Adaptive compression of large vectors

Abstract

Numerical algorithms for elliptic partial differential equations frequently employ error estimators and adaptive mesh refinement strategies in order to reduce the computational cost. We can extend these techniques to general vectors by splitting the vectors into a hierarchically organized partition of subsets and using appropriate bases to represent the corresponding parts of the vectors. This leads to the concept of \emph{hierarchical vectors}. A hierarchical vector with $m$ subsets and bases of rank $k$ requires $mk$ units of storage, and typical operations like the evaluation of norms and inner products or linear updates can be carried out in $\mathcal{O}(mk^2)$ operations. Using an auxiliary basis, the product of a hierarchical vector and an $\mathcal{H}^2$-matrix can also be computed in $\mathcal{O}(mk^2)$ operations, and if the result admits an approximation with $\widetilde m$ subsets in the original basis, this approximation can be obtained in $\mathcal{O}((m+\widetilde m)k^2)$ operations. Since it is possible to compute the corresponding approximation error exactly, sophisticated error control strategies can be used to ensure the optimal compression. Possible applications of hierarchical vectors include the approximation of eigenvectors and the solution of time-dependent problems with moving local irregularities.