Wavelet sparse approximate inverse preconditioners

We show how to use wavelet compression ideas to improve the performance of approximate inverse preconditioners. Our main idea is to first transform the inverse of the coefficient matrix into a wavelet basis, before applying standard approximate inverse techniques. In this process, smoothness in the entries of A(-1) are converted into small wavelet coefficients, thus allowing a more efficient approximate inverse approximation. We shall justify theoretically and numerically that our approach is effective for matrices with smooth inverses.