Taylor's Ode Solver: A Multi-Level Adolc Example and Test
Keywords: taylor's ode solver: multi-level adolc example test purpose solution derivative method using ad base_adolc.hpp memory management configuration requirement source

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Taylor's Ode Solver: A Multi-Level Adolc Example and Test
Purpose
This is a realistic example using two levels of AD; see mul_level . The first level uses Adolc's `adouble` type to tape the solution of an ordinary differential equation. This solution is then differentiated with respect to a parameter vector. The second level uses CppAD's type `AD<adouble>` 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 mul_level_ode.cpp computes the same values using `AD<double>` and `AD< AD<double> >`. The example ode_taylor.cpp is a simpler applications of Taylor's method for solving an ODE.
ODE
For this example the ODE's are defined by the function h : R n × R n → R n where h [ x , y ( t , x ) ] = ( x 0 x 1 y 0 ( t , x ) ⋮ x n -1 y n -2 ( t , x ) ) = ( ∂ t y 0 ( t , x ) ∂ t y 1 ( t , x ) ⋮ ∂ t y n -1 ( t , x ) ) 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 → R n is used to define the ODE where g ( y ) = ( x 0 x 1 y 0 ⋮ x n -1 y n -2 )
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 ) = ( x 0 t x 1 x 0 t 2 / 2 ⋮ x n -1 x n -2 … x 0 t n / n ! )
Derivative of ODE Solution
Differentiating the solution above, with respect to the parameter vector x , we notice that ∂ x y ( t , x ) = ( y 0 ( t , x ) / x 0 0 ⋯ 0 y 1 ( t , x ) / x 0 y 1 ( t , x ) / x 1 0 ⋮ ⋮ ⋮ ⋱ 0 y n -1 ( t , x ) / x 0 y n -1 ( t , x ) / x 1 ⋯ y n -1 ( t , x ) / x n -1 )
Taylor's Method Using AD
An `m`-th order Taylor method for approximating the solution of an ordinary differential equations is y ( t + Δ t , x ) ≈ ∑ k = 0 p ∂ t k y ( t , x ) Δ t k k ! = y ( 0 ) ( t , x ) + y ( 1 ) ( t , x ) Δ t + ⋯ + y ( p ) ( t , x ) Δ t p where the Taylor coefficients y ( k ) ( t , x ) are defined by y ( k ) ( t , x ) = ∂ 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 ∂ t y ( t , x ) = z ( t , x ) ∂ t k + 1 y ( t , x ) = ∂ t k z ( t , x ) y ( k + 1 ) ( t , x ) = z ( k ) ( t , x ) / ( k + 1 ) 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 ) , … , 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 ) .
base_adolc.hpp
The file base_adolc.hpp is implements the Base type requirements where `Base` is `adolc`.
Memory Management
Adolc uses raw memory arrays that depend on the number of dependent and independent variables. The thread_alloc memory management utilities create_array and delete_array are used to manage this memory allocation.
Configuration Requirement
This example will be compiled and tested provided that the value ipopt_prefix is specified on the cmake command line.
Source `

    # include <adolc/adouble.h>
    # include <adolc/taping.h>
    # include <adolc/drivers/drivers.h>

    // definitions not in Adolc distribution and required to use CppAD::AD<adouble>
    # include <cppad/example/base_adolc.hpp>

    # include <cppad/cppad.hpp>
    // ==========================================================================
    namespace { // BEGIN empty namespace
    // define types for each level
    typedef adouble           a1type;
    typedef CppAD::AD<a1type> a2type;

    // -------------------------------------------------------------------------
    // 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_TESTVECTOR(a1type) a1x_;
    public:
    	// constructor
    	Ode(const CPPAD_TESTVECTOR(a1type)& a1x) : a1x_(a1x)
    	{ }
    	// the function g(y) is evaluated with two levels of taping
    	CPPAD_TESTVECTOR(a2type) operator()
    	( const CPPAD_TESTVECTOR(a2type)& a2y) const
    	{	size_t n = a2y.size();
    		CPPAD_TESTVECTOR(a2type) a2g(n);
    		size_t i;
    		a2g[0] = a1x_[0];
    		for(i = 1; i < n; i++)
    			a2g[i] = a1x_[i] * a2y[i-1];

    		return a2g;
    	}
    };

    // -------------------------------------------------------------------------
    // Routine that uses Taylor's method to solve ordinary differential equaitons
    // and allows for algorithmic differentiation of the solution.
    CPPAD_TESTVECTOR(a1type) taylor_ode_adolc(
    	Ode                            G       ,  // function that defines the ODE
    	size_t                         order   ,  // order of Taylor's method used
    	size_t                         nstep   ,  // number of steps to take
    	const a1type                   &a1dt   ,  // Delta t for each step
    	const CPPAD_TESTVECTOR(a1type) &a1y_ini)  // y(t) at the initial time
    {
    	// some temporary indices
    	size_t i, k, ell;

    	// number of variables in the ODE
    	size_t n = a1y_ini.size();

    	// copies of x and g(y) with two levels of taping
    	CPPAD_TESTVECTOR(a2type)   a2y(n), Z(n);

