feynman_kac_1d.cpp

# include <cstdlib>
# include <iostream>
# include <iomanip>
# include <cmath>
# include <ctime>

using namespace std;

int main ( int argc, char **argv );
double potential ( double a, double x );
double r8_abs ( double x );
double r8_uniform_01 ( int *seed );
void timestamp ( );

//****************************************************************************80

int main ( int argc, char **argv )

//****************************************************************************80
//
//  Purpose:
//
//    MAIN is the main program for FEYNMAN_KAC_1D.
//
//  Discussion:
//
//    This program is derived from section 2.5, exercise 2.2 of Petersen 
//    and Arbenz.
//
//    The problem is to determine the solution U(X) of the following 
//    partial differential equation:
//
//      (1/2) Laplacian U - V(X) * U = 0,
//
//    inside the domain D:
//
//      D = { X | (X/A)^2 <= 1 }
// 
//    with the boundary condition U(boundary(D)) = 1.
//
//    V(X) is the potential function:
//
//      V = 2 * ( (X/A^2)^2 + 1/A^2.
//
//    The analytic solution of this problem is already known:
//
//      U(X) = exp ( (X/A)^2 - 1 ).
//
//    Our method is via the Feynman-Kac Formula.
//
//    The idea is to start from any x in D, and
//    compute x+Wx(t) where 1D Brownian motion
//    Wx is updated each step by sqrt(h)*z,
//    each z is an independent approximately Gaussian 
//    random variable with zero mean and variance 1. 
//
//    Each x1(t) is advanced until x1(t) exits the domain D.  
//
//    Upon its first exit from D, the sample path x1 is stopped and a 
//    new sample path at x is started until N such paths are completed.
//
//    The Feynman-Kac formula gives the solution here as
//
//      U(X) = (1/N) sum(1 <= I <= N) Y(tau_i),
//
//    where
//
//      Y(tau) = exp( -int(s=0..tau) v(x1(s)) ds),
//
//    and tau = first exit time for path x1. 
//
//    The integration procedure is a second order weak accurate method:
//
//      X(t+h)  = x1(t) + sqrt ( h ) * z
//
//    Here Z is an approximately normal univariate Gaussian. 
//
//    An Euler predictor approximates Y at the end of the step
//
//      Y_e     = (1 - h*v(X(t)) * Y(t), 
//
//    A trapezoidal rule completes the step:
//
//      Y(t+h)  = Y(t) - (h/2)*[v(X(t+h))*Y_e + v(X(t))*Y(t)].
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    29 May 2012
//
//  Author:
//
//    Original C 3D version by Wesley Petersen.
//    C++ 1D version by John Burkardt.
//
//  Reference:
//
//    Peter Arbenz, Wesley Petersen,
//    Introduction to Parallel Computing:
//    A Practical Guide with Examples in C,
//    Oxford, 2004,
//    ISBN: 0-19-851577-4,
//    LC: QA76.59.P47.
//
{
  double a = 2.0;
  double chk;
  double dx;
  double err;
  double h = 0.0001;
  int i;
  int it;
  int j;
  int k;
  int n1;
  int n = 10000;
  int n_int;
  int ni;
  double rth;
  int seed = 123456789;
  int steps;
  int steps_ave;
  double sum;
  double test;
  double us;
  double vh;
  double vs;
  double x;
  double x1;
  double w;
  double w_exact;
  double we;
  double wt;

  timestamp ( );

  cout << "\n";
  cout << "FEYNMAN_KAC_1D:\n";
  cout << "  C++ version.\n";
  cout << "\n";
  cout << "  Program parameters:\n";
  cout << "\n";
  cout << "  The calculation takes place inside an interval.\n";
  cout << "  The solution will be estimated at points\n";
  cout << "  on a regular spaced grid within the interval.\n";
  cout << "  Each solution will be estimated by computing " << n << " trajectories\n";
  cout << "  from the point to the boundary.\n";
  cout << "\n";
  cout << "    (X/A)^2 = 1\n";
  cout << "\n";
  cout << "  The interval parameter A is:\n";
  cout << "\n";
  cout << "    A = " << a << "\n";
  cout << "\n";
  cout << "  Path stepsize H = " << h << "\n";
//
//  Choose the spacing so we have about ni points on or in the interval.
//
  ni = 21;

