Hierarchical a posteriori error estimation of Bank-Weiser type in the FEniCS Project

in Computers and Mathematics with Applications (2023), 131

In the seminal paper of Bank and Weiser [Math. Comp., 44 (1985), pp.283-301] a new a posteriori estimator was introduced. This estimator requires the solution of a local Neumann problem on every cell of ... [more ▼]

In the seminal paper of Bank and Weiser [Math. Comp., 44 (1985), pp.283-301] a new a posteriori estimator was introduced. This estimator requires the solution of a local Neumann problem on every cell of the finite element mesh. Despite the promise of Bank-Weiser type estimators, namely locality, computational efficiency, and asymptotic sharpness, they have seen little use in practical computational problems. The focus of this contribution is to describe a novel implementation of hierarchical estimators of the Bank-Weiser type in a modern high-level finite element software with automatic code generation capabilities. We show how to use the estimator to drive (goal-oriented) adaptive mesh refinement and to mixed approximations of the nearly-incompressible elasticity problems. We provide comparisons with various other used estimators. An open-source implementation based on the FEniCS Project finite element software is provided as supplementary material. [less ▲]

Convergence Analysis of Evolutionary Algorithms in the Presence of Crash-Faults and Cheaters

in Computers and Mathematics with Applications (2012), 64(12), 3805-3819

This paper analyzes the fault-tolerance nature of Evolutionary Algorithms (EAs) when executed in a distributed environment subjected to malicious acts. More precisely, the inherent resilience of EAs ... [more ▼]

This paper analyzes the fault-tolerance nature of Evolutionary Algorithms (EAs) when executed in a distributed environment subjected to malicious acts. More precisely, the inherent resilience of EAs against two types of failures is considered: (1) crash faults, typically due to resource volatility which lead to data loss and part of the computation loss; (2) cheating faults, a far more complex kind of fault that can be modeled as the alteration of output values produced by some or all tasks of the program being executed. This last type of failure is due to the presence of cheaters on the computing platform. Most often in Global Computing (GC) systems such as BOINC, cheaters are attracted by the various incentives provided to stimulate the volunteers to share their computing resources: cheaters typically seek to obtain rewards with little or no contribution to the system. In this paper, the Algorithm-Based Fault Tolerance (ABFT) aspects of EAs against the above types of faults is characterized. Whereas the inherent resilience of EAs has been previously observed in the literature, for the first time, a formal analysis of the impact of the considered faults over the executed EA including a proof of convergence is proposed in this article. By the variety of problems addressed by EAs, this study will hopefully promote their usage in the future developments around distributed computing platform such as Desktop Grids and Volunteer Computing Systems or Cloud systems where the resources cannot be fully trusted. [less ▲]