FlowEquation.h

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#include "MUQ/Modeling/ModPiece.h"
 
#include <boost/property_tree/ptree.hpp>
 
#include <Eigen/Core>
#include <Eigen/Sparse>
 
/***
## Class Definition
 
This class solves the 1D elliptic PDE of the form
    $$
    -\frac{\partial}{\partial x}\cdot(K(x) \frac{\partial h}{\partial x}) = f(x).
    $$
    over $x\in[0,1]$ with boundary conditions $h(0)=0$ and
    $\partial h/\partial x =0$ at $x=1$.  This equation is a basic model of steady
    state fluid flow in a porous media, where $h(x)$ is the hydraulic head, $K(x)$
    is the hydraulic conductivity, and $f(x)$ is the recharge.
 
    This ModPiece uses linear finite elements on a uniform grid. There is a single input,
    the conductivity $k(x)$, which is represented as piecewise constant within each
    of the $N$ cells.   There is a single output of this ModPiece: the head $h(x)$ at the
    $N+1$ nodes in the discretization.
 
*/
class FlowEquation : public muq::Modeling::ModPiece
{
public:
 
  FlowEquation(Eigen::VectorXd const& sourceTerm) : muq::Modeling::ModPiece({int(sourceTerm.size())},
                                                                            {int(sourceTerm.size()+1)})
  {
    numCells = sourceTerm.size();
    numNodes = sourceTerm.size()+1;
 
    xs = Eigen::VectorXd::LinSpaced(numNodes,0,1);
    dx = xs(1)-xs(0);
 
    rhs = BuildRhs(sourceTerm);
  };
 
protected:
 
  void EvaluateImpl(muq::Modeling::ref_vector<Eigen::VectorXd> const& inputs) override
  {
    // Extract the conductivity vector from the inptus
    auto& cond = inputs.at(0).get();
 
    // Build the stiffness matrix
    auto K = BuildStiffness(cond);
 
    // Solve the sparse linear system and store the solution in the outputs vector
    Eigen::SimplicialLDLT<Eigen::SparseMatrix<double>> solver;
    solver.compute(K);
 
    outputs.resize(1);
    outputs.at(0) = solver.solve(rhs);
  };
 
  virtual void GradientImpl(unsigned int outWrt,
                            unsigned int inWrt,
                            muq::Modeling::ref_vector<Eigen::VectorXd> const& inputs,
                            Eigen::VectorXd const& sens) override
  {
    // Extract the conductivity vector from the inptus
    auto& cond = inputs.at(0).get();
 
    // Build the stiffness matrix
    auto A = BuildStiffness(cond);
 
    // Factor the stiffness matrix for forward and adjoint solves
    Eigen::SimplicialLDLT<Eigen::SparseMatrix<double>> solver;
    solver.compute(A);
 
    // Solve the forward problem
    Eigen::VectorXd sol = solver.solve(rhs);
 
    // Solve the adjoint problem
    Eigen::VectorXd adjRhs = BuildAdjointRHS(sens);
    Eigen::VectorXd adjSol = solver.solve(adjRhs);
 
    // Compute the gradient from the adjoint solution
    gradient = (1.0/dx)*(sol.tail(numCells)-sol.head(numCells)).array() * (adjSol.tail(numCells) - adjSol.head(numCells)).array();
  }
 
 
  /***
    This function computes the application of the model Jacobian's matrix $J$
    on a vector $v$.  In addition to the model parameter $k$, this function also
    requires the vector $v$.
 
    The gradient with respect to the conductivity field is computed by solving
    the forward model to get $h(x)$ and then using the tangent linear approach
    described above to obtain the Jacobian action.
 
    @param[in] outWrt For a model with multiple outputs, this would be the index
                      of the output list that corresponds to the sensitivity vector.
                      Since this ModPiece only has one output, the outWrt argument
                      is not used in the GradientImpl function.
 
    @param[in] inWrt Specifies the index of the input for which we want to compute
                     the gradient.  For inWrt==0, then the Jacobian with respect
                     to the conductivity is used.  Since this ModPiece only has one
                     input, 0 is the only valid value for inWrt.
 
    @param[in] inputs Just like the EvalauteImpl function, this is a list of vector-valued inputs.
 
    @param[in] vec A vector with the same size of inputs[0].  The Jacobian will be applied to this vector.
 
    @return This function returns nothing.  It stores the result in the private
            ModPiece::jacobianAction variable that is then returned by the `Jacobian` function.
 
  */
  virtual void ApplyJacobianImpl(unsigned int outWrt,
                                 unsigned int inWrt,
                                 muq::Modeling::ref_vector<Eigen::VectorXd> const& inputs,
                                 Eigen::VectorXd const& vec) override
  {
    // Extract the conductivity vector from the inptus
    auto& cond = inputs.at(0).get();
 
    // Build the stiffness matrix
    auto A = BuildStiffness(cond);
 
    // Factor the stiffness matrix for forward and tangent linear solves
    Eigen::SimplicialLDLT<Eigen::SparseMatrix<double>> solver;
    solver.compute(A);
 
    // Solve the forward system
    Eigen::VectorXd sol = solver.solve(rhs);
 
    // Build the tangent linear rhs
    Eigen::VectorXd incrRhs = Eigen::VectorXd::Zero(numNodes);
 
    Eigen::VectorXd dh_dx = (sol.tail(numCells)-sol.head(numCells))/dx; // Spatial derivative of solution
 
    incrRhs.head(numCells) += ( vec.array()*dh_dx.array() ).matrix();
    incrRhs.tail(numCells) -= ( vec.array()*dh_dx.array() ).matrix();
 
