Forward Mode Hessian Sparsity Patterns

Loading [MathJax]/extensions/TeX/newcommand.js

$\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }$ This is cppad-20221105 documentation. Here is a link to its current documentation . Forward Mode Hessian Sparsity Patterns
Syntax

f.for_hes_sparsity(

    select_domain, select_range, internal_bool, pattern_out

)

Purpose
We use

$F : \B{R}^n \rightarrow \B{R}^m$ to denote the AD function corresponding to the operation sequence stored in f . Fix a diagonal matrix

$D \in \B{R}^{n \times n}$ , a vector

$s \in \B{R}^m$ and define the function

$H(x) = D ( s^\R{T} F )^{(2)} ( x ) D$ Given the sparsity for

$D$ and

$s$ , for_hes_sparsity computes a sparsity pattern for

$H(x)$ .

x
Note that the sparsity pattern

$H(x)$ corresponds to the operation sequence stored in f and does not depend on the argument x .

BoolVector
The type BoolVector is a SimpleVector class with elements of type bool.

SizeVector
The type SizeVector is a SimpleVector class with elements of type size_t.

f
The object f has prototype



    ADFun<Base> f

select_domain
The argument select_domain has prototype



    const BoolVector& select_domain

It has size

$n$ and specifies which components of the diagonal of

$D$ are non-zero; i.e., select_domain[j] is true if and only if

$D_{j,j}$ is possibly non-zero.

select_range
The argument select_range has prototype



    const BoolVector& select_range

It has size

$m$ and specifies which components of the vector

$s$ are non-zero; i.e., select_range[i] is true if and only if

$s_i$ is possibly non-zero.

internal_bool
If this is true, calculations are done with sets represented by a vector of boolean values. Otherwise, a vector of sets of integers is used.

pattern_out
This argument has prototype



    sparse_rc<SizeVector>& pattern_out

This input value of pattern_out does not matter. Upon return pattern_out is a sparsity pattern for

$H(x)$ .

Sparsity for Entire Hessian
Suppose that

$R$ is the

$n \times n$ identity matrix. In this case, pattern_out is a sparsity pattern for

$(s^\R{T} F) F^{(2)} ( x )$ .

Algorithm
See Algorithm II in Computing sparse Hessians with automatic differentiation by Andrea Walther. Note that s provides the information so that 'dead ends' are not included in the sparsity pattern.

Example
The file for_hes_sparsity.cpp contains an example and test of this operation.

Input File: include/cppad/core/for_hes_sparsity.hpp