twoD_diffusion_problem.cpp

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#include "twoD_diffusion_problem.hpp"
 
#include "Stokhos_KL_ExponentialRandomField.hpp"
#include "EpetraExt_MatrixMatrix.h"
#include "Teuchos_Assert.hpp"
#include "Stokhos_PreconditionerFactory.hpp"
 
namespace {
 
  // Functor representing a diffusion function given by a KL expansion
  struct KL_Diffusion_Func {
    double mean;
    mutable Teuchos::Array<double> point;
    Teuchos::RCP< Stokhos::KL::ExponentialRandomField<double> > rf;
    
    KL_Diffusion_Func(double xyLeft, double xyRight, 
          double mean_, double std_dev, 
          double L, int num_KL) : mean(mean_), point(2)
    {
      Teuchos::ParameterList rfParams;
      rfParams.set("Number of KL Terms", num_KL);
      rfParams.set("Mean", mean);
      rfParams.set("Standard Deviation", std_dev);
      int ndim = 2;
      Teuchos::Array<double> domain_upper(ndim), domain_lower(ndim), 
  correlation_length(ndim);
      for (int i=0; i<ndim; i++) {
  domain_upper[i] = xyRight;
  domain_lower[i] = xyLeft;
  correlation_length[i] = L;
      }
      rfParams.set("Domain Upper Bounds", domain_upper);
      rfParams.set("Domain Lower Bounds", domain_lower);
      rfParams.set("Correlation Lengths", correlation_length);
 
      rf = 
  Teuchos::rcp(new Stokhos::KL::ExponentialRandomField<double>(rfParams));
    }
 
    double operator() (double x, double y, int k) const {
      if (k == 0)
  return mean;
      point[0] = x;
      point[1] = y;
      return rf->evaluate_eigenfunction(point, k-1);
    }
  };
 
  // Functor representing a diffusion function given by a KL expansion
  // where the basis is normalized
  template <typename DiffusionFunc>
  struct Normalized_KL_Diffusion_Func {
    const DiffusionFunc& func;
    int d;
    Teuchos::Array<double> psi_0, psi_1;
    
    Normalized_KL_Diffusion_Func(
      const DiffusionFunc& func_,
      const Stokhos::OrthogPolyBasis<int,double>& basis) : 
      func(func_),
      d(basis.dimension()), 
      psi_0(basis.size()), 
      psi_1(basis.size())    
    {
      Teuchos::Array<double> zero(d), one(d);
      for(int k=0; k<d; k++) {
  zero[k] = 0.0;
  one[k] = 1.0;
      }
      basis.evaluateBases(zero, psi_0);
      basis.evaluateBases(one, psi_1);
    }
 
    double operator() (double x, double y, int k) const {
      if (k == 0) {
  double val = func(x, y, 0);
  for (int i=1; i<=d; i++)
    val -= psi_0[i]/(psi_1[i]-psi_0[i])*func(x, y, i);
  val /= psi_0[0];
  return val;
      }
      else
  return 1.0/(psi_1[k]-psi_0[k])*func(x, y, k);
    }
  };
 
  // Functor representing a diffusion function given by a log-normal PC expansion
  template <typename DiffusionFunc>
  struct LogNormal_Diffusion_Func {
    double mean;
    const DiffusionFunc& func;
    Teuchos::RCP<const Stokhos::ProductBasis<int, double> > prodbasis;
 
    LogNormal_Diffusion_Func(
      double mean_,
      const DiffusionFunc& func_,
      const Teuchos::RCP<const Stokhos::ProductBasis<int, double> > prodbasis_) 
      : mean(mean_), func(func_), prodbasis(prodbasis_) {}
    
    double operator() (double x, double y, int k) const {
      int d = prodbasis->dimension();
      const Teuchos::Array<double>& norms = prodbasis->norm_squared();
      Teuchos::Array<int> multiIndex = prodbasis->getTerm(k);
      double sum_g = 0.0, efval;
      for (int l=0; l<d; l++) {
  sum_g += std::pow(func(x,y,l+1),2);
      }
      efval = std::exp(mean + 0.5*sum_g)/norms[k];
      for (int l=0; l<d; l++) {
  efval *= std::pow(func(x,y,l+1),multiIndex[l]); 
      }
      return efval;
    }
  };
 
