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Taylor's Ode Solver: An Example and Test

Purpose
This is a realistic example using two levels of taping (see mul_level ). The first level of taping uses AD<double> to tape the solution of an ordinary differential equation. This solution is then differentiated with respect to a parameter vector. The second level of taping uses AD< AD<double> > to take derivatives during the solution of the differential equation. These derivatives are used in the application of Taylor's method to the solution of the ODE. The example ode_taylor_adolc.cpp computes the same values using Adolc's type adouble and CppAD's type AD<adouble>.

ODE
For this example the ODE's are defined by the function  h : \R^n \times \R^n \rightarrow \R^n where  \[
     h[ x, y(t, x) ] = 
     \left( \begin{array}{c}
               x_0                     \\
               x_1 y_0 (t, x)          \\
               \vdots                  \\
               x_{n-1} y_{n-2} (t, x)
     \end{array} \right)
     = 
     \left( \begin{array}{c}
               \partial_t y_0 (t , x)      \\
               \partial_t y_1 (t , x)      \\
               \vdots                      \\
               \partial_t y_{n-1} (t , x) 
     \end{array} \right)
\] 
and the initial condition  y(0, x) = 0 . The value of  x is fixed during the solution of the ODE and the function  g : \R^n \rightarrow \R^n is used to define the ODE where  \[
     g(y) = 
     \left( \begin{array}{c}
               x_0     \\
               x_1 y_0 \\
               \vdots  \\
               x_{n-1} y_{n-2} 
     \end{array} \right)
\] 


ODE Solution
The solution for this example can be calculated by starting with the first row and then using the solution for the first row to solve the second and so on. Doing this we obtain  \[
     y(t, x ) =
     \left( \begin{array}{c}
          x_0 t                  \\
          x_1 x_0 t^2 / 2        \\
          \vdots                 \\
          x_{n-1} x_{n-2} \ldots x_0 t^n / n !
     \end{array} \right)
\] 


Derivative of ODE Solution
Differentiating the solution above, with respect to the parameter vector  x , we notice that  \[
\partial_x y(t, x ) =
\left( \begin{array}{cccc}
y_0 (t,x) / x_0      & 0                   & \cdots & 0      \\
y_1 (t,x) / x_0      & y_1 (t,x) / x_1     & 0      & \vdots \\
\vdots               & \vdots              & \ddots & 0      \\
y_{n-1} (t,x) / x_0  & y_{n-1} (t,x) / x_1 & \cdots & y_{n-1} (t,x) / x_{n-1}
\end{array} \right)
\] 



An m-th order Taylor method for approximating the solution of an ordinary differential equations is  \[
     y(t + \Delta t , x) 
     \approx 
     \sum_{k=0}^p \partial_t^k y(t , x ) \frac{ \Delta t^k }{ k ! }
     =
     y^{(0)} (t , x ) + 
     y^{(1)} (t , x ) \Delta t + \cdots + 
     y^{(p)} (t , x ) \Delta t^p
\] 
where the Taylor coefficients  y^{(k)} (t, x) are defined by  \[
     y^{(k)} (t, x) = \partial_t^k y(t , x ) / k !
\] 
We define the function  z(t, x) by the equation  \[
     z ( t , x ) = g[ y ( t , x ) ] = h [ x , y( t , x ) ]
\] 
It follows that  \[
\begin{array}{rcl}
     \partial_t y(t, x) & = & z (t , x) 
     \\
      \partial_t^{k+1} y(t , x) & = & \partial_t^k z (t , x)
     \\
     y^{(k+1)} ( t , x) & = & z^{(k)} (t, x) / (k+1) 
\end{array}
\] 
where   z^{(k)} (t, x) is the k-th order Taylor coefficient for  z(t, x) . In the example below, the Taylor coefficients  \[
     y^{(0)} (t , x) , \ldots , y^{(k)} ( t , x )
\] 
are used to calculate the Taylor coefficient  z^{(k)} ( t , x ) which in turn gives the value for   y^{(k+1)} y ( t , x) .

# include <cppad/cppad.hpp>

// =========================================================================
// define types for each level
namespace { // BEGIN empty namespace
typedef CppAD::AD<double>     ADdouble;
typedef CppAD::AD< ADdouble > ADDdouble;

// -------------------------------------------------------------------------
// class definition for C++ function object that defines ODE
class Ode {
private:
     // copy of a that is set by constructor and used by g(y)
     CPPAD_TEST_VECTOR< ADdouble > x_; 
public:
     // constructor
     Ode( CPPAD_TEST_VECTOR< ADdouble > x) : x_(x)
     { }
     // the function g(y) is evaluated with two levels of taping
     CPPAD_TEST_VECTOR< ADDdouble > operator()
     ( const CPPAD_TEST_VECTOR< ADDdouble > &y) const
     {    size_t n = y.size();
          CPPAD_TEST_VECTOR< ADDdouble > g(n);
          size_t i;
          g[0] = x_[0];
          for(i = 1; i < n; i++)
               g[i] = x_[i] * y[i-1];

          return g;
     }
};

