SG++

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 nonzero 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  
GridStorage &  storage 
Basic algorithm for getting all affected basis functions.
This implicitly assumes a tensorproduct 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:
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 \(d\) onedimensional basis functions each.
A better way is to (recursively) look at each subspace, as only one basis function per subspace can be nonzero (partially disjunct supports):
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 onedimensional basis functions, see the following figure.
Descending recursively in the dth dimension, one can propagate the value of the intermediate function evaluation for the first d1 dimensions that have already been looked at.

inlineexplicit 

inline 

inline 
Returns evaluations of all basis functions that are nonzero 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 nonzero. If one wants to evaluate \(f_N(x)\), one only has to compute
\[ \sum_{r\in\mathbf{result}} \alpha[r\rightarrow\mathbf{first}] \cdot r\rightarrow\mathbf{second}. \]
basis  a sparse grid basis 
point  evaluation point within the domain 
alpha  the sparse grid's coefficients 
References 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().

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 nonzero.
basis  a sparse grid basis 
point  evaluation point within the domain 
current_dim  the dimension currently looked at (recursion parameter) 
value  the value of the evaluation of the current basis function up to (excluding) dimension current_dim (product of the evaluations of the onedimensional ones) 
working  iterator working on the GridStorage of the basis 
source  array of indices for each dimension (identifying the indices of the current grid point) 
alpha  the spars grid's ansatzfunctions coefficients 
result  reference 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()().

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