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Up-> CppAD Appendix deprecated cppad_ipopt_nlp ipopt_nlp_ode ipopt_nlp_ode_fast

deprecated-> include_deprecated FunDeprecated CompareChange omp_max_thread TrackNewDel omp_alloc memory_leak epsilon test_vector cppad_ipopt_nlp old_atomic zdouble autotools

cppad_ipopt_nlp-> ipopt_nlp_get_started.cpp ipopt_nlp_ode ipopt_ode_speed.cpp

ipopt_nlp_ode-> ipopt_nlp_ode_problem ipopt_nlp_ode_simple ipopt_nlp_ode_fast ipopt_nlp_ode_run.hpp ipopt_nlp_ode_check.cpp

ipopt_nlp_ode_fast-> ipopt_nlp_ode_fast.hpp

Headings-> Purpose Objective Function ---..Range Indices I(k,0) ---..Domain Indices J(k,0) Initial Condition ---..Range Indices I(k,0) ---..Domain Indices J(k,0) Trapezoidal Approximation ---..Range Indices I(k,0) ---..Domain Indices J(k,0) Source

@(@\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }@)@ ODE Fitting Using Fast Representation
Purpose
In this section we represent a more complex representation of the simultaneous forward and reverse ODE fitting problem (described above). The representation defines the problem using simpler functions that are faster to differentiate (either by hand coding or by using AD).

Objective Function
We use the following representation for the objective function : For @(@ k = 0 , \ldots , Nz - 1 @)@, we define the function @(@ r^k : \B{R}^{Ny+Na} \rightarrow \B{R} @)@ by @[@ \begin{array}{rcl} fg_0 (x) & = & \sum_{i=1}^{Nz} H_i ( y^{S(i)} , a ) \\ fg_0 (x) & = & \sum_{k=0}^{Nz-1} r^k ( u^{k,0} ) \end{array} @]@ where for @(@ k = 0 , \ldots , Nz-1 @)@, @(@ u^{k,0} \in \B{R}^{Ny + Na} @)@ is defined by @(@ u^{k,0} = ( y^{S(k+1)} , a ) @)@

Range Indices I(k,0)
For @(@ k = 0 , \ldots , Nz - 1 @)@, the range index in the vector @(@ fg (x) @)@ corresponding to @(@ r^k ( u^{k,0} ) @)@ is 0. Thus, the range indices are given by @(@ I(k,0) = \{ 0 \} @)@ for @(@ k = 0 , \ldots , Nz-1 @)@.

Domain Indices J(k,0)
For @(@ k = 0 , \ldots , Nz - 1 @)@, the components of the vector @(@ x @)@ corresponding to the vector @(@ u^{k,0} @)@ are @[@ \begin{array}{rcl} u^{k,0} & = & ( y^{S(k+1} , a ) \\ & = & ( x_{Ny * S(k+1)} \; , \; \ldots \; , \; x_{Ny * S(k+1) + Ny - 1} \; , \; x_{Ny * S(Nz) + Ny } \; , \; \ldots \; , \; x_{Ny * S(Nz) + Ny + Na - 1} ) \end{array} @]@ Thus, the domain indices are given by @[@ J(k,0) = \{ Ny * S(k+1) \; , \; \ldots \; , \; Ny * S(k+1) + Ny - 1 \; , \; Ny * S(Nz) + Ny \; , \; \ldots \; , \; Ny * S(Nz) + Ny + Na - 1 \} @]@

Initial Condition
We use the following representation for the initial condition constraint : For @(@ k = Nz @)@ we define the function @(@ r^k : \B{R}^{Ny} \times \B{R}^{Na + Ny} @)@ by @[@ \begin{array}{rcl} 0 & = & fg_i ( x ) = y_i^0 - F_i (a) \\ 0 & = & r_{i-1}^k ( u^{k,0} ) = y_i^0 - F_i(a) \end{array} @]@ where @(@ i = 1 , \ldots , Ny @)@ and where @(@ u^{k,0} \in \B{R}^{Ny + Na} @)@ is defined by @(@ u^{k,0} = ( y^0 , a) @)@.

Range Indices I(k,0)
For @(@ k = Nz @)@, the range index in the vector @(@ fg (x) @)@ corresponding to @(@ r^k ( u^{k,0} ) @)@ are @(@ I(k,0) = \{ 1 , \ldots , Ny \} @)@.

Domain Indices J(k,0)
For @(@ k = Nz @)@, the components of the vector @(@ x @)@ corresponding to the vector @(@ u^{k,0} @)@ are @[@ \begin{array}{rcl} u^{k,0} & = & ( y^0 , a) \\ & = & ( x_0 \; , \; \ldots \; , \; x_{Ny-1} \; , \; x_{Ny * S(Nz) + Ny } \; , \; \ldots \; , \; x_{Ny * S(Nz) + Ny + Na - 1} ) \end{array} @]@ Thus, the domain indices are given by @[@ J(k,0) = \{ 0 \; , \; \ldots \; , \; Ny - 1 \; , \; Ny * S(Nz) + Ny \; , \; \ldots \; , \; Ny * S(Nz) + Ny + Na - 1 \} @]@

Trapezoidal Approximation
We use the following representation for the trapezoidal approximation constraint : For @(@ k = 1 , \ldots , Nz @)@, we define the function @(@ r^{Nz+k} : \B{R}^{2*Ny+Na} \rightarrow \B{R}^{Ny} @)@ by @[@ r^{Nz+k} ( y , w , a ) = y - w - [ G( y , a ) + G( w , a ) ] * \frac{ \Delta t_k }{ 2 } @]@ For @(@ \ell = 0 , \ldots , N(k)-1 @)@, using the notation @(@ i = Ny * S(k-1) + \ell + 1 @)@, the corresponding trapezoidal approximation is represented by @[@ \begin{array}{rcl} 0 & = & fg_{Ny+i} (x) \\ & = & y^i - y^{i-1} - \left[ G( y^i , a ) + G( y^{i-1} , a ) \right] * \frac{\Delta t_k }{ 2 } \\ & = & r^{Nz+k} ( u^{Nz+k , \ell} ) \end{array} @]@ where @(@ u^{Nz+k,\ell} \in \B{R}^{2*Ny + Na} @)@ is defined by @(@ u^{Nz+k,\ell} = ( y^{i-1} , y^i , a) @)@.

Range Indices I(k,0)
For @(@ k = Nz + 1, \ldots , 2*Nz @)@, and @(@ \ell = 0 , \ldots , N(k)-1 @)@, the range index in the vector @(@ fg (x) @)@ corresponding to @(@ r^k ( u^{k,\ell} ) @)@ are @(@ I(k,\ell) = \{ Ny + i , \ldots , 2*Ny + i - 1 \} @)@ where @(@ i = Ny * S(k-1) + \ell + 1 @)@.

Domain Indices J(k,0)
For @(@ k = Nz + 1, \ldots , 2*Nz @)@, and @(@ \ell = 0 , \ldots , N(k)-1 @)@, define @(@ i = Ny * S(k-1) + \ell + 1 @)@. The components of the vector @(@ x @)@ corresponding to the vector @(@ u^{k,\ell} @)@ are (and the function @(@ fg (x) @)@ in cppad_ipopt_nlp ) @[@ \begin{array}{rcl} u^{k, \ell} & = & ( y^{i-1} , y^i , a ) \\ & = & ( x_{Ny * (i-1)} \; , \; \ldots \; , \; x_{Ny * (i+1) - 1} \; , \; x_{Ny * S(Nz) + Ny } \; , \; \ldots \; , \; x_{Ny * S(Nz) + Ny + Na - 1} ) \end{array} @]@ Thus, the domain indices are given by @[@ J(k,\ell) = \{ Ny * (i-1) \; , \; \ldots \; , \; Ny * (i+1) - 1 \; , \; Ny * S(Nz) + Ny \; , \; \ldots \; , \; Ny * S(Nz) + Ny + Na - 1 \} @]@

Source
The file ipopt_nlp_ode_fast.hpp contains source code for this representation of the objective and constraints.

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Input File: cppad_ipopt/example/ode2.omh