document: Adaptive wavelet methods . II . Beyond the elliptic case This paper is concerned with the design and analysis of adaptive wavelet methods for systems of operator equations . Its main accomplishment is to extend the range of applicability of the adaptive wavelet-based method developed previously for symmetric positive definite problems to indefinite or unsymmetric systems of operator equations . This is accomplished by first introducing techniques -LRB- such as the least squares formulation developed previously -RRB- that transform the original -LRB- continuous -RRB- problem into an equivalent infinite system of equations which is now well-posed in the Euclidean metric . It is then shown how to utilize adaptive techniques to solve the resulting infinite system of equations . It is shown that for a wide range of problems , this new adaptive method performs with asymptotically optimal complexity , i.e. , it recovers an approximate solution with desired accuracy at a computational expense that stays proportional to the number of terms in a corresponding wavelet-best N-term approximation . An important advantage of this adaptive approach is that it automatically stabilizes the numerical procedure so that , for instance , compatibility constraints on the choice of trial spaces , like the LBB condition , no longer arise
doc_bio_tags: B I I O O O O O B I O O O O O O O O O O B I I O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
extractive_keyphrases: adaptive wavelet methods elliptic case operator equations least squares formulation euclidean metric asymptotically optimal complexity
abstractive_keyphrases: n-term approximation