MatrixInterpolator.cpp

 
// Description: Implementation of the MatrixInterpolator algorithm.
 
#include "MatrixInterpolator.h"
 
#include <limits.h>
#include <cmath>
#include "linalg/Matrix.h"
#include "linalg/scalapack_wrapper.h"
#include "mpi.h"
 
/* Use C++11 built-in shared pointers if available; else fallback to Boost. */
#if __cplusplus >= 201103L
#include <memory>
#else
#include <boost/shared_ptr.hpp>
#endif
 
/* Use automatically detected Fortran name-mangling scheme */
#define dposv CAROM_FC_GLOBAL(dposv, DPOSV)
 
extern "C" {
    // Solve a system of linear equations.
    void dposv(char*, int*, int*, double*, int*, double*, int*, int*);
}
 
using namespace std;
 
namespace CAROM {
 
MatrixInterpolator::MatrixInterpolator(const std::vector<Vector> &
                                       parameter_points,
                                       const std::vector<std::shared_ptr<Matrix>> & rotation_matrices,
                                       const std::vector<std::shared_ptr<Matrix>> & reduced_matrices,
                                       int ref_point,
                                       std::string matrix_type,
                                       std::string rbf,
                                       std::string interp_method,
                                       double closest_rbf_val,
                                       bool compute_gradients) :
    Interpolator(parameter_points,
                 rotation_matrices,
                 ref_point,
                 rbf,
                 interp_method,
                 closest_rbf_val,
                 compute_gradients)
{
    CAROM_VERIFY(reduced_matrices.size() == rotation_matrices.size());
    CAROM_VERIFY(matrix_type == "SPD" || matrix_type == "NS" || matrix_type == "R"
                 || matrix_type == "B");
    d_matrix_type = matrix_type;
 
    // Rotate the reduced matrices
    for (int i = 0; i < reduced_matrices.size(); i++)
    {
        // The ref_point does not need to be rotated
        if (i == d_ref_point)
        {
            d_rotated_reduced_matrices.push_back(reduced_matrices[i]);
        }
        else
        {
            if (reduced_matrices[i]->numRows() == rotation_matrices[i]->numRows()
                    && reduced_matrices[i]->numColumns() == rotation_matrices[i]->numRows())
            {
                std::unique_ptr<Matrix> Q_tA = rotation_matrices[i]->transposeMult(
                                                   *reduced_matrices[i]);
                std::shared_ptr<Matrix> Q_tAQ = Q_tA->mult(*rotation_matrices[i]);
                d_rotated_reduced_matrices.push_back(Q_tAQ);
            }
            else if (reduced_matrices[i]->numRows() == rotation_matrices[i]->numRows())
            {
                std::shared_ptr<Matrix> Q_tA = rotation_matrices[i]->transposeMult(
                                                   *reduced_matrices[i]);
                d_rotated_reduced_matrices.push_back(Q_tA);
            }
            else if (reduced_matrices[i]->numColumns() == rotation_matrices[i]->numRows())
            {
                std::shared_ptr<Matrix> AQ = reduced_matrices[i]->mult(*rotation_matrices[i]);
                d_rotated_reduced_matrices.push_back(AQ);
            }
            else
            {
                d_rotated_reduced_matrices.push_back(reduced_matrices[i]);
            }
        }
 
    }
}
 
std::shared_ptr<Matrix> MatrixInterpolator::interpolate(const Vector & point,
        bool orthogonalize)
{
    std::shared_ptr<Matrix> interpolated_matrix;
    if (d_matrix_type == "SPD")
    {
        if (d_compute_gradients)
        {
            CAROM_ERROR("Gradients are only implemented for B or G");
        }
        interpolated_matrix = interpolateSPDMatrix(point);
    }
    else if (d_matrix_type == "NS")
    {
        if (d_compute_gradients)
        {
            CAROM_ERROR("Gradients are only implemented for B or G");
        }
        interpolated_matrix = interpolateNonSingularMatrix(point);
    }
    else
    {
        interpolated_matrix = interpolateMatrix(point);
    }
 