  cout << "\n";
  cout << "  X coordinate discretized by " << ni + 2 << " points\n";
//
//  RTH is the scaled stepsize.
//
  rth = sqrt ( h );

  err = 0.0;
//
//  Loop over the points.
//
  cout << "\n";
  cout << "     I     K       X           W exact";
  cout << "      W Approx        Error      Ave Steps  Test\n";
  cout << "\n";

  k = 0;
  n_int = 0;

  for ( i = 0; i <= ni + 1; i++ )
  {
    x = ( ( double ) ( ni - i     ) * ( - a ) 
        + ( double ) (      i - 1 ) *     a ) 
        / ( double ) ( ni     - 1 );

    k = k + 1;

    test = a * a - x * x;

    if ( test < 0.0 )
    {
      w_exact = 1.0;
      wt = 1.0;
      steps_ave = 0;
      cout << "  " << setw(4) << i
           << "  " << setw(4) << k
           << "  " << setw(12) << x
           << "  " << setw(12) << w_exact
           << "  " << setw(12) << wt
           << "  " << setw(12) << r8_abs ( w_exact - wt )
           << "  " << setw(8) << steps_ave
           << "  " << setw(8) << test << "\n";
      continue;
    }

    n_int = n_int + 1;
//
//  Compute the exact solution at this point (x,y,z).
//
    w_exact = exp ( pow ( x / a, 2 ) - 1.0 );
//
//  Now try to estimate the solution at this point.
//
    wt = 0.0;
    steps = 0;

    for ( it = 1; it <= n; it++ )
    {

      x1 = x;
// 
//  W = exp(-int(s=0..t) v(X)ds) 
//
      w = 1.0;
//
//  CHK is < 1.0 while the point is inside the interval.
//
      chk = 0.0;

      while ( chk < 1.0 )
      {
//
//  Determine DX.
//
        us = r8_uniform_01 ( &seed ) - 0.5;
        if ( us < 0.0 )
        {
          dx = - rth;
        }
        else
        {
          dx = + rth;
        }

        vs = potential ( a, x1 );
//
//  Move to the new point.
//
        x1 = x1 + dx;

        steps = steps + 1;

        vh = potential ( a, x1 );

        we = ( 1.0 - h * vs ) * w;
        w = w - 0.5 * h * ( vh * we + vs * w );

        chk = pow ( x1 / a, 2 );
      }
      wt = wt + w;
    }
//
//   WT is the average of the sum of the different trials.
//
    wt = wt / ( double ) ( n );
    steps_ave = steps / n;
//
//  Add error in WT to the running L2 error in the solution.
//
    err = err + pow ( w_exact - wt, 2 );

    cout << "  " << setw(4) << i
         << "  " << setw(4) << k
         << "  " << setw(12) << x
         << "  " << setw(12) << w_exact
         << "  " << setw(12) << wt
         << "  " << setw(12) << r8_abs ( w_exact - wt )
         << "  " << setw(8) << steps_ave
         << "  " << setw(8) << test << "\n";
  }
//
//  Compute the RMS error for all the points.
//
  err = sqrt ( err / ( double ) ( n_int ) );

  cout << "\n";
  cout << "  RMS absolute error in solution = " << err << "\n";
//
//  Terminate.
//
  cout << "\n";
  cout << "FEYNMAN_KAC_1D:\n";
  cout << "  Normal end of execution.\n";
  cout << "\n";

  timestamp ( );

  return 0;
}
//****************************************************************************80

double potential ( double a, double x )

//****************************************************************************80
//
//  Purpose:
//
//    POTENTIAL evaluates the potential function V(X).
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    29 May 2012
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double A, the parameter that defines the region.
//
//    Input, double X, the coordinate of the point.
//
//    Output, double POTENTIAL, the value of the potential function.
//
{
  double value;

  value = 2.0 * pow ( x / a / a, 2 ) + 1.0 / a / a;

  return value;
}
//****************************************************************************80

double r8_abs ( double x )

