SG++-Doxygen-Documentation
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Class representing a prewavelet base. More...
#include <PrewaveletBasis.hpp>
Public Member Functions | |
double | eval (LT level, IT index, double p) override |
Evaluate a basis function. More... | |
size_t | getDegree () const override |
Returns the polynomial degree of the basis. More... | |
double | getIntegral (LT level, IT index) override |
returns the integal of the current basis function More... | |
~PrewaveletBasis () override | |
Destructor. More... | |
Public Member Functions inherited from sgpp::base::Basis< LT, IT > | |
virtual | ~Basis () |
Destructor. More... | |
Class representing a prewavelet base.
A prewavelet \(\psi\) is a combination of 5 normal hat functions \(\phi\) with a \(\left[\frac{1}{10}, -\frac{6}{10}, 1, -\frac{6}{10}, \frac{1}{10}\right]\) stamp:
\[ \psi_{(l,i)} = \frac{1}{10}\phi_{(l,i-2)} - \frac{6}{10}\phi_{(l,i-1)} + \phi_{(l,i)} - \frac{6}{10} \phi_{(l,i+1)} + \frac{1}{10}\phi_{(l,i+2)} \]
Next to the border:
\[ \psi_{(l,1)} = \frac{9}{10}\phi_{(l,1)} - \frac{6}{10}\phi_{(l,2)} + \frac{1}{10}\phi_{(l,3)} \]
For level \( l = 1\) the prewavelet and linear basis are equivalent.
bold dots indicate the position of other prewavelet basis functions on the same level, the thin dots representing grid points missing in the sparse grid on that level. Please note that the left and right TWO neighbors are interfering with a specific prewavelet base." The prewavelets form a semi-orthogonal basis. That means \(<\psi_{(i,l)},\psi_{(j,k)}> = 0\) if \(l\neq k\). On the same level, the prewavelets are not orthogonal. This property will ease the calculation of some specific operations, but on the other hand, this advantage is bought with a wider support of the ansatzfunctions.
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inlineoverride |
Destructor.
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inlineoverridevirtual |
Evaluate a basis function.
Has a dependence on the absolute position of grid point and support.
Implements sgpp::base::Basis< LT, IT >.
References level.
Referenced by python.uq.analysis.asgc.ASGCAnalysis.ASGCAnalysis::estimateDensity(), and sgpp::base::GetAffectedBasisFunctions< PrewaveletBasis< unsigned int, unsigned int > >::rec().
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inlineoverridevirtual |
Returns the polynomial degree of the basis.
Implements sgpp::base::Basis< LT, IT >.
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inlineoverridevirtual |
returns the integal of the current basis function
level | level of the basis function |
index | index of the basis function |
Implements sgpp::base::Basis< LT, IT >.