    // Solve the tangent linear model for the jacobian action
    jacobianAction = solver.solve(incrRhs);
  }
 
  virtual void ApplyHessianImpl(unsigned int outWrt,
                                 unsigned int inWrt1,
                                 unsigned int inWrt2,
                                 muq::Modeling::ref_vector<Eigen::VectorXd> const& inputs,
                                 Eigen::VectorXd const& sens,
                                 Eigen::VectorXd const& vec) override
  {
    // Extract the conductivity vector from the inptus
    auto& cond = inputs.at(0).get();
 
    // Build the stiffness matrix
    auto K = BuildStiffness(cond);
 
    // Factor the stiffness matrix for forward and adjoint solves
    Eigen::SimplicialLDLT<Eigen::SparseMatrix<double>> solver;
    solver.compute(K);
 
    // Solve the forward problem
    Eigen::VectorXd sol = solver.solve(rhs);
 
    // If we're using the Hessian $\nabla_{kk} J$
    if((inWrt1==0)&&(inWrt2==0)){
 
      // Solve the adjoint problem
      Eigen::VectorXd adjRhs = BuildAdjointRHS(sens);
      Eigen::VectorXd adjSol = solver.solve(adjRhs);
 
      // Solve the incremental forward problem
      Eigen::VectorXd incrForRhs = BuildIncrementalRhs(sol, vec);
      Eigen::VectorXd incrForSol = solver.solve(incrForRhs);
 
      // Solve the incremental adjoint problem
      Eigen::VectorXd incrAdjRhs = BuildIncrementalRhs(adjSol, vec);
      Eigen::VectorXd incrAdjSol = solver.solve(incrAdjRhs);
 
      // Construct the Hessian action
      auto solDeriv = (sol.tail(numCells)-sol.head(numCells))/dx;
      auto adjDeriv = (adjSol.tail(numCells)-adjSol.head(numCells))/dx;
      auto incrForDeriv = (incrForSol.tail(numCells) - incrForSol.head(numCells))/dx;
      auto incrAdjDeriv = (incrAdjSol.tail(numCells) - incrAdjSol.head(numCells))/dx;
 
      hessAction = -(incrAdjDeriv.array() * solDeriv.array() + incrForDeriv.array() * adjDeriv.array());
 
    // If we're using the mixed Hessian $\nabla_{ks} J$
    }else if((inWrt1==0)&&(inWrt2==1)){
 
      Eigen::VectorXd temp = solver.solve(vec);
      auto solDeriv = (sol.tail(numCells) - sol.head(numCells))/dx;
      auto tempDeriv = (temp.tail(numCells)-temp.head(numCells))/dx;
 
      hessAction = -dx * solDeriv.array() * tempDeriv.array();
 
    // We should never see any other options...
    }else{
      assert(false);
    }
  }
 
  Eigen::VectorXd BuildRhs(Eigen::VectorXd const& sourceTerm) const
  {
    Eigen::VectorXd rhs = Eigen::VectorXd::Zero(numNodes);
 
    rhs.segment(1,numNodes-2) = 0.5*dx*(sourceTerm.tail(numNodes-2) + sourceTerm.head(numNodes-2));
    rhs(numNodes-1) = 0.5*dx*sourceTerm(numNodes-2);
 
    return rhs;
  }
 
  Eigen::VectorXd BuildAdjointRHS(Eigen::VectorXd const& sensitivity) const
  {
    Eigen::VectorXd rhs = -1.0*sensitivity;
    rhs(0) = 0.0; // <- To enforce Dirichlet BC
    return rhs;
  }
 
  Eigen::VectorXd BuildIncrementalRhs(Eigen::VectorXd const& sol, Eigen::VectorXd const& khat)
  {
    // Compute the derivative of the solution in each cell
    Eigen::VectorXd solGrad = (sol.tail(numCells)-sol.head(numCells))/dx;
    Eigen::VectorXd rhs = Eigen::VectorXd::Zero(numNodes);
 
    unsigned int leftNode, rightNode;
    for(unsigned int cellInd=0; cellInd<numCells; ++cellInd)
    {
      leftNode = cellInd;
      rightNode = cellInd + 1;
 
      rhs(leftNode) -= dx*khat(cellInd)*solGrad(cellInd);
      rhs(rightNode) += dx*khat(cellInd)*solGrad(cellInd);
    }
 
    rhs(0) = 0.0; // <- To enforce Dirichlet BC at x=0
    return rhs;
  }
 
  Eigen::SparseMatrix<double> BuildStiffness(Eigen::VectorXd const& condVals) const{
 
    typedef Eigen::Triplet<double> T;
    std::vector<T> nzVals;
 
    // Add a large number to K[0,0] to enforce Dirichlet BC
    nzVals.push_back( T(0,0,1e10) );
 
    unsigned int leftNode, rightNode;
    for(unsigned int cellInd=0; cellInd<numCells; ++cellInd)
    {
      leftNode  = cellInd;
      rightNode = cellInd+1;
 
      nzVals.push_back( T(leftNode,  rightNode, -condVals(cellInd)/dx) );
      nzVals.push_back( T(rightNode, leftNode,  -condVals(cellInd)/dx) );
      nzVals.push_back( T(rightNode, rightNode,  condVals(cellInd)/dx) );
      nzVals.push_back( T(leftNode,  leftNode,   condVals(cellInd)/dx) );
    }
 
    Eigen::SparseMatrix<double> stiffMat(numNodes,numNodes);
    stiffMat.setFromTriplets(nzVals.begin(), nzVals.end());
 
    return stiffMat;
  }
 
 
private:
 
  // Store "mesh" information.  xs contains the node locations and dx is the uniform spacing between nodes
  Eigen::VectorXd xs;
  double dx;
  unsigned int numCells;
  unsigned int numNodes;
 
  // Will store the precomputed RHS for the forward problem
  Eigen::VectorXd rhs;
 
}; // end of class SimpleModel