    // Orthogonalize the interpolated matrix if requested.
    if (orthogonalize)
    {
        interpolated_matrix->orthogonalize();
    }
    return interpolated_matrix;
}
 
void MatrixInterpolator::obtainLambda()
{
    if (d_interp_method == "LS")
    {
        // Solving f = B*lambda
        Matrix* f_T = new Matrix(d_gammas[0]->numRows() * d_gammas[0]->numColumns(),
                                 d_gammas.size(), false);
        for (int i = 0; i < f_T->numRows(); i++)
        {
            for (int j = 0; j < f_T->numColumns(); j++)
            {
                f_T->item(i, j) = d_gammas[j]->getData()[i];
            }
        }
 
        // Obtain B matrix by calculating RBF.
        Matrix* B = new Matrix(d_gammas.size(), d_gammas.size(), false);
        for (int i = 0; i < B->numRows(); i++)
        {
            B->item(i, i) = 1.0;
            for (int j = i + 1; j < B->numColumns(); j++)
            {
                double res = obtainRBF(d_rbf, d_epsilon, d_parameter_points[i],
                                       d_parameter_points[j]);
                B->item(i, j) = res;
                B->item(j, i) = res;
            }
        }
 
        char uplo = 'U';
        int gamma_size = d_gammas.size();
        int num_elements = d_gammas[0]->numRows() * d_gammas[0]->numColumns();
        int info;
 
        dposv(&uplo, &gamma_size, &num_elements, B->getData(),  &gamma_size,
              f_T->getData(), &gamma_size, &info);
        if (info != 0)
        {
            std::cout << "Linear solve failed. Please choose a different epsilon value." <<
                      std::endl;
        }
        CAROM_VERIFY(info == 0);
 
        delete B;
 
        d_lambda_T = std::unique_ptr<Matrix>(f_T);
    }
}
 
std::unique_ptr<Matrix> MatrixInterpolator::obtainLogInterpolatedMatrix(
    std::vector<double>& rbf)
{
    Matrix* log_interpolated_matrix = new Matrix(
        d_rotated_reduced_matrices[d_ref_point]->numRows(),
        d_rotated_reduced_matrices[d_ref_point]->numColumns(),
        d_rotated_reduced_matrices[d_ref_point]->distributed());
    if (d_interp_method == "LS")
    {
        for (int i = 0; i < d_lambda_T->numRows(); i++)
        {
            for (int j = 0; j < rbf.size(); j++)
            {
                log_interpolated_matrix->getData()[i] += d_lambda_T->item(i, j) * rbf[j];
            }
        }
    }
    else if (d_interp_method == "IDW")
    {
        double sum = rbfWeightedSum(rbf);
        int num_elements = d_rotated_reduced_matrices[d_ref_point]->numRows() *
                           d_rotated_reduced_matrices[d_ref_point]->numColumns();
        for (int i = 0; i < num_elements; i++)
        {
            for (int j = 0; j < rbf.size(); j++)
            {
                log_interpolated_matrix->getData()[i] += d_gammas[j]->getData()[i] * rbf[j];
            }
            log_interpolated_matrix->getData()[i] /= sum;
        }
    }
    else if (d_interp_method == "LP")
    {
        int num_elements = d_rotated_reduced_matrices[d_ref_point]->numRows() *
                           d_rotated_reduced_matrices[d_ref_point]->numColumns();
        for (int i = 0; i < num_elements; i++)
        {
            for (int j = 0; j < rbf.size(); j++)
            {
                log_interpolated_matrix->getData()[i] += d_gammas[j]->getData()[i] * rbf[j];
            }
        }
    }
    return std::unique_ptr<Matrix>(log_interpolated_matrix);
}
 
std::shared_ptr<Matrix> MatrixInterpolator::interpolateSPDMatrix(
    const Vector & point)
{
    if (d_gammas.size() == 0)
    {
        // Diagonalize X to work towards obtaining X^-1/2
        EigenPair ref_reduced_matrix_eigenpair = SymmetricRightEigenSolve(
                    *d_rotated_reduced_matrices[d_ref_point]);
        Matrix* ref_reduced_matrix_ev = ref_reduced_matrix_eigenpair.ev;
 
        Matrix* ref_reduced_matrix_sqrt_eigs = new Matrix(
            ref_reduced_matrix_eigenpair.eigs.size(),
            ref_reduced_matrix_eigenpair.eigs.size(), false);
        for (int i = 0; i < ref_reduced_matrix_eigenpair.eigs.size(); i++)
        {
            ref_reduced_matrix_sqrt_eigs->item(i, i) =
                std::sqrt(ref_reduced_matrix_eigenpair.eigs[i]);
        }
 
