Introduction

Thanks to the achievements of numerical analysis and scientific computing in the last decades, numerical simulations in engineering and applied sciences have gained an ever increasing importance. In several fields, from aerospace and mechanical engineering to life sciences, numerical simulations of partial differential equations (PDEs) currently provide a virtual platform ancillary to material/mechanics testing or

in vitro

experiments. These are in turn useful either for (

i

) the prediction of input/output response or (

ii

) the design and optimization of a system [237, 238, 216]. The constant increase of available computational power, accompanied by the progressive improvement of algorithms for solving large linear systems, make nowadays possible the numerical simulation of complex, multiscale and multiphysics phenomena by means of

high-fidelity

(or

full-order

) approximation techniques such as the finite element method, finite volumes, finite differences or spectral methods. However, this might be quite demanding, because it involves up to

O

(10

6

– 109

9

) degrees of freedom and several hours (or even days) of CPU time, also on powerful hardware parallel architectures.