}
 
twoD_diffusion_problem::
twoD_diffusion_problem(
  const Teuchos::RCP<const Epetra_Comm>& comm, int n, int d, 
  double s, double mu, 
  const Teuchos::RCP<const Stokhos::OrthogPolyBasis<int,double> >& basis_,
  bool log_normal_,
  bool eliminate_bcs_) :
  mesh(n*n),
  basis(basis_),
  log_normal(log_normal_),
  eliminate_bcs(eliminate_bcs_)
{
  // Construct the mesh.  
  // The mesh is uniform and the nodes are numbered
  // LEFT to RIGHT, DOWN to UP.
  //
  // 5-6-7-8-9
  // | | | | |
  // 0-1-2-3-4
  double xyLeft = -.5;
  double xyRight = .5;
  h = (xyRight - xyLeft)/((double)(n-1));
  Teuchos::Array<int> global_dof_indices;
  for (int j=0; j<n; j++) {
    double y = xyLeft + j*h;
    for (int i=0; i<n; i++) {
      double x = xyLeft + i*h;
      int idx = j*n+i;
      mesh[idx].x = x;
      mesh[idx].y = y;
      if (i == 0 || i == n-1 || j == 0 || j == n-1)
  mesh[idx].boundary = true;
      if (i != 0)
  mesh[idx].left = idx-1;
      if (i != n-1)
  mesh[idx].right = idx+1;
      if (j != 0)
  mesh[idx].down = idx-n;
      if (j != n-1)
  mesh[idx].up = idx+n;
      if (!(eliminate_bcs && mesh[idx].boundary))
  global_dof_indices.push_back(idx);
    }
  }
  
  // Solution vector map
  int n_global_dof = global_dof_indices.size();
  int n_proc = comm->NumProc();
  int proc_id = comm->MyPID();
  int n_my_dof = n_global_dof / n_proc;
  if (proc_id == n_proc-1)
    n_my_dof += n_global_dof % n_proc;
  int *my_dof = global_dof_indices.getRawPtr() + proc_id*(n_global_dof / n_proc);
  x_map = 
    Teuchos::rcp(new Epetra_Map(n_global_dof, n_my_dof, my_dof, 0, *comm));
 
  // Initial guess, initialized to 0.0
  x_init = Teuchos::rcp(new Epetra_Vector(*x_map));
  x_init->PutScalar(0.0);
 
  // Parameter vector map
  p_map = Teuchos::rcp(new Epetra_LocalMap(d, 0, *comm));
 
  // Response vector map
  g_map = Teuchos::rcp(new Epetra_LocalMap(1, 0, *comm));
 
  // Initial parameters
  p_init = Teuchos::rcp(new Epetra_Vector(*p_map));
  p_init->PutScalar(0.0);
 
  // Parameter names
  p_names = Teuchos::rcp(new Teuchos::Array<std::string>(d));
  for (int i=0;i<d;i++) {
    std::stringstream ss;
    ss << "KL Random Variable " << i+1;
    (*p_names)[i] = ss.str(); 
  }
 
  // Build Jacobian graph
  int NumMyElements = x_map->NumMyElements();
  int *MyGlobalElements = x_map->MyGlobalElements();
  graph = Teuchos::rcp(new Epetra_CrsGraph(Copy, *x_map, 5));
  for (int i=0; i<NumMyElements; ++i ) {
 
    // Center
    int global_idx = MyGlobalElements[i];
    graph->InsertGlobalIndices(global_idx, 1, &global_idx);
 
    if (!mesh[global_idx].boundary) {
      // Down
      if (!(eliminate_bcs && mesh[mesh[global_idx].down].boundary))
  graph->InsertGlobalIndices(global_idx, 1, &mesh[global_idx].down);
 
      // Left
      if (!(eliminate_bcs && mesh[mesh[global_idx].left].boundary))
  graph->InsertGlobalIndices(global_idx, 1, &mesh[global_idx].left);
 