        Matrix ref_reduced_matrix_ev_inv(*ref_reduced_matrix_ev);
        ref_reduced_matrix_ev->inverse(ref_reduced_matrix_ev_inv);
 
        // Obtain X^1/2
        std::unique_ptr<Matrix> ref_reduced_matrix_ev_mult_sqrt_eig =
            ref_reduced_matrix_ev->mult(*ref_reduced_matrix_sqrt_eigs);
        d_x_half_power = ref_reduced_matrix_ev_mult_sqrt_eig->mult(
                             ref_reduced_matrix_ev_inv);
 
        Matrix x_half_power_inv(*d_x_half_power);
 
        // Obtain X^-1/2
        d_x_half_power->inverse(x_half_power_inv);
 
        delete ref_reduced_matrix_ev;
        delete ref_reduced_matrix_sqrt_eigs;
 
        // Obtain gammas for all points in the database.
        for (int i = 0; i < d_parameter_points.size(); i++)
        {
            // For the ref point, gamma is the zero matrix
            if (i == d_ref_point)
            {
                std::shared_ptr<Matrix> gamma(new Matrix(x_half_power_inv.numRows(),
                                              x_half_power_inv.numColumns(),
                                              x_half_power_inv.distributed()));
                d_gammas.push_back(gamma);
            }
            else
            {
                std::unique_ptr<Matrix> x_half_power_inv_mult_y = x_half_power_inv.mult(
                            *d_rotated_reduced_matrices[i]);
 
                // Obtain X^-1/2*Y*X^-1/2
                std::unique_ptr<Matrix> x_half_power_inv_mult_y_mult_x_half_power_inv =
                    x_half_power_inv_mult_y->mult(x_half_power_inv);
 
                // Diagonalize X^-1/2*Y*X^-1/2 to obtain the log of this matrix.
                // Diagonalize YX^-1 to obtain log of this matrix.
                // Following https://en.wikipedia.org/wiki/Logarithm_of_a_matrix
                // 1. Diagonalize M to obtain M' = V^-1*M*V. M' are the eigenvalues
                // of M and V are the eigenvectors of M.
                // 2. log M = V(log M')V^-1
                EigenPair log_eigenpair = SymmetricRightEigenSolve(
                                              *x_half_power_inv_mult_y_mult_x_half_power_inv);
                Matrix* log_ev = log_eigenpair.ev;
 
                Matrix* log_eigs = new Matrix(log_eigenpair.eigs.size(),
                                              log_eigenpair.eigs.size(), false);
                for (int i = 0; i < log_eigenpair.eigs.size(); i++)
                {
                    if (log_eigenpair.eigs[i] < 0)
                    {
                        if (d_rank == 0) std::cout << "Some eigenvalues of this matrix are negative, "
                                                       <<
                                                       "which leads to NaN values when taking the log. Aborting." << std::endl;
                        CAROM_VERIFY(log_eigenpair.eigs[i] > 0);
                    }
                    log_eigs->item(i, i) = std::log(log_eigenpair.eigs[i]);
                }
 
                // Invert matrix.
                Matrix log_ev_inv(*log_ev);
                log_ev->inverse(log_ev_inv);
 
                // Perform log mapping.
                std::unique_ptr<Matrix> log_ev_mult_log_eig = log_ev->mult(*log_eigs);
                std::shared_ptr<Matrix> gamma = log_ev_mult_log_eig->mult(log_ev_inv);
                d_gammas.push_back(gamma);
 
                delete log_ev;
                delete log_eigs;
            }
        }
 
        // Obtain lambda for the P interpolation matrix
        obtainLambda();
    }
 
    // Obtain distances from database points to new point
    std::vector<double> rbf = obtainRBFToTrainingPoints(d_parameter_points,
                              d_interp_method,
                              d_rbf, d_epsilon, point);
 
    // Interpolate gammas to get gamma for new point
    std::unique_ptr<Matrix> log_interpolated_matrix = obtainLogInterpolatedMatrix(
                rbf);
 
