ODE Fitting Using Fast Representation
Keywords: ode fitting using fast representation cppad_ipopt_nlp example source

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ODE Fitting Using Fast Representation
`

    # include "ode_problem.hpp"

    namespace {
    	using namespace cppad_ipopt;

    	class FG_fast : public cppad_ipopt_fg_info
    	{
    	private:
    		bool       retape_;
    		SizeVector N_;
    		SizeVector S_;
    	public:
    		// derived class part of constructor
    		FG_fast(bool retape_in, const SizeVector& N)
    		: retape_ (retape_in), N_(N)
    		{	assert( N_[0] == 0 );
    			S_.resize( N_.size() );
    			S_[0] = 0;
    			for(size_t i = 1; i < N_.size(); i++)
    				S_[i] = S_[i-1] + N_[i];
    		}
    		// r^k for k = 0, 1, ..., Nz-1 used for measurements
    		// r^k for k = Nz              use for initial condition
    		// r^k for k = Nz+1, ..., 2*Nz used for trapezoidal approx
    		size_t number_functions(void)
    		{	return Nz + 1 + Nz; }
    		ADVector eval_r(size_t k, const ADVector &u)
    		{	count_eval_r();

    			size_t j;
    			ADVector y(Ny), a(Na);
    			// objective function --------------------------------
    			if( k < Nz )
    			{	// used for measurement with index k+1
    				ADVector r(1); // return value is a scalar
    				// u is [y( s[k+1] ) , a]
    				for(j = 0; j < Ny; j++)
    					y[j] = u[j];
    				for(j = 0; j < Na; j++)
    					a[j] = u[Ny + j];
    				r[0] = eval_H<ADNumber>(k+1, y, a);
    				return r;
    			}
    			// initial condition ---------------------------------
    			if( k == Nz )
    			{	ADVector r(Ny), F(Ny);
    				// u is [y(t), a] at t = 0
    				for(j = 0; j < Ny; j++)
    					y[j] = u[j];
    				for(j = 0; j < Na; j++)
    					a[j] = u[Ny + j];
    				F    = eval_F(a);
    				for(j = 0; j < Ny; j++)
    					r[j]   = y[j] - F[j];
    				return  r;
    			}
    			// trapezoidal approximation -------------------------
    			ADVector ym(Ny), G(Ny), Gm(Ny), r(Ny);
    			// r^k for k = Nz+1, ... , 2*Nz
    			// interval between data samples
    			Number T = s[k-Nz] - s[k-Nz-1];
    			// integration step size
    			Number dt = T / Number( N_[k-Nz] );
    			// u = [ y(t[i-1], a) , y(t[i], a), a )
    			for(j = 0; j < Ny; j++)
    			{	ym[j] = u[j];
    				y[j]  = u[Ny + j];
    			}
    			for(j = 0; j < Na; j++)
    				a[j] = u[2 * Ny + j];
    			Gm  = eval_G(ym, a);
    			G   = eval_G(y,  a);
    			for(j = 0; j < Ny; j++)
    				r[j] = y[j] - ym[j] - (G[j] + Gm[j]) * dt / 2.;
    			return r;
    		}
    		// The operations sequence for r_eval does not depend on u,
    		// hence retape = false should work and be faster.
    		bool retape(size_t k)
    		{	return retape_; }
    		// size of the vector u in eval_r
    		size_t domain_size(size_t k)
    		{	if( k < Nz )
    				return Ny + Na;   // objective function
    			if( k == Nz )
    				return Ny + Na;  // initial value constraint
    			return 2 * Ny + Na;      // trapezodial constraints
    		}
    		// size of the return value from eval_r
    		size_t range_size(size_t k)
    		{	if( k < Nz )
    				return 1;
    			return Ny;
    		}
    		// number of terms that use this value of k
    		size_t number_terms(size_t k)
    		{	if( k <= Nz )
    				return 1;  // r^k used once for k <= Nz
    			// r^k used N_[k-Nz] times for k > Nz
    			return N_[k-Nz];
    		}
    		void index(size_t k, size_t ell, SizeVector& I, SizeVector& J)
    		{	size_t i, j;
    			// # of components of x corresponding to values for y
    			size_t ny_inx = (S_[Nz] + 1) * Ny;
    			// objective function -------------------------------
    			if( k < Nz )
    			{	// index in fg corresponding to objective
    				I[0] = 0;
    				// u = [ y(t, a) , a ]
    				// The first Ny components of u is y(t) at
    				//	t = s[k+1] = t[S_[k+1]]
    				// x indices corresponding to this value of y
    				for(j = 0; j < Ny; j++)
    					J[j] = S_[k + 1] * Ny + j;
    				// components of x correspondig to a
    				for(j = 0; j < Na; j++)
    					J[Ny + j] = ny_inx + j;
    				return;
    			}
    			// initial conditions --------------------------------
    			if( k == Nz )
    			{	// index in fg for inidial condition constraint
    				for(j = 0; j < Ny; j++)
    					I[j] = 1 + j;
    				// u = [ y(t, a) , a ] where t = 0
    				// x indices corresponding to this value of y
    				for(j = 0; j < Ny; j++)
    					J[j] = j;
    				// following that, u contains the vector a
    				for(j = 0; j < Na; j++)
    					J[Ny + j] = ny_inx + j;
    				return;
    			}
    			// trapoziodal approximation -------------------------
    			// index of first grid point in this approximation
    			i = S_[k - Nz - 1]  + ell;
    			// There are Ny difference equations for each time
    			// point.  Add one for the objective function, and Ny
    			// for the initial value constraints.
    			for(j = 0; j < Ny; j++)
    				I[j] = 1 + Ny + i * Ny + j;
    			// u = [ y(t, a) , y(t+dt, a) , a ] at t = t[i]
    			for(j = 0; j < Ny; j++)
    			{	J[j]      = i * Ny  + j; // y^i indices
    				J[Ny + j] = J[j] + Ny;   // y^{i+1} indices
    			}
    			for(j = 0; j < Na; j++)
    				J[2 * Ny + j] = ny_inx + j; // a indices
    		}
    	};

    }

`

Input File: cppad_ipopt/example/ode_fast.hpp