Using Adolc with Multiple Levels of Taping: Example and Test

mul_level_adolc.cpp

@(@\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}} }@)@ This is cppad-20221105 documentation. Here is a link to its current documentation . Using Adolc with Multiple Levels of Taping: Example and Test
Purpose
In this example, we use AD< adouble> > (level two taping), the compute values of the function @(@ f : \B{R}^n \rightarrow \B{R} @)@ where @[@ f(x) = \frac{1}{2} \left( x_0^2 + \cdots + x_{n-1}^2 \right) @]@ We then use Adolc's adouble (level one taping) to compute the directional derivative @[@ f^{(1)} (x) * v = x_0 v_0 + \cdots + x_{n-1} v_{n-1} @]@. where @(@ v \in \B{R}^n @)@. We then use double (no taping) to compute @[@ \frac{d}{dx} \left[ f^{(1)} (x) * v \right] = v @]@ This is only meant as an example of multiple levels of taping. The example hes_times_dir.cpp computes the same value more efficiently by using the identity: @[@ \frac{d}{dx} \left[ f^{(1)} (x) * v \right] = f^{(2)} (x) * v @]@ The example mul_level.cpp computes the same values using AD< AD<double> > and AD<double>.

Memory Management
Adolc uses raw memory arrays that depend on the number of dependent and independent variables. The memory management utility thread_alloc is used to manage this memory allocation.

Configuration Requirement
This example will be compiled and tested provided include_adolc is true on the cmake command line.

Source


// suppress conversion warnings before other includes
# include <cppad/wno_conversion.hpp>
//

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

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

# include <cppad/cppad.hpp>

namespace {
    // f(x) = |x|^2 / 2 = .5 * ( x[0]^2 + ... + x[n-1]^2 )
    template <class Type>
    Type f(const CPPAD_TESTVECTOR(Type)& x)
    {   Type sum;

        sum  = 0.;
        size_t i = size_t(x.size());
        for(i = 0; i < size_t(x.size()); i++)
            sum += x[i] * x[i];

        return .5 * sum;
    }
}

bool mul_level_adolc(void)
{   bool ok = true;                // initialize test result
    using CppAD::thread_alloc;        // The CppAD memory allocator

    typedef adouble           a1type;  // for first level of taping
    typedef CppAD::AD<a1type> a2type; // for second level of taping
    size_t n = 5;                          // number independent variables
    size_t j;

    // 10 times machine epsilon
    double eps = 10. * std::numeric_limits<double>::epsilon();

    CPPAD_TESTVECTOR(double) x(n);
    CPPAD_TESTVECTOR(a1type) a1x(n);
    CPPAD_TESTVECTOR(a2type) a2x(n);

    // Values for the independent variables while taping the function f(x)
    for(j = 0; j < n; j++)
        a2x[j] = double(j);
    // Declare the independent variable for taping f(x)
    CppAD::Independent(a2x);

    // Use AD<adouble> to tape the evaluation of f(x)
    CPPAD_TESTVECTOR(a2type) a2y(1);
    a2y[0] = f(a2x);

    // Declare a1f as the corresponding ADFun<adouble> function f(x)
    // (make sure we do not run zero order forward during constructor)
    CppAD::ADFun<a1type> a1f;
    a1f.Dependent(a2x, a2y);

    // Value of the independent variables whitle taping f'(x) * v
    short tag = 0;
    int keep = 1;
    trace_on(tag, keep);
    for(j = 0; j < n; j++)
        a1x[j] <<= double(j);

    // set the argument value x for computing f'(x) * v
    a1f.Forward(0, a1x);

    // compute f'(x) * v
    CPPAD_TESTVECTOR(a1type) a1v(n);
    CPPAD_TESTVECTOR(a1type) a1df(1);
    for(j = 0; j < n; j++)
        a1v[j] = double(n - j);
    a1df = a1f.Forward(1, a1v);

    // declare Adolc function corresponding to f'(x) * v
    double df;
    a1df[0] >>= df;
    trace_off();

    // compute the d/dx of f'(x) * v = f''(x) * v
    size_t m      = 1;                     // # dependent in f'(x) * v

    // w = new double[capacity] where capacity >= m
    size_t capacity;
    double* w  = thread_alloc::create_array<double>(m, capacity);

    // dw = new double[capacity] where capacity >= n
    double* dw = thread_alloc::create_array<double>(n, capacity);

    w[0]  = 1.;
    fos_reverse(tag, int(m), int(n), w, dw);

    for(j = 0; j < n; j++)
    {   double vj = a1v[j].value();
        ok &= CppAD::NearEqual(dw[j], vj, eps, eps);
    }

    // make memory avaialble for other use by this thread
    thread_alloc::delete_array(w);
    thread_alloc::delete_array(dw);
    return ok;
}

Input File: example/general/mul_level_adolc.cpp