    // Diagonalize the new gamma so we can exponentiate it
    // Diagonalize X to obtain exp(X) of this matrix.
    // Following https://en.wikipedia.org/wiki/Matrix_exponential
    // 1. Diagonalize M to obtain M' = V^-1*M*V. M' are the eigenvalues
    // of M and V are the eigenvectors of M.
    // 2. exp M = V(exp M')V^-1
    EigenPair exp_eigenpair = SymmetricRightEigenSolve(*log_interpolated_matrix);
 
    Matrix* exp_ev = exp_eigenpair.ev;
 
    Matrix* exp_eigs = new Matrix(exp_eigenpair.eigs.size(),
                                  exp_eigenpair.eigs.size(), false);
    for (int i = 0; i < exp_eigenpair.eigs.size(); i++)
    {
        exp_eigs->item(i, i) = std::exp(exp_eigenpair.eigs[i]);
    }
 
    Matrix exp_ev_inv(*exp_ev);
    exp_ev->inverse(exp_ev_inv);
    std::unique_ptr<Matrix> exp_ev_mult_exp_eig = exp_ev->mult(*exp_eigs);
 
    // Exponentiate gamma
    std::unique_ptr<Matrix> exp_gamma = exp_ev_mult_exp_eig->mult(exp_ev_inv);
 
    delete exp_ev;
    delete exp_eigs;
 
    // Obtain exp mapping by doing X^1/2*exp(gamma)*X^1/2
    std::unique_ptr<Matrix> x_half_power_mult_exp_gamma = d_x_half_power->mult(
                *exp_gamma);
    std::shared_ptr<Matrix> interpolated_matrix = x_half_power_mult_exp_gamma->mult(
                *d_x_half_power);
 
    return interpolated_matrix;
 
}
 
std::shared_ptr<Matrix> MatrixInterpolator::interpolateNonSingularMatrix(
    const Vector & point)
{
    if (d_gammas.size() == 0)
    {
        // Invert X
        Matrix ref_matrix_inv(*d_rotated_reduced_matrices[d_ref_point]);
        d_rotated_reduced_matrices[d_ref_point]->inverse(ref_matrix_inv);
 
        for (int i = 0; i < d_parameter_points.size(); i++)
        {
            // For ref_point, gamma is the zero matrix
            if (i == d_ref_point)
            {
                std::shared_ptr<Matrix> gamma(new Matrix(ref_matrix_inv.numRows(),
                                              ref_matrix_inv.numColumns(), ref_matrix_inv.distributed()));
                d_gammas.push_back(gamma);
            }
            else
            {
                std::unique_ptr<Matrix> y_mult_ref_matrix_inv =
                    d_rotated_reduced_matrices[i]->mult(ref_matrix_inv);
 
                // Diagonalize YX^-1 to obtain log of this matrix.
                // Following https://en.wikipedia.org/wiki/Logarithm_of_a_matrix
                // 1. Diagonalize M to obtain M' = V^-1*M*V. M' are the eigenvalues
                // of M and V are the eigenvectors of M.
                // 2. log M = V(log M')V^-1
                EigenPair log_eigenpair = SymmetricRightEigenSolve(*y_mult_ref_matrix_inv);
                Matrix* log_ev = log_eigenpair.ev;
 
                Matrix* log_eigs = new Matrix(log_eigenpair.eigs.size(),
                                              log_eigenpair.eigs.size(), false);
                for (int i = 0; i < log_eigenpair.eigs.size(); i++)
                {
                    if (log_eigenpair.eigs[i] < 0)
                    {
                        if (d_rank == 0) std::cout << "Some eigenvalues of this matrix are " <<
                                                       "negative, which leads to NaN values when taking the log. Aborting." <<
                                                       std::endl;
                        CAROM_VERIFY(log_eigenpair.eigs[i] > 0);
                    }
                    log_eigs->item(i, i) = std::log(log_eigenpair.eigs[i]);
                }
 
                Matrix log_ev_inv(log_ev->numRows(), log_ev->numColumns(), false);
                log_ev->inverse(log_ev_inv);
 
                // Perform log mapping.
                std::unique_ptr<Matrix> log_ev_mult_log_eig = log_ev->mult(*log_eigs);
                std::shared_ptr<Matrix> gamma = log_ev_mult_log_eig->mult(log_ev_inv);
                d_gammas.push_back(gamma);
 
                delete log_ev;
                delete log_eigs;
            }
        }
 
        // Obtain lambda for the P interpolation matrix
        obtainLambda();
    }
 
    // Obtain distances from database points to new point
    std::vector<double> rbf = obtainRBFToTrainingPoints(d_parameter_points,
                              d_interp_method, d_rbf, d_epsilon, point);
 
