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The volume contains a selection of original contributions by the participants of an INDAM workshop, held in Rome in May 2016. The book is dedicated to interactions between Numerical Algorithmics and Mathematical Analysis of PDE's. A wide breadth of applications, ranging from post-Newtonian approximations of General Relativity to mathematical modeling in biology, is preferred. Moreover, topics including practical two-dimensional computation of macroscopic equations, inverse design algorithms relying on filtering techniques and sound-wave propagation in rarefied gases are studied as well. The book may be considered as a state of the art on certain subjects, and a snapshot of recent research advances on other ones.