Adaptive methods are well suited for approximating solutions with local singularities. Though adaptivity exploits sparsity features in pre-specified dictionaries and the resulting solutions are optimal in a sense, the "classic" adaptive approaches still scale exponentially with the dimension. Recent developments in the field of structured tensor formats and applications to high-dimensional equations suggest that certain problems can be well approximated over sparse tensor manifolds, potentially reducing the complexity in the dimension to (almost) linear.
In this talk I will introduce general notions of adaptive/non-linear approximation, specifically properties of wavelet bases and why they are well suited for adaptivity. I will discuss the basic ideas of using structured tensor formats to remedy the "curse of dimensionality". Finally, I will present some recent developments in adaptive high-dimensional approximation.
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