  State of the art domain decomposition algorithms for large-scale boundary
value problems (with $M\gg 1$ degrees of freedom) suffer from bounded strong
scalability because they involve the synchronisation and communication of
workers inherent to iterative linear algebra. Here, we introduce PDDSparse, a
different approach to scientific supercomputing which relies on a "Feynman-Kac
formula for domain decomposition". Concretely, the interfacial values (only)
are determined by a stochastic, highly sparse linear system $G(\omega){\vec
u}={\vec b}(\omega)$ of size ${\cal O}(\sqrt{M})$, whose coefficients are
constructed with Monte Carlo simulations-hence embarrassingly in parallel. In
addition to a wider scope for strong scalability in the deep supercomputing
regime, PDDSparse has built-in fault tolerance and is ideally suited for GPUs.
A proof of concept example with up to 1536 cores is discussed in detail.
