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sgpp::base::AlgorithmEvaluation< BASIS > Class Template Reference

Basic algorithm for getting all affected basis functions. More...

#include <AlgorithmEvaluation.hpp>

Public Member Functions

 AlgorithmEvaluation (GridStorage &storage)
 
double operator() (BASIS &basis, const DataVector &point, const DataVector &alpha)
 Returns evaluations of all basis functions that are non-zero at a given evaluation point. More...
 
 ~AlgorithmEvaluation ()
 

Protected Member Functions

void rec (BASIS &basis, const DataVector &point, size_t current_dim, double value, GridStorage::grid_iterator &working, index_t *source, const DataVector &alpha, double &result)
 Recursive traversal of the "tree" of basis functions for evaluation, used in operator(). More...
 

Protected Attributes

GridStoragestorage
 

Detailed Description

template<class BASIS>
class sgpp::base::AlgorithmEvaluation< BASIS >

Basic algorithm for getting all affected basis functions.

This implicitly assumes a tensor-product approach and local support. No grid points on the border are supported.

The main idea behind this algorithm is to spend as few function evaluations as possible. Assume a regular sparse grid level 3 in two dimensions with the sparse grid basis \(\Phi:=\{\phi_i(x), i=1,\ldots,N\}\). Then the tableau of subspaces looks as follows:

GetAffectedBasisFunctions_subspaces.png
Tableau of subspaces for a regular sparse grid level 3

You could evaluate the function \( f_N(x) = \sum_{i=1}^N \alpha_i \phi_i(x)\) for all basis functions \(\phi_i(x)\), multiply them with the surplus and add them up. In \(d\) dimensions this would lead to \(N\) evaluations of \(d\) one-dimensional basis functions each.

A better way is to (recursively) look at each subspace, as only one basis function per subspace can be non-zero (partially disjunct supports):

GetAffectedBasisFunctions_subspaces_affectedBasisFunctions.png
Traversal of subspaces for evaluation

This can be done recursively in both the dimension and the level. In each subspace the basis function concerned can be identified via a few index calculations and evaluated at the given point in the domain.

Even better would be to save further function evaluations and to reuse intermediate values obtained by the evaluation of one-dimensional basis functions, see the following figure.

GetAffectedBasisFunctions_subspaces_affectedBasisFunctions_recursive.png
Minimize the number of evaluations

Descending recursively in the d-th dimension, one can propagate the value of the intermediate function evaluation for the first d-1 dimensions that have already been looked at.

Constructor & Destructor Documentation

template<class BASIS >
sgpp::base::AlgorithmEvaluation< BASIS >::AlgorithmEvaluation ( GridStorage storage)
inlineexplicit
template<class BASIS >
sgpp::base::AlgorithmEvaluation< BASIS >::~AlgorithmEvaluation ( )
inline

Member Function Documentation

template<class BASIS >
double sgpp::base::AlgorithmEvaluation< BASIS >::operator() ( BASIS &  basis,
const DataVector point,
const DataVector alpha 
)
inline

Returns evaluations of all basis functions that are non-zero at a given evaluation point.

For a given evaluation point \(x\), it stores tuples (std::pair) of \((i,\phi_i(x))\) in the result vector for all basis functions that are non-zero. If one wants to evaluate \(f_N(x)\), one only has to compute

\[ \sum_{r\in\mathbf{result}} \alpha[r\rightarrow\mathbf{first}] \cdot r\rightarrow\mathbf{second}. \]

Parameters
basisa sparse grid basis
pointevaluation point within the domain
alphathe sparse grid's coefficients
Returns
result result of the function evaluation

References chess::dim, sgpp::base::HashGridStorage::getBoundingBox(), sgpp::base::HashGridStorage::getDimension(), sgpp::base::BoundingBox::isContainingPoint(), sgpp::base::AlgorithmEvaluation< BASIS >::rec(), sgpp::base::AlgorithmEvaluation< BASIS >::storage, and sgpp::base::BoundingBox::transformPointToUnitCube().

template<class BASIS >
void sgpp::base::AlgorithmEvaluation< BASIS >::rec ( BASIS &  basis,
const DataVector point,
size_t  current_dim,
double  value,
GridStorage::grid_iterator working,
index_t source,
const DataVector alpha,
double &  result 
)
inlineprotected

Recursive traversal of the "tree" of basis functions for evaluation, used in operator().

For a given evaluation point \(x\), it stores tuples (std::pair) of \((i,\phi_i(x))\) in the result vector for all basis functions that are non-zero.

Parameters
basisa sparse grid basis
pointevaluation point within the domain
current_dimthe dimension currently looked at (recursion parameter)
valuethe value of the evaluation of the current basis function up to (excluding) dimension current_dim (product of the evaluations of the one-dimensional ones)
workingiterator working on the GridStorage of the basis
sourcearray of indices for each dimension (identifying the indices of the current grid point)
alphathe spars grid's ansatzfunctions coefficients
resultreference to a double into which the result should be stored

References sgpp::base::HashGridIterator::get(), sgpp::base::HashGridStorage::getDimension(), sgpp::base::HashGridIterator::hint(), sgpp::base::HashGridStorage::isInvalidSequenceNumber(), sgpp::base::HashGridIterator::leftChild(), sgpp::base::HashGridIterator::resetToLevelOne(), sgpp::base::HashGridIterator::rightChild(), and sgpp::base::HashGridIterator::seq().

Referenced by sgpp::base::AlgorithmEvaluation< BASIS >::operator()().

Member Data Documentation

template<class BASIS >
GridStorage& sgpp::base::AlgorithmEvaluation< BASIS >::storage
protected

The documentation for this class was generated from the following file: