Real RBEvaluation::compute_residual_dual_norm(const unsigned int N)
{
START_LOG("compute_residual_dual_norm()", "RBEvaluation");
const RBParameters& mu = get_parameters();
// Use the stored representor inner product values
// to evaluate the residual norm
Number residual_norm_sq = 0.;
unsigned int q=0;
for(unsigned int q_f1=0; q_f1<rb_theta_expansion->get_n_F_terms(); q_f1++)
{
for(unsigned int q_f2=q_f1; q_f2<rb_theta_expansion->get_n_F_terms(); q_f2++)
{
Real delta = (q_f1==q_f2) ? 1. : 2.;
residual_norm_sq += delta * libmesh_real(
rb_theta_expansion->eval_F_theta(q_f1, mu)
* libmesh_conj(rb_theta_expansion->eval_F_theta(q_f2, mu)) * Fq_representor_innerprods[q] );
q++;
}
}
for(unsigned int q_f=0; q_f<rb_theta_expansion->get_n_F_terms(); q_f++)
{
for(unsigned int q_a=0; q_a<rb_theta_expansion->get_n_A_terms(); q_a++)
{
for(unsigned int i=0; i<N; i++)
{
Real delta = 2.;
residual_norm_sq +=
delta * libmesh_real( rb_theta_expansion->eval_F_theta(q_f, mu) *
libmesh_conj(rb_theta_expansion->eval_A_theta(q_a, mu)) *
libmesh_conj(RB_solution(i)) * Fq_Aq_representor_innerprods[q_f][q_a][i] );
}
}
}
q=0;
for(unsigned int q_a1=0; q_a1<rb_theta_expansion->get_n_A_terms(); q_a1++)
{
for(unsigned int q_a2=q_a1; q_a2<rb_theta_expansion->get_n_A_terms(); q_a2++)
{
Real delta = (q_a1==q_a2) ? 1. : 2.;
for(unsigned int i=0; i<N; i++)
{
for(unsigned int j=0; j<N; j++)
{
residual_norm_sq +=
delta * libmesh_real( libmesh_conj(rb_theta_expansion->eval_A_theta(q_a1, mu)) *
rb_theta_expansion->eval_A_theta(q_a2, mu) *
libmesh_conj(RB_solution(i)) * RB_solution(j) * Aq_Aq_representor_innerprods[q][i][j] );
}
}
q++;
}
}
if(libmesh_real(residual_norm_sq) < 0.)
{
// libMesh::out << "Warning: Square of residual norm is negative "
// << "in RBSystem::compute_residual_dual_norm()" << std::endl;
// Sometimes this is negative due to rounding error,
// but when this occurs the error is on the order of 1.e-10,
// so shouldn't affect error bound much...
residual_norm_sq = std::abs(residual_norm_sq);
}
STOP_LOG("compute_residual_dual_norm()", "RBEvaluation");
return std::sqrt( libmesh_real(residual_norm_sq) );
}
Real RBEIMEvaluation::rb_solve(unsigned int N)
{
// Short-circuit if we are using the same parameters and value of N
if( (_previous_parameters == get_parameters()) &&
(_previous_N == N) )
{
return _previous_error_bound;
}
// Otherwise, update _previous parameters, _previous_N
_previous_parameters = get_parameters();
_previous_N = N;
LOG_SCOPE("rb_solve()", "RBEIMEvaluation");
if(N > get_n_basis_functions())
libmesh_error_msg("ERROR: N cannot be larger than the number of basis functions in rb_solve");
if(N==0)
libmesh_error_msg("ERROR: N must be greater than 0 in rb_solve");
// Get the rhs by sampling parametrized_function
// at the first N interpolation_points
DenseVector<Number> EIM_rhs(N);
for(unsigned int i=0; i<N; i++)
{
EIM_rhs(i) = evaluate_parametrized_function(interpolation_points_var[i],
interpolation_points[i],
*interpolation_points_elem[i]);
}
DenseMatrix<Number> interpolation_matrix_N;
interpolation_matrix.get_principal_submatrix(N, interpolation_matrix_N);
interpolation_matrix_N.lu_solve(EIM_rhs, RB_solution);
// Optionally evaluate an a posteriori error bound. The EIM error estimate
// recommended in the literature is based on using "next" EIM point, so
// we skip this if N == get_n_basis_functions()
if(evaluate_RB_error_bound && (N != get_n_basis_functions()))
{
// Compute the a posteriori error bound
// First, sample the parametrized function at x_{N+1}
Number g_at_next_x = evaluate_parametrized_function(interpolation_points_var[N],
interpolation_points[N],
*interpolation_points_elem[N]);
// Next, evaluate the EIM approximation at x_{N+1}
Number EIM_approx_at_next_x = 0.;
for(unsigned int j=0; j<N; j++)
{
EIM_approx_at_next_x += RB_solution(j) * interpolation_matrix(N,j);
}
Real error_estimate = std::abs(g_at_next_x - EIM_approx_at_next_x);
_previous_error_bound = error_estimate;
return error_estimate;
}
else // Don't evaluate an error bound
{
_previous_error_bound = -1.;
return -1.;
}
}
Real RBEIMEvaluation::rb_solve(unsigned int N)
{
// Short-circuit if we are using the same parameters and value of N
if( (_previous_parameters == get_parameters()) &&
(_previous_N == N) )
{
return _previous_error_bound;
}
// Otherwise, update _previous parameters, _previous_N
_previous_parameters = get_parameters();
_previous_N = N;
START_LOG("rb_solve()", "RBEIMEvaluation");
if(N > get_n_basis_functions())
{
libMesh::err << "ERROR: N cannot be larger than the number "
<< "of basis functions in rb_solve" << std::endl;
libmesh_error();
}
if(N==0)
{
libMesh::err << "ERROR: N must be greater than 0 in rb_solve" << std::endl;
libmesh_error();
}
// Get the rhs by sampling parametrized_function
// at the first N interpolation_points
DenseVector<Number> EIM_rhs(N);
for(unsigned int i=0; i<N; i++)
{
EIM_rhs(i) = evaluate_parametrized_function(interpolation_points_var[i], interpolation_points[i]);
}
DenseMatrix<Number> interpolation_matrix_N;
interpolation_matrix.get_principal_submatrix(N, interpolation_matrix_N);
interpolation_matrix_N.lu_solve(EIM_rhs, RB_solution);
// Evaluate an a posteriori error bound
if(evaluate_RB_error_bound)
{
// Compute the a posteriori error bound
// First, sample the parametrized function at x_{N+1}
Number g_at_next_x;
if(N == get_n_basis_functions())
g_at_next_x = evaluate_parametrized_function(extra_interpolation_point_var, extra_interpolation_point);
else
g_at_next_x = evaluate_parametrized_function(interpolation_points_var[N], interpolation_points[N]);
// Next, evaluate the EIM approximation at x_{N+1}
Number EIM_approx_at_next_x = 0.;
for(unsigned int j=0; j<N; j++)
{
if(N == get_n_basis_functions())
{
EIM_approx_at_next_x += RB_solution(j) * extra_interpolation_matrix_row(j);
}
else
{
EIM_approx_at_next_x += RB_solution(j) * interpolation_matrix(N,j);
}
}
Real error_estimate = std::abs(g_at_next_x - EIM_approx_at_next_x);
STOP_LOG("rb_solve()", "RBEIMEvaluation");
_previous_error_bound = error_estimate;
return error_estimate;
}
else // Don't evaluate an error bound
{
STOP_LOG("rb_solve()", "RBEIMEvaluation");
_previous_error_bound = -1.;
return -1.;
}
}