolverBuilder::createSolver("amesos.xml");
RCP<ObjectiveBase> obj = rcp(new LinearPDEConstrainedObj(
Lagrangian, u, u0, lambda, lambda0, alpha, alpha0,
solver));
Vector<double> xInit = obj->getInit();
bool doFDCheck = false;
if (doFDCheck)
{
Out::root() << "Doing FD check of gradient..." << endl;
bool fdOK = obj->fdCheck(xInit, 1.0e-6, 2);
if (fdOK)
{
Out::root() << "FD check OK" << endl;
}
else
{
Out::root() << "FD check FAILED" << endl;
}
}
RCP<UnconstrainedOptimizerBase> opt
= OptBuilder::createOptimizer("basicLMBFGS.xml");
opt->setVerb(2);
OptState state = opt->run(obj, xInit);
bool ok = true;
if (state.status() != Opt_Converged)
{
Out::root()<< "optimization failed: " << state.status() << endl;
TEUCHOS_TEST_FOR_EXCEPT(state.status() != Opt_Converged);
}
Out::root() << "opt converged: " << state.iter() << " iterations"
<< endl;
Out::root() << "exact solution: U0=" << U0 << " A0=" << A0 << endl;
FieldWriter w = new VTKWriter("PoissonSourceInversion");
w.addMesh(mesh);
w.addField("u", new ExprFieldWrapper(u0));
w.addField("alpha", new ExprFieldWrapper(alpha0));
w.addField("lambda", new ExprFieldWrapper(lambda0));
w.write();
double uErr = L2Norm(mesh, interior, u0-U0*uStar, q4);
double aErr = L2Norm(mesh, interior, alpha0-A0*uStar, q4);
Out::root() << "error in u = " << uErr << endl;
Out::root() << "error in alpha = " << aErr << endl;
double tol = 0.01;
Sundance::passFailTest(uErr + aErr, tol);
}
catch(std::exception& e)
{
cerr << "main() caught exception: " << e.what() << endl;
}
Sundance::finalize();
return Sundance::testStatus();
}
bool NonlinReducedIntegration()
{
int np = MPIComm::world().getNProc();
int n = 4;
bool increaseProbSize = true;
if ( (np % 4)==0 ) increaseProbSize = false;
Array<double> h;
Array<double> errQuad;
Array<double> errReduced;
for (int i=0; i<4; i++)
{
n *= 2;
int nx = n;
int ny = n;
VectorType<double> vecType = new EpetraVectorType();
MeshType meshType = new BasicSimplicialMeshType();
int npx = -1;
int npy = -1;
PartitionedRectangleMesher::balanceXY(np, &npx, &npy);
TEUCHOS_TEST_FOR_EXCEPT(npx < 1);
TEUCHOS_TEST_FOR_EXCEPT(npy < 1);
TEUCHOS_TEST_FOR_EXCEPT(npx * npy != np);
if (increaseProbSize)
{
nx = nx*npx;
ny = ny*npy;
}
MeshSource mesher = new PartitionedRectangleMesher(0.0, 1.0, nx, npx,
0.0, 1.0, ny, npy, meshType);
Mesh mesh = mesher.getMesh();
WatchFlag watchMe("watch eqn");
watchMe.setParam("integration setup", 0);
watchMe.setParam("integration", 0);
watchMe.setParam("fill", 0);
watchMe.setParam("evaluation", 0);
watchMe.deactivate();
WatchFlag watchBC("watch BCs");
watchBC.setParam("integration setup", 0);
watchBC.setParam("integration", 0);
watchBC.setParam("fill", 0);
watchBC.setParam("evaluation", 0);
watchBC.deactivate();
CellFilter interior = new MaximalCellFilter();
CellFilter edges = new DimensionalCellFilter(1);
CellFilter left = edges.subset(new CoordinateValueCellPredicate(0,0.0));
CellFilter right = edges.subset(new CoordinateValueCellPredicate(0,1.0));
