This book is a collection of lecture notes for the CIME course on
"Multiscale and Adaptivity: Modeling, Numerics and Applications," held
in Cetraro (Italy), in July 2009. Complex systems arise in several
physical, chemical, and biological processes, in which length and time
scales may span several orders of magnitude. Traditionally, scientists
have focused on methods that are particularly applicable in only one
regime, and knowledge of the system on one scale has been transferred to
another scale only indirectly. Even with modern computer power, the
complexity of such systems precludes their being treated directly with
traditional tools, and new mathematical and computational instruments
have had to be developed to tackle such problems. The outstanding and
internationally renowned lecturers, coming from different areas of
Applied Mathematics, have themselves contributed in an essential way to
the development of the theory and techniques that constituted the
subjects of the courses.