As computers reach exascale and beyond, the incidence of faults will increase. Solutions to this problem are an active research topic. We focus on strategies to make the preconditioned conjugate gradient (PCG) solver resilient against node failures, speci cally, the exact state reconstruction (ESR) method, which exploits redundancies in PCG. Reducing the frequency at which redundant information is stored lessens the runtime overhead. However, after the node failure, the solver must restart from the last iteration for which redundant information was stored, which increases recovery overhead. This formulation highlights the method's similarities to checkpoint-restart (CR). Thus, this method, which we call ESR with periodic storage (ESRP), can be considered a form of algorithm-based checkpointrestart. The state is stored implicitly, by exploiting redundancy inherent to the algorithm, rather than explicitly as in CR. We also minimize the amount of data to be stored and retrieved compared to CR, but additional computation is required to reconstruct the solver's state. In this paper, we describe the necessary modi cations to ESR to convert it into ESRP, and perform an experimental evaluation. We compare ESRP experimentally with previously-existing ESR and application-level in-memory CR. Our results con rm that the overhead for ESR is reduced signi cantly, both in the failure-free case, and if node failures are introduced. In the former case, the overhead of ESRP is usually lower than that of CR. However, CR is faster if node failures happen. We claim that these di erences can be alleviated by the implementation of more appropriate preconditioners.

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The observed and expected continued growth in the number of nodes in large-scale parallel computers gives rise to two major challenges: global communication operations are becoming major bottlenecks due to their limited scalability, and the likelihood of node failures is increasing. We study an approach for addressing these challenges in the context of solving large sparse linear systems. In particular, we focus on the pipelined preconditioned conjugate gradient (PPCG) method, which has been shown to successfully deal with the first of these challenges. In this paper, we address the second challenge. We present extensions to the PPCG solver and two of its variants which make them resilient against the failure of a compute node while fully preserving their communication-hiding properties and thus their scalability. The basic idea is to efficiently communicate a few redundant copies of local vector elements to neighboring nodes with very little overhead. In case a node fails, these redundant copies are gathered at a replacement node, which can then accurately reconstruct the lost parts of the solver's state. After that, the parallel solver can continue as in the failurefree scenario. Experimental evaluations of our approach illustrate on average very low runtime overheads compared to the standard non-resilient algorithms. This shows that scalable algorithmic resilience can be achieved at low extra cost.

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This chapter describes a data flow implementation of the image processing algorithms Erosion and Dilation. Erosion and Dilation are basic image processing algorithms which are used to reduce or increase the size of objects in images, respectively, and which are used in a wide number of image processing applications. The chapter first describes the control flow versions of the algorithms in detail. Subsequently, the translation of these algorithms to the Data Flow paradigm is examined, and the details of the data flow implementation as well as possible optimizations are discussed.