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Probabilistic domain decomposition (PDD) is an alternative paradigm for solving boundary value problems (BVPs) in parallel with excellent scalability properties, thanks to its reliance on stochastic representations of the BVP. However, there are cases when the latter is less numerically convenient, or unknown. Semilinear elliptic BVPs and the Helmholtz equation are prominent examples of either class. In this paper, we overcome this issue by designing suitable iterative schemes for either problem. These schemes not only retain the desirable properties of PDD but also are optimally suited for pathwise variance reduction, resulting in a systematic, nearly cost-free reduction of the statistical error through the iterations. Numerical tests carried out on the supercomputer Marconi100 are presented. [ABSTRACT FROM AUTHOR]

We present an iterative scheme, reminiscent of the Multigrid method, to solve large boundary value problems with Probabilistic Domain Decomposition (PDD). In it, increasingly accurate approximations to the solution are used as control variates in order to reduce the Monte Carlo error of the following iterates—resulting in an overall acceleration of PDD for a given error tolerance. The key feature of the proposed algorithm is the ability to approximately predict the speedup with little computational overhead and in parallel. Besides, the theoretical framework allows to explore other aspects of PDD, such as stability. One numerical example is worked out, yielding an improvement between one and two orders of magnitude over the previous version of PDD. [ABSTRACT FROM AUTHOR]

We consider the numerical solution of nonlinear elliptic boundary value problems with Kansa’s method. We derive analytic formulas for the Jacobian and Hessian of the resulting nonlinear collocation system and exploit them within the framework of the trust-region algorithm. This ansatz is tested on semilinear, quasilinear and fully nonlinear elliptic PDEs (including Plateau’s problem, Hele–Shaw flow and the Monge–Ampère equation) with excellent results. The new approach distinctly outperforms previous ones based on linearization or finite-difference Jacobians. [ABSTRACT FROM AUTHOR]