    // Interpolate gammas to get gamma for new point
    std::unique_ptr<Matrix> log_interpolated_matrix = obtainLogInterpolatedMatrix(
                rbf);
 
    // Diagonalize the new gamma so we can exponentiate it
    // Diagonalize X to obtain exp(X) of this matrix.
    // Following https://en.wikipedia.org/wiki/Matrix_exponential
    // 1. Diagonalize M to obtain M' = V^-1*M*V. M' are the eigenvalues
    // of M and V are the eigenvectors of M.
    // 2. exp M = V(exp M')V^-1
    EigenPair exp_eigenpair = SymmetricRightEigenSolve(*log_interpolated_matrix);
    Matrix* exp_ev = exp_eigenpair.ev;
    Matrix* exp_ev_inv = NULL;
 
    Matrix* exp_eigs = new Matrix(exp_eigenpair.eigs.size(),
                                  exp_eigenpair.eigs.size(), false);
    for (int i = 0; i < exp_eigenpair.eigs.size(); i++)
    {
        exp_eigs->item(i, i) = std::exp(exp_eigenpair.eigs[i]);
    }
 
    // Invert matrix.
    exp_ev->inverse(*exp_ev_inv);
 
    // Perform log mapping.
    std::unique_ptr<Matrix> exp_ev_mult_exp_eig = exp_ev->mult(*exp_eigs);
 
    // Exponentiate gamma
    std::unique_ptr<Matrix> exp_gamma = exp_ev_mult_exp_eig->mult(*exp_ev_inv);
 
    delete exp_ev;
    delete exp_ev_inv;
    delete exp_eigs;
 
    // Obtain exp mapping by doing exp(gamma)*X
    std::shared_ptr<Matrix> interpolated_matrix = exp_gamma->mult(
                *d_rotated_reduced_matrices[d_ref_point]);
 
    return interpolated_matrix;
}
 
std::shared_ptr<Matrix> MatrixInterpolator::interpolateMatrix(
    const Vector & point)
{
    if (d_gammas.size() == 0)
    {
        for (int i = 0; i < d_parameter_points.size(); i++)
        {
            // For ref point, gamma is the zero matrix.
            if (i == d_ref_point)
            {
                std::shared_ptr<Matrix> gamma(new Matrix(
                                                  d_rotated_reduced_matrices[d_ref_point]->numRows(),
                                                  d_rotated_reduced_matrices[d_ref_point]->numColumns(),
                                                  d_rotated_reduced_matrices[d_ref_point]->distributed()));
                d_gammas.push_back(gamma);
            }
            else
            {
                // Gamma is Y - X
                std::shared_ptr<Matrix> gamma(new Matrix(*d_rotated_reduced_matrices[i]));
                *gamma -= *d_rotated_reduced_matrices[d_ref_point];
                d_gammas.push_back(gamma);
            }
        }
 
        // Obtain lambda for the P interpolation matrix
        obtainLambda();
    }
    // Obtain distances from database points to new point
    std::vector<double> rbf = obtainRBFToTrainingPoints(d_parameter_points,
                              d_interp_method,
                              d_rbf, d_epsilon, point);
 
    // Interpolate gammas to get gamma for new point
    std::shared_ptr<Matrix> interpolated_matrix(obtainLogInterpolatedMatrix(rbf));
 
    if (d_compute_gradients)
    {
        if(d_interp_method == "LS")
        {
            for (int i = 0; i < point.dim(); i++)
            {
                std::vector<double> rbf = obtainRBFGradientToTrainingPoints(d_parameter_points,
                                          d_interp_method,
                                          d_rbf, d_epsilon, point, i);
 
                std::shared_ptr<Matrix> gradient_matrix(obtainLogInterpolatedMatrix(rbf));
                d_interpolation_gradient.push_back(gradient_matrix);
            }
        }
        else
        {
            CAROM_ERROR("Interpolated gradients are only implemented for \"LS\"");
        }
    }
 
    // The exp mapping is X + the interpolated gamma
    *interpolated_matrix += *d_rotated_reduced_matrices[d_ref_point];
    return interpolated_matrix;
}
}