//****************************************************************************80
//
//  Purpose:
//
//    R8_ABS returns the absolute value of an R8.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    14 November 2006
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double X, the quantity whose absolute value is desired.
//
//    Output, double R8_ABS, the absolute value of X.
//
{
  double value;

  if ( 0.0 <= x )
  {
    value = + x;
  } 
  else
  {
    value = - x;
  }
  return value;
}
//****************************************************************************80

double r8_uniform_01 ( int *seed )

//****************************************************************************80
//
//  Purpose:
//
//    R8_UNIFORM_01 returns a unit pseudorandom R8.
//
//  Discussion:
//
//    This routine implements the recursion
//
//      seed = ( 16807 * seed ) mod ( 2^31 - 1 )
//      u = seed / ( 2^31 - 1 )
//
//    The integer arithmetic never requires more than 32 bits,
//    including a sign bit.
//
//    If the initial seed is 12345, then the first three computations are
//
//      Input     Output      R8_UNIFORM_01
//      SEED      SEED
//
//         12345   207482415  0.096616
//     207482415  1790989824  0.833995
//    1790989824  2035175616  0.947702
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    11 August 2004
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Paul Bratley, Bennett Fox, Linus Schrage,
//    A Guide to Simulation,
//    Second Edition,
//    Springer, 1987,
//    ISBN: 0387964673,
//    LC: QA76.9.C65.B73.
//
//    Bennett Fox,
//    Algorithm 647:
//    Implementation and Relative Efficiency of Quasirandom
//    Sequence Generators,
//    ACM Transactions on Mathematical Software,
//    Volume 12, Number 4, December 1986, pages 362-376.
//
//    Pierre L'Ecuyer,
//    Random Number Generation,
//    in Handbook of Simulation,
//    edited by Jerry Banks,
//    Wiley, 1998,
//    ISBN: 0471134031,
//    LC: T57.62.H37.
//
//    Peter Lewis, Allen Goodman, James Miller,
//    A Pseudo-Random Number Generator for the System/360,
//    IBM Systems Journal,
//    Volume 8, Number 2, 1969, pages 136-143.
//
//  Parameters:
//
//    Input/output, int *SEED, the "seed" value.  Normally, this
//    value should not be 0.  On output, SEED has been updated.
//
//    Output, double R8_UNIFORM_01, a new pseudorandom variate, 
//    strictly between 0 and 1.
//
{
  int i4_huge = 2147483647;
  int k;
  double r;

  if ( *seed == 0 )
  {
    cerr << "\n";
    cerr << "R8_UNIFORM_01 - Fatal error!\n";
    cerr << "  Input value of SEED = 0.\n";
    exit ( 1 );
  }

  k = *seed / 127773;

  *seed = 16807 * ( *seed - k * 127773 ) - k * 2836;

  if ( *seed < 0 )
  {
    *seed = *seed + i4_huge;
  }
//
//  Although SEED can be represented exactly as a 32 bit integer,
//  it generally cannot be represented exactly as a 32 bit real number.
//
  r = ( double ) ( *seed ) * 4.656612875E-10;

  return r;
}
//****************************************************************************80

void timestamp ( )

//****************************************************************************80
//
//  Purpose:
//
//    TIMESTAMP prints the current YMDHMS date as a time stamp.
//
//  Example:
//
//    31 May 2001 09:45:54 AM
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    08 July 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    None
//
{
# define TIME_SIZE 40

  static char time_buffer[TIME_SIZE];
  const struct std::tm *tm_ptr;
  size_t len;
  std::time_t now;

  now = std::time ( NULL );
  tm_ptr = std::localtime ( &now );

  len = std::strftime ( time_buffer, TIME_SIZE, "%d %B %Y %I:%M:%S %p", tm_ptr );

  std::cout << time_buffer << "\n";

  return;
# undef TIME_SIZE
}