    	// y, y^{(k)} , z^{(k)}, and y^{(k+1)}
    	CPPAD_TESTVECTOR(a1type)  a1y(n), a1y_k(n), a1z_k(n), a1y_kp(n);

    	// initialize x
    	for(i = 0; i < n; i++)
    		a1y[i] = a1y_ini[i];

    	// loop with respect to each step of Taylors method
    	for(ell = 0; ell < nstep; ell++)
    	{	// prepare to compute derivatives using a1type
    		for(i = 0; i < n; i++)
    			a2y[i] = a1y[i];
    		CppAD::Independent(a2y);

    		// evaluate ODE using a2type
    		Z = G(a2y);

    		// define differentiable version of g: X -> Y
    		// that computes its derivatives using a1type
    		CppAD::ADFun<a1type> a1g(a2y, Z);

    		// Use Taylor's method to take a step
    		a1y_k            = a1y;     // initialize y^{(k)}
    		a1type dt_kp = a1dt;    // initialize dt^(k+1)
    		for(k = 0; k <= order; k++)
    		{	// evaluate k-th order Taylor coefficient of y
    			a1z_k = a1g.Forward(k, a1y_k);

    			for(i = 0; i < n; i++)
    			{	// convert to (k+1)-Taylor coefficient for x
    				a1y_kp[i] = a1z_k[i] / a1type(k + 1);

    				// add term for to this Taylor coefficient
    				// to solution for y(t, x)
    				a1y[i]    += a1y_kp[i] * dt_kp;
    			}
    			// next power of t
    			dt_kp *= a1dt;
    			// next Taylor coefficient
    			a1y_k   = a1y_kp;
    		}
    	}
    	return a1y;
    }
    } // END empty namespace
    // ==========================================================================
    // Routine that tests algorithmic differentiation of solutions computed
    // by the routine taylor_ode.
    bool mul_level_adolc_ode(void)
    {	bool ok = true;
    	double eps = 100. * std::numeric_limits<double>::epsilon();

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

    	// some temporary indices
    	size_t i, j;

    	// set up for thread_alloc memory allocator
    	using CppAD::thread_alloc; // the allocator
    	size_t capacity;           // capacity of an allocation

    	// the vector x with lenght n (or greater) in double
    	double* x = thread_alloc::create_array<double>(n, capacity);

    	// the vector x with lenght n in a1type
    	CPPAD_TESTVECTOR(a1type) a1x(n);
    	for(i = 0; i < n; i++)
    		a1x[i] = x[i] = double(i + 1);

    	// declare the parameters as the independent variable
    	int tag = 0;                     // Adolc setup
    	int keep = 1;
    	trace_on(tag, keep);
    	for(i = 0; i < n; i++)
    		a1x[i] <<= double(i + 1);  // a1x is independent for adouble type

    	// arguments to taylor_ode_adolc
    	Ode G(a1x);                // function that defines the ODE
    	size_t   order = n;      // order of Taylor's method used
    	size_t   nstep = 2;      // number of steps to take
    	a1type   a1dt  = 1.;     // Delta t for each step
    	// value of y(t, x) at the initial time
    	CPPAD_TESTVECTOR(a1type) a1y_ini(n);
    	for(i = 0; i < n; i++)
    		a1y_ini[i] = 0.;

    	// integrate the differential equation
    	CPPAD_TESTVECTOR(a1type) a1y_final(n);
    	a1y_final = taylor_ode_adolc(G, order, nstep, a1dt, a1y_ini);

    	// declare the differentiable fucntion f : x -> y_final
    	// (corresponding to the tape of adouble operations)
    	double* y_final = thread_alloc::create_array<double>(n, capacity);
    	for(i = 0; i < n; i++)
    		a1y_final[i] >>= y_final[i];
    	trace_off();

    	// check function values
    	double check = 1.;
    	double t     = nstep * a1dt.value();
    	for(i = 0; i < n; i++)
    	{	check *= x[i] * t / double(i + 1);
    		ok &= CppAD::NearEqual(y_final[i], check, eps, eps);
    	}

    	// memory where Jacobian will be returned
    	double* jac_ = thread_alloc::create_array<double>(n * n, capacity);
    	double** jac = thread_alloc::create_array<double*>(n, capacity);
    	for(i = 0; i < n; i++)
    		jac[i] = jac_ + i * n;

    	// evaluate Jacobian of h at a
    	size_t m = n;              // # dependent variables
    	jacobian(tag, int(m), int(n), x, jac);

    	// check Jacobian
    	for(i = 0; i < n; i++)
    	{	for(j = 0; j < n; j++)
    		{	if( i < j )
    				check = 0.;
    			else	check = y_final[i] / x[j];
    			ok &= CppAD::NearEqual(jac[i][j], check, eps, eps);
    		}
    	}

    	// make memroy avaiable for other use by this thread
    	thread_alloc::delete_array(x);
    	thread_alloc::delete_array(y_final);
    	thread_alloc::delete_array(jac_);
    	thread_alloc::delete_array(jac);
    	return ok;
    }

`

Input File: example/mul_level_adolc_ode.cpp