// -------------------------------------------------------------------------
// Routine that uses Taylor's method to solve ordinary differential equaitons
// and allows for algorithmic differentiation of the solution. 
CPPAD_TEST_VECTOR < ADdouble > taylor_ode(
     Ode                     G       ,  // function that defines the ODE
     size_t                  order   ,  // order of Taylor's method used
     size_t                  nstep   ,  // number of steps to take
     ADdouble                &dt     ,  // Delta t for each step
     CPPAD_TEST_VECTOR< ADdouble > &y_ini  )  // y(t) at the initial time
{
     // some temporary indices
     size_t i, k, ell;

     // number of variables in the ODE
     size_t n = y_ini.size();

     // copies of x and g(y) with two levels of taping
     CPPAD_TEST_VECTOR< ADDdouble >   Y(n), Z(n);

     // y, y^{(k)} , z^{(k)}, and y^{(k+1)}
     CPPAD_TEST_VECTOR< ADdouble >  y(n), y_k(n), z_k(n), y_kp(n);
     
     // initialize x
     for(i = 0; i < n; i++)
          y[i] = y_ini[i];

     // loop with respect to each step of Taylors method
     for(ell = 0; ell < nstep; ell++)
     {    // prepare to compute derivatives of in ADdouble
          for(i = 0; i < n; i++)
               Y[i] = y[i];
          CppAD::Independent(Y);

          // evaluate ODE in ADDdouble
          Z = G(Y);

          // define differentiable version of g: X -> Y
          // that computes its derivatives in ADdouble
          CppAD::ADFun<ADdouble> g(Y, Z);

          // Use Taylor's method to take a step
          y_k            = y;     // initialize y^{(k)}
          ADdouble dt_kp = dt;    // initialize dt^(k+1)
          for(k = 0; k <= order; k++)
          {    // evaluate k-th order Taylor coefficient of y
               z_k = g.Forward(k, y_k);
 
               for(i = 0; i < n; i++)
               {    // convert to (k+1)-Taylor coefficient for x
                    y_kp[i] = z_k[i] / ADdouble(k + 1);

                    // add term for to this Taylor coefficient
                    // to solution for y(t, x)
                    y[i]    += y_kp[i] * dt_kp;
               }
               // next power of t
               dt_kp *= dt;
               // next Taylor coefficient
               y_k   = y_kp;
          }
     }
     return y;
}
} // END empty namespace
// ==========================================================================
// Routine that tests alogirhtmic differentiation of solutions computed
// by the routine taylor_ode.
bool ode_taylor(void)
{    // initialize the return value as true  
     bool ok = true;

     // number of components in differential equation
     size_t n = 4;

     // some temporary indices
     size_t i, j;

     // parameter vector in both double and ADdouble
     CPPAD_TEST_VECTOR<double>   x(n);
     CPPAD_TEST_VECTOR<ADdouble> X(n);
     for(i = 0; i < n; i++)
          X[i] = x[i] = double(i + 1);

     // declare the parameters as the independent variable
     CppAD::Independent(X);

     // arguments to taylor_ode 
     Ode G(X);                // function that defines the ODE
     size_t   order = n;      // order of Taylor's method used
     size_t   nstep = 2;      // number of steps to take
     ADdouble DT    = 1.;     // Delta t for each step
     // value of y(t, x) at the initial time
     CPPAD_TEST_VECTOR< ADdouble > Y_INI(n);
     for(i = 0; i < n; i++)
          Y_INI[i] = 0.;

     // integrate the differential equation
     CPPAD_TEST_VECTOR< ADdouble > Y_FINAL(n);
     Y_FINAL = taylor_ode(G, order, nstep, DT, Y_INI);

     // define differentiable fucntion object f : A -> Y_FINAL
     // that computes its derivatives in double
     CppAD::ADFun<double> f(X, Y_FINAL);

     // check function values
     double check = 1.;
     double t     = nstep * Value(DT);
     for(i = 0; i < n; i++)
     {    check *= x[i] * t / double(i + 1);
          ok &= CppAD::NearEqual(Value(Y_FINAL[i]), check, 1e-10, 1e-10);
     }

     // evaluate the Jacobian of h at a
     CPPAD_TEST_VECTOR<double> jac = f.Jacobian(x);

     // check Jacobian 
     for(i = 0; i < n; i++)
     {    for(j = 0; j < n; j++)
          {    double jac_ij = jac[i * n + j]; 
               if( i < j )
                    check = 0.;
               else check = Value( Y_FINAL[i] ) / x[j];
               ok &= CppAD::NearEqual(jac_ij, check, 1e-10, 1e-10);
          }
     }
     return ok;
}


Input File: example/ode_taylor.cpp