Speaker
Description
Hierarchical matrices or $\mathcal{H}$-matrices are the block low-rank representation of the original matrices and are widely used in fast matrix computations. In this talk, we show that the low-rank blocks of $\mathcal{H}$-matrices can be represented in low precision (precision lower than the working precision) without degrading the overall approximation quality. We provide an explicit rule to decide which precision should be chosen for a particular low-rank block. We propose an adaptive mixed precision algorithm for constructing and storing $\mathcal{H}$-matrices. We also show that use of mixed precision does not compromise the numerical stability and accuracy of the $\mathcal{H}$-matrix-vector product. We perform a wide range of numerical experiments to validate our theoretical results. Our numerical results illustrate that adaptive mixed precision $\mathcal{H}$-matrices can achieve significant storage reductions compared to uniform precision $\mathcal{H}$-matrices, without compromising accuracy.