CellFilter top = edges.subset(new CoordinateValueCellPredicate(1,1.0));
CellFilter bottom = edges.subset(new CoordinateValueCellPredicate(1,0.0));
BasisFamily basis = new Lagrange(1);
Expr u = new UnknownFunction(basis, "u");
Expr v = new TestFunction(basis, "v");
Expr dx = new Derivative(0);
Expr dy = new Derivative(1);
Expr grad = List(dx, dy);
Expr x = new CoordExpr(0);
Expr y = new CoordExpr(1);
QuadratureFamily quad = new ReducedQuadrature();
QuadratureFamily quad2 = new GaussianQuadrature(2);
/* Define the weak form */
const double pi = 4.0*atan(1.0);
Expr c = cos(pi*x);
Expr s = sin(pi*x);
Expr ch = cosh(y);
Expr sh = sinh(y);
Expr s2 = s*s;
Expr c2 = c*c;
Expr sh2 = sh*sh;
Expr ch2 = ch*ch;
Expr pi2 = pi*pi;
Expr uEx = s*ch;
Expr eu = exp(uEx);
Expr f = -(ch*eu*(-1 + pi2)*s) + ch2*(c2*eu*pi2 - s2) + eu*s2*sh2;
Expr eqn = Integral(interior, exp(u)*(grad*u)*(grad*v)
+ v*f + v*u*u, quad, watchMe)
+ Integral(right, v*exp(u)*pi*cosh(y), quad,watchBC);
/* Define the Dirichlet BC */
Expr bc = EssentialBC(left+top, v*(u-uEx), quad, watchBC);
Expr eqn2 = Integral(interior, exp(u)*(grad*u)*(grad*v)
+ v*f + v*u*u, quad2, watchMe)
+ Integral(right, v*exp(u)*pi*cosh(y), quad2,watchBC);
/* Define the Dirichlet BC */
Expr bc2 = EssentialBC(left+top, v*(u-uEx), quad2, watchBC);
DiscreteSpace discSpace(mesh, new Lagrange(1), vecType);
Expr soln1 = new DiscreteFunction(discSpace, 0.0, "u0");
Expr soln2 = new DiscreteFunction(discSpace, 0.0, "u0");
L2Projector proj(discSpace, uEx);
Expr uEx0 = proj.project();
NonlinearProblem nlp(mesh, eqn, bc, v, u, soln1, vecType);
NonlinearProblem nlp2(mesh, eqn2, bc2, v, u, soln2, vecType);
ParameterXMLFileReader reader("nox-aztec.xml");
ParameterList noxParams = reader.getParameters();
NOXSolver solver(noxParams);
nlp.solve(solver);
nlp2.solve(solver);
FieldWriter w = new VTKWriter("NonlinReduced-n" + Teuchos::toString(n));
w.addMesh(mesh);
w.addField("soln1", new ExprFieldWrapper(soln1[0]));
w.addField("soln2", new ExprFieldWrapper(soln2[0]));
w.addField("exact", new ExprFieldWrapper(uEx0[0]));
w.write();
Expr err1 = uEx - soln1;
Expr errExpr1 = Integral(interior,
err1*err1,
new GaussianQuadrature(4));
Expr err2 = uEx - soln2;
Expr errExpr2 = Integral(interior,
err2*err2,
new GaussianQuadrature(4));
Expr err12 = soln2 - soln1;
Expr errExpr12 = Integral(interior,
err12*err12,
new GaussianQuadrature(4));
double error1 = ::sqrt(evaluateIntegral(mesh, errExpr1));
double error2 = ::sqrt(evaluateIntegral(mesh, errExpr2));
double error12 = ::sqrt(evaluateIntegral(mesh, errExpr12));
Out::root() << "final result: " << n << " " << error1 << " " << error2 << " " << error12
<< endl;
h.append(1.0/((double) n));
errQuad.append(error2);
errReduced.append(error1);
}
double pQuad = fitPower(h, errQuad);
double pRed = fitPower(h, errReduced);
Out::root() << "exponent (reduced integration) " << pRed << endl;
Out::root() << "exponent (full integration) " << pQuad << endl;
return SundanceGlobal::checkTest(::fabs(pRed-2.0), 0.1);
}