Failure Mitigation in Linear, Sesquilinear and Bijective Operations On Integer Data Streams Via Numerical Entanglement

A new roll-forward technique is proposed that recovers from any single fail-stop failure in $M$ integer data streams ($M\geq3$) when undergoing linear, sesquilinear or bijective (LSB) operations, such as: scaling, additions/subtractions, inner or outer vector products and permutations. In the proposed approach, the $M$ input integer data streams are linearly superimposed to form $M$ numerically entangled integer data streams that are stored in-place of the original inputs. A series of LSB operations can then be performed directly using these entangled data streams. The output results can be extracted from any $M-1$ entangled output streams by additions and arithmetic shifts, thereby guaranteeing robustness to a fail-stop failure in any single stream computation. Importantly, unlike other methods, the number of operations required for the entanglement, extraction and recovery of the results is linearly related to the number of the inputs and does not depend on the complexity of the performed LSB operations. We have validated our proposal in an Intel processor (Haswell architecture with AVX2 support) via convolution operations. Our analysis and experiments reveal that the proposed approach incurs only $1.8\%$ to $2.8\%$ reduction in processing throughput in comparison to the failure-intolerant approach. This overhead is 9 to 14 times smaller than that of the equivalent checksum-based method. Thus, our proposal can be used in distributed systems and unreliable processor hardware, or safety-critical applications, where robustness against fail-stop failures becomes a necessity.Comment: Proc. 21st IEEE International On-Line Testing Symposium (IOLTS 2015), July 2015, Halkidiki, Greec

Reliable Linear, Sesquilinear and Bijective Operations On Integer Data Streams Via Numerical Entanglement

A new technique is proposed for fault-tolerant linear, sesquilinear and bijective (LSB) operations on $M$ integer data streams ($M\geq3$), such as: scaling, additions/subtractions, inner or outer vector products, permutations and convolutions. In the proposed method, the $M$ input integer data streams are linearly superimposed to form $M$ numerically-entangled integer data streams that are stored in-place of the original inputs. A series of LSB operations can then be performed directly using these entangled data streams. The results are extracted from the $M$ entangled output streams by additions and arithmetic shifts. Any soft errors affecting any single disentangled output stream are guaranteed to be detectable via a specific post-computation reliability check. In addition, when utilizing a separate processor core for each of the $M$ streams, the proposed approach can recover all outputs after any single fail-stop failure. Importantly, unlike algorithm-based fault tolerance (ABFT) methods, the number of operations required for the entanglement, extraction and validation of the results is linearly related to the number of the inputs and does not depend on the complexity of the performed LSB operations. We have validated our proposal in an Intel processor (Haswell architecture with AVX2 support) via fast Fourier transforms, circular convolutions, and matrix multiplication operations. Our analysis and experiments reveal that the proposed approach incurs between $0.03\%$ to $7\%$ reduction in processing throughput for a wide variety of LSB operations. This overhead is 5 to 1000 times smaller than that of the equivalent ABFT method that uses a checksum stream. Thus, our proposal can be used in fault-generating processor hardware or safety-critical applications, where high reliability is required without the cost of ABFT or modular redundancy.Comment: to appear in IEEE Trans. on Signal Processing, 201

Generalized Numerical Entanglement For Reliable Linear, Sesquilinear And Bijective Operations On Integer Data Streams

We propose a new technique for the mitigation of fail-stop failures and/or silent data corruptions (SDCs) within linear, sesquilinear or bijective (LSB) operations on M integer data streams (M ⩾ 3). In the proposed approach, the M input streams are linearly superimposed to form M numerically entangled integer data streams that are stored in-place of the original inputs, i.e., no additional (aka. “checksum”) streams are used. An arbitrary number of LSB operations can then be performed in M processing cores using these entangled data streams. The output results can be extracted from any (M-K) entangled output streams by additions and arithmetic shifts, thereby mitigating K fail-stop failures (K ≤ ⌊(M-1)/2 ⌋ ), or detecting up to K SDCs per M-tuple of outputs at corresponding in-stream locations. Therefore, unlike other methods, the number of operations required for the entanglement, extraction and recovery of the results is linearly related to the number of the inputs and does not depend on the complexity of the performed LSB operations. Our proposal is validated within an Amazon EC2 instance (Haswell architecture with AVX2 support) via integer matrix product operations. Our analysis and experiments for failstop failure mitigation and SDC detection reveal that the proposed approach incurs 0.75% to 37.23% reduction in processing throughput in comparison to the equivalent errorintolerant processing. This overhead is found to be up to two orders of magnitude smaller than that of the equivalent checksum-based method, with increased gains offered as the complexity of the performed LSB operations is increasing. Therefore, our proposal can be used in distributed systems, unreliable multicore clusters and safety-critical applications, where robustness against failures and SDCs is a necessity

Reliable Linear, Sesquilinear, and Bijective Operations on Integer Data Streams Via Numerical Entanglement

A new technique is proposed for fault-tolerant linear, sesquilinear and bijective (LSB) operations on $M$ integer data streams ( $M \geq 3$), such as: scaling, additions/subtractions, inner or outer vector products, permutations and convolutions. In the proposed method, $M$ input integer data streams are linearly superimposed to form $M$ numerically-entangled integer data streams that are stored in-place of the original inputs. LSB operations can then be performed directly using these entangled data streams. The results are extracted from the $M$ entangled output streams by additions and arithmetic shifts. Any soft errors affecting one disentangled output stream are guaranteed to be detectable via a post-computation reliability check. Additionally, when utilizing a separate processor core for each stream, our approach can recover all outputs after any single fail-stop failure. Importantly, unlike algorithm-based fault tolerance (ABFT) methods, the number of operations required for the entire process is linearly related to the number of inputs and does not depend on the complexity of the performed LSB operations. We have validated our proposal in an Intel processor via several types of operations: fast Fourier transforms, convolutions, and matrix multiplication operations. Our analysis and experiments reveal that the proposed approach incurs between 0.03% to 7% reduction in processing throughput for numerous LSB operations. This overhead is 5 to 1000 times smaller than that of the equivalent ABFT method that uses a