Choosing an appropriate overlapping decomposition of the given domain, a suitabl e frame system can be constructed easily. Applying it to the given continuous pr oblem yields a discrete, bi-infinite nonlinear system of equations, which is sho wn to be solvable by a damped Richardson iteration method. We then successively introduce all building blocks for the numerical implementation of the iteration method. Here, we concentrate on the evaluation of the discrete nonlinearity, whe re we show that the previously developed auxiliary of tree-structured index sets can be generalized to the wavelet frame setting in a proper way.
This allows an effective numerical treatment of the nonlinearity by so-called aggregated trees . Choosing the error tolerances appropriately, we show that our adaptive scheme is asymptotically optimal with respect to aggregated tree-structured index sets, i.e., it realizes the same convergence rate as the sequence of best N-term fram e approximations of the solution respecting aggregated trees. Moreover, under th e assumption of a sufficiently precise numerical quadrature method, the computat ional cost of our algorithm stays the same order as the number of wavelets used by it.
The theoretical results are widely confirmed by one- and two-dimensional test pr oblems over non-trivial bounded domains.
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