New algorithms in computational mathematics suited for next-generation supercomputers

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. In practice, Monte Carlo as well as quasi-Monte Carlo methods are used to generate only few interfacial values in two-dimensional domains where boundary-value elliptic problems are formulated. This allows for a fully domain decomposition of the problem.