We introduce multilevel augmentation methods for solving operator equa-tions based on direct sum decompositions of the range space of the operator and the solution space of the operator equation and a matrix sphtting scheme. We establish a general setting/or the analysis of these methods, showing that the methods yield ap-proximate solutions ol the same convergence order as the best approximation from the subspace. These augmentation methods allow US to develop, fast, accurate and stable nonconventional numerical algorithms for solving operator equations. In particular. For second kind equations, special splitting techniques are proposed to develop such algo-rithms. These algorithms are then applied to solve the linear systems resulting form matrix compression schemes using wavelet-like functions|or solving Fredholm integral equations of the second kind. For this special case. a complete analysis for computa-tional complexity and convergence order is presented. Numerical examples are included to demonstrate the efficiency and accuracy of the methods. In these examples we use the proposed augmentation method to solve large scale linear systems resulting form the recently developed wavelet Galerkin methods and fast collocation methods applied to integral equations of the second kind. Our numerical results confirm that this aug-mentation method is particularly efficient for solving large scale linear systems induced from wavelet compression schemes.