Abstract: To solve a large-scale boundary value problem on a supercomputer, the problem domain is partitioned into smaller subdomains that can be tackled by individual processors. Since those subdomain-restricted problems are not well posed, an updating procedure is required which, by virtue of Amdahl's law, sets an upper bound to the attainable strong scalability. One implication is that many applications (for instance diagnosis by medical imaging) can be solved only so fast regardless of the available hardware. In this talk, I will be presenting PDDSparse, a novel algorithm for this task which is significantly less affected by communication-induced stalling. A "domain decomposition" Feynman-Kac formula can be derived which results in a much smaller linear system requiring no further updates. The numerics are interesting because stochastic representations become intertwined with linear algebra. I will discuss results concerning scalability, error control, application to nonlinear elliptic equations, and to equations for which the Feynman-Kac formula is unsuitable, such as the Helmholtz equation.