      // Right
      if (!(eliminate_bcs && mesh[mesh[global_idx].right].boundary))
  graph->InsertGlobalIndices(global_idx, 1, &mesh[global_idx].right);
 
      // Up
      if (!(eliminate_bcs && mesh[mesh[global_idx].up].boundary))
  graph->InsertGlobalIndices(global_idx, 1, &mesh[global_idx].up);
    }
  }
  graph->FillComplete();
  graph->OptimizeStorage();
 
  KL_Diffusion_Func klFunc(xyLeft, xyRight, mu, s, 1.0, d);
  if (!log_normal) {
    // Fill coefficients of KL expansion of operator
    if (basis == Teuchos::null) { 
      fillMatrices(klFunc, d+1);
    }
    else {
      Normalized_KL_Diffusion_Func<KL_Diffusion_Func> nklFunc(klFunc, *basis);
      fillMatrices(nklFunc, d+1);
    }
  }
  else {
    // Fill coefficients of PC expansion of operator
    int sz = basis->size();
    Teuchos::RCP<const Stokhos::ProductBasis<int, double> > prodbasis =
      Teuchos::rcp_dynamic_cast<const Stokhos::ProductBasis<int, double> >(
  basis, true);
    LogNormal_Diffusion_Func<KL_Diffusion_Func> lnFunc(mu, klFunc, prodbasis);
    fillMatrices(lnFunc, sz);
  }
 
  // Construct deterministic operator
  A = Teuchos::rcp(new Epetra_CrsMatrix(Copy, *graph));
 
  // Construct the RHS vector.
  b = Teuchos::rcp(new Epetra_Vector(*x_map));
  for( int i=0 ; i<NumMyElements; ++i ) {
    int global_idx = MyGlobalElements[i];
    if (mesh[global_idx].boundary)
      (*b)[i] = 0;
    else 
      (*b)[i] = 1;
  }
 
  if (basis != Teuchos::null) {
    point.resize(d);
    basis_vals.resize(basis->size());
  }
}
 
// Overridden from EpetraExt::ModelEvaluator
 
Teuchos::RCP<const Epetra_Map>
twoD_diffusion_problem::
get_x_map() const
{
  return x_map;
}
 
Teuchos::RCP<const Epetra_Map>
twoD_diffusion_problem::
get_f_map() const
{
  return x_map;
}
 
Teuchos::RCP<const Epetra_Map>
twoD_diffusion_problem::
get_p_map(int l) const
{
  TEUCHOS_TEST_FOR_EXCEPTION(l != 0, 
         std::logic_error,
                     std::endl << 
                     "Error!  twoD_diffusion_problem::get_p_map():  " <<
                     "Invalid parameter index l = " << l << std::endl);
 
  return p_map;
}
 
Teuchos::RCP<const Epetra_Map>
twoD_diffusion_problem::
get_g_map(int l) const
{
  TEUCHOS_TEST_FOR_EXCEPTION(l != 0, 
         std::logic_error,
                     std::endl << 
                     "Error!  twoD_diffusion_problem::get_g_map():  " <<
                     "Invalid parameter index l = " << l << std::endl);
 
  return g_map;
}
 
Teuchos::RCP<const Teuchos::Array<std::string> >
twoD_diffusion_problem::
get_p_names(int l) const
{
  TEUCHOS_TEST_FOR_EXCEPTION(l != 0, 
         std::logic_error,
                     std::endl << 
                     "Error!  twoD_diffusion_problem::get_p_names():  " <<
                     "Invalid parameter index l = " << l << std::endl);
 
  return p_names;
}
 
Teuchos::RCP<const Epetra_Vector>
twoD_diffusion_problem::
get_x_init() const
{
  return x_init;
}
 
