Fault Resilient Domain Decomposition Preconditioner for PDEs

Abstract

The move towards extreme-scale computing platforms challenges scientific simulations in many ways. Given the recent tendencies in computer architecture development, one needs to reformulate legacy codes in order to cope with large amounts of communication, system faults and requirements of low-memory usage per core. In this work, we develop a novel framework for solving partial differential equations (PDEs) via domain decomposition that reformulates the solution as a state-of-knowledge with a probabilistic interpretation. Such reformulation allows resiliency with respect to potential faults without having to apply fault detection, avoids unnecessary communication and is generally well-suited for rigorous uncertainty quantification studies that target improvements of predictive fidelity of scientific models. We demonstrate our algorithm for one-dimensional PDE examples where artificial faults have been implemented as bit-flips in the binary representation of subdomain solutions.

Discrete A Priori Bounds for the Detection of Corrupted PDE Solutions in Exascale Computations

“…Proof. Let α as in (33), then it follows from Lemmas 9 and 10 that the conditions of Theorem 7 are satisfied with λ i "κ d i for any i P t1, . .…”

“…Depending on the magnitude of the error caused by the faults and on the sharpness of the bounds, checking that a PDE solution lies between these bounds may not be sufficient to overcome the occurrence of faults. Building on an existing domain decomposition approach for solving elliptic PDEs in a resilient fashion [33], we shall see how complementing it by checking bounds improves its resilience capabilities.…”

“…Domain decomposition methods consist in dividing the domain over which the problem needs to be solved into subdomains on which one solves smaller, and thus cheaper, subproblems. In the resilient approach developed in [33,32] and summarized in Figure 1, the domain Ω is split into N overlapping subdomains tΩ i u N i"1 with corresponding boundaries…”

“…To address this issue, we have recently developed a resilient solver for elliptic partial differential equations (PDEs); see [33,32]. The solver is based on an overlapping domain decomposition method (see, e.g., [30,43]) and the resilient update involves the solution of local problems (independent from one subdomain to another), which is well suited for massive parallelism.…”

“…In the approaches of [33,32], the resilient update is achieved through robust regression. In this work, we explore the possibility of further improving the resilience capabilities of such techniques by checking the admissibility of the local solutions before they are fed to the regression.…”