Abstract:

Enormous progress has been made in the construction and analysis of adaptive wavelet methods in the recent years. Cohen, Dahmen, and DeVore showed that such methods converge for a wide class of operator equations, both linear and nonlinear. Moreover, they showed that the rate of convergence is asymptotically optimal and that the methods are asymptotically optimally efficient.
The quantitative behaviour of such methods of course depends on the choice of the wavelet bases used, in particular on the condition number of these bases. It has been observed that compactly supported biorthogonal spline wavelets (for which these adaptive methods are designed for) give rise to condition numbers that cause serious problems in practical applications. An alternative would be the use of semiorthogonal wavelets which are known to have good condition numbers. However, the above mentioned methods do require compactly supported dual wavelets, which in general is not the case for semiorthogonal spline wavelets.
In this paper, we focus on a core ingredient of adaptive wavelet methods for nonlinear problems, namely the adaptive evaluation of nonlinear functions. We present an efficient adaptive method for approximately evaluating nonlinear functions of wavelet expansions using semiorthogonal spline wavelets. This is achieved by modifying and extending a method for compactly supported biorthogonal wavelets by Dahmen, Schneider and Xu.
Using the semiorthogonality, we only need compact support of the primal basis functions. Starting with an adaptive quasi-interpolant in terms of the primal scaling functions, we perform then a fast change of basis into a linear combination of dual scaling functions. Finally, a fast decomposition algorithm is performed, which uses only the finitely supported primal refinement coefficients, to obtain the desired representation in terms of the dual wavelets.
In particular, this paper shows that semiorthogonal spline wavelets can be used in the above mentioned framework of adaptive wavelet schemes for operator equations.

BiBTeX entry:

@article{BiUr04,
    author = {K. Bittner and K. Urban},
    title  = {Adaptive Wavelet Methods using Semiorthogonal Spline Wavelets:  
                   Sparse Evaluation of Nonlinear Functions},
    journal= {},
    volume = {},
    pages  = {},
    year   = {} 
}