Supercomputing applications to the numerical modeling of industrial and applied mathematics problems

Present and future supercomputers offer many opportunities and advantages to attack complex and demanding industrial and applied mathematical problems, but provide also new challenges. In the Peta-Flops regime, these concern both, the way to exploit the increasingly available power and the need of designing algorithms which are scalable and fault-tolerant at the same time. An example of a probabilistic domain decomposition method, which is indeed scalable and naturally fault-tolerant, is presented. Grid computing should also be mentioned as an increasingly popular way to perform massively distributed computing: it represents a way to exploit computing power, aside the existing supercomputers. Beyond classical supercomputers there is the prospective quantum computer, in view of which it is advisable to start now a search for suitable algorithms for certain classes of problems.