Teuchos::RCP<const Epetra_Vector>
twoD_diffusion_problem::
get_p_init(int l) const
{
  TEUCHOS_TEST_FOR_EXCEPTION(l != 0, 
         std::logic_error,
                     std::endl << 
                     "Error!  twoD_diffusion_problem::get_p_init():  " <<
                     "Invalid parameter index l = " << l << std::endl);
  
  return p_init;
}
 
Teuchos::RCP<Epetra_Operator>
twoD_diffusion_problem::
create_W() const
{
  Teuchos::RCP<Epetra_CrsMatrix> AA = 
    Teuchos::rcp(new Epetra_CrsMatrix(Copy, *graph));
  AA->FillComplete();
  AA->OptimizeStorage();
  return AA;
}
 
void 
twoD_diffusion_problem::
computeResidual(const Epetra_Vector& x,
    const Epetra_Vector& p,
    Epetra_Vector& f)
{
  // f = A*x - b
  compute_A(p);
  A->Apply(x,f);
  f.Update(-1.0, *b, 1.0);
}
 
void 
twoD_diffusion_problem::
computeJacobian(const Epetra_Vector& x,
    const Epetra_Vector& p,
    Epetra_Operator& J)
{
  // J = A
  compute_A(p);
  Epetra_CrsMatrix& jac = dynamic_cast<Epetra_CrsMatrix&>(J);
  jac = *A;
  jac.FillComplete();
  jac.OptimizeStorage();
}
 
void 
twoD_diffusion_problem::
computeResponse(const Epetra_Vector& x,
    const Epetra_Vector& p,
    Epetra_Vector& g)
{
  // g = average of x
  x.MeanValue(&g[0]);
  g[0] *= double(x.GlobalLength()) / double(mesh.size());
}
 
void 
twoD_diffusion_problem::
computeSGResidual(const Stokhos::EpetraVectorOrthogPoly& x_sg,
      const Stokhos::EpetraVectorOrthogPoly& p_sg,
      Stokhos::OrthogPolyExpansion<int,double>& expn,
      Stokhos::EpetraVectorOrthogPoly& f_sg)
{
  // Get stochastic expansion data
  typedef Stokhos::Sparse3Tensor<int,double> Cijk_type;
  Teuchos::RCP<const Cijk_type> Cijk = expn.getTripleProduct(); 
  const Teuchos::Array<double>& norms = basis->norm_squared();
 
  if (sg_kx_vec_all.size() != basis->size()) {
    sg_kx_vec_all.resize(basis->size());
    for (int i=0;i<basis->size();i++) {
      sg_kx_vec_all[i] = Teuchos::rcp(new Epetra_Vector(*x_map));
    }
  }
  f_sg.init(0.0);
 
  Cijk_type::k_iterator k_begin = Cijk->k_begin();
  Cijk_type::k_iterator k_end = Cijk->k_end();
  for (Cijk_type::k_iterator k_it=k_begin; k_it!=k_end; ++k_it) {
    int k = Stokhos::index(k_it);
    for (Cijk_type::kj_iterator j_it = Cijk->j_begin(k_it); 
   j_it != Cijk->j_end(k_it); ++j_it) {
      int j = Stokhos::index(j_it);
      A_k[k]->Apply(x_sg[j],*(sg_kx_vec_all[j]));
    }
    for (Cijk_type::kj_iterator j_it = Cijk->j_begin(k_it); 
   j_it != Cijk->j_end(k_it); ++j_it) {
      int j = Stokhos::index(j_it);
      for (Cijk_type::kji_iterator i_it = Cijk->i_begin(j_it);
     i_it != Cijk->i_end(j_it); ++i_it) {
  int i = Stokhos::index(i_it);
  double c = Stokhos::value(i_it);  // C(i,j,k)
  f_sg[i].Update(1.0*c/norms[i],*(sg_kx_vec_all[j]),1.0);
      }
    }
  }
  f_sg[0].Update(-1.0,*b,1.0);
}
 
void 
twoD_diffusion_problem::
computeSGJacobian(const Stokhos::EpetraVectorOrthogPoly& x_sg,
      const Stokhos::EpetraVectorOrthogPoly& p_sg,
      Stokhos::EpetraOperatorOrthogPoly& J_sg)
{
  for (int i=0; i<J_sg.size(); i++) {
    Teuchos::RCP<Epetra_CrsMatrix> jac = 
      Teuchos::rcp_dynamic_cast<Epetra_CrsMatrix>(J_sg.getCoeffPtr(i), true);
    *jac = *A_k[i];
    jac->FillComplete();
    jac->OptimizeStorage();
  }
}
 
void 
twoD_diffusion_problem::
computeSGResponse(const Stokhos::EpetraVectorOrthogPoly& x_sg,
      const Stokhos::EpetraVectorOrthogPoly& p_sg,
      Stokhos::EpetraVectorOrthogPoly& g_sg)
{
  int sz = x_sg.size();
  for (int i=0; i<sz; i++) {
    x_sg[i].MeanValue(&g_sg[i][0]);
    g_sg[i][0] *= double(x_sg[i].GlobalLength()) / double(mesh.size());
  }
}
 
template <typename FuncT>
void
twoD_diffusion_problem::
fillMatrices(const FuncT& func, int sz)
{
  int NumMyElements = x_map->NumMyElements();
  int *MyGlobalElements = x_map->MyGlobalElements();
  double h2 = h*h;
  double val;
 
  A_k.resize(sz);
  for (int k=0; k<sz; k++) {
    A_k[k] = Teuchos::rcp(new Epetra_CrsMatrix(Copy, *graph));
    for( int i=0 ; i<NumMyElements; ++i ) {
 
      // Center
      int global_idx = MyGlobalElements[i];
      if (mesh[global_idx].boundary) {
  if (k == 0)
    val = 1.0; //Enforce BC in mean matrix
  else
    val = 0.0;
  A_k[k]->ReplaceGlobalValues(global_idx, 1, &val, &global_idx);
      }
      else {
  double a_down = 
    -func(mesh[global_idx].x, mesh[global_idx].y-h/2.0, k)/h2;
  double a_left = 
    -func(mesh[global_idx].x-h/2.0, mesh[global_idx].y, k)/h2;
  double a_right = 
    -func(mesh[global_idx].x+h/2.0, mesh[global_idx].y, k)/h2;
  double a_up = 
    -func(mesh[global_idx].x, mesh[global_idx].y+h/2.0, k)/h2;
 
  // Center
  val = -(a_down + a_left + a_right + a_up);
  A_k[k]->ReplaceGlobalValues(global_idx, 1, &val, &global_idx);
 
  // Down
  if (!(eliminate_bcs && mesh[mesh[global_idx].down].boundary))
    A_k[k]->ReplaceGlobalValues(global_idx, 1, &a_down, 
              &mesh[global_idx].down);
 
  // Left
  if (!(eliminate_bcs && mesh[mesh[global_idx].left].boundary))
    A_k[k]->ReplaceGlobalValues(global_idx, 1, &a_left, 
              &mesh[global_idx].left);
 
  // Right
  if (!(eliminate_bcs && mesh[mesh[global_idx].right].boundary))
    A_k[k]->ReplaceGlobalValues(global_idx, 1, &a_right, 
              &mesh[global_idx].right);
 
  // Up
  if (!(eliminate_bcs && mesh[mesh[global_idx].up].boundary))
    A_k[k]->ReplaceGlobalValues(global_idx, 1, &a_up, 
              &mesh[global_idx].up);
      }
    }
    A_k[k]->FillComplete();
    A_k[k]->OptimizeStorage();
  }
}
 
void
twoD_diffusion_problem::
compute_A(const Epetra_Vector& p)
{
  if (basis != Teuchos::null) {
    for (int i=0; i<point.size(); i++)
      point[i] = p[i];
    basis->evaluateBases(point, basis_vals);
    A->PutScalar(0.0);
    for (int k=0;k<A_k.size();k++)
      EpetraExt::MatrixMatrix::Add((*A_k[k]), false, basis_vals[k], *A, 1.0);
  }
  else {
    *A = *(A_k[0]);
    for (int k=1;k<A_k.size();k++)
      EpetraExt::MatrixMatrix::Add((*A_k[k]), false, p[k-1], *A, 1.0);
  }
  A->FillComplete();
  A->OptimizeStorage();
}