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Evaluation of Integral Expressions using Orthogonal Daubechies Wavelets

The last example shows how to evaluate integral expressions

$$f({\bf x})=\int_{\bf y} K({\bf x},{\bf y}) u({\bf y}) d{\bf y}$$

with smooth kernel $K$ in an efficient way. Numerical analysis [7] shows that regular sparse grids lead to optimal trial spaces for the approximations of $K$ as well as $u$ if a high accuracy of $f$ is the primal goal. Currently, the member function for the involved matrix-vector multiply is implemented for LevelAdaptiveData and orthogonal (Daubechies-) wavelets, only. The restriction to orthogonal wavelets comes from mathematical reasons: $K$ and $u$ must be represented by biorthogonal, $L^2$-integrable pairs of bases. Therefore, the Interpolets are ruled out. In principle, Lifting-Interpolets would be an alternative. But then we would have to provide full support (filter masks, quadrature formulae, ...) for the dual wavelets of the Lifting-Interpolets as well. On the other hand, since the Daubechies wavelets are not fully supported by e.g. AdaptiveData, it didn't made much sense to implement the matrix-vector multiply there; even though this would be relatively easy.

The code for the present example was used to assess the theoretically predicted accuracy of our optimized scheme. The integral expression is evaluated on a sequence of finer and finer regular sparse grids, and compared to the true result. As expected, the error decays with a rate of $2^3=8$ which is optimal for the employed Daubechies wavelets of order 3. The source code is SparseIntegralError.cc.