The Interaction of Numerics and Machines

Hardware and software advances stimulate development of computing concepts and devices. Parallelism in computation is a popular contemporary theme of this interaction. Several studies developing new aspects of this are presented. The first study, noisy sort, shows how intensive use of memories and probably analogue (say holomorphic) versions of associative memories may be employed in the problem of sorting. Then a new notion called synergy and a measure of synergy is described. Synergy characterizes the cooperation or benefit thereof in a parallel mode of operation. Multilevel algorithms (aggregation/disaggregation algorithms) and their parallel implementation are then presented. Two examples of an ultra-fine grain approach to parallelism follow. The first is for certain elliptic spectral problems, where normally the computation of an eigenvalue depends on calculating all the previous ones. By using a special functional analytic property, this dependence is dissolved and each eigenvalue is viewed independently. The second relates to the path integrals of quantum mechanics. An approach to solving differential equations using path integrals, an ultra-fine grain approach to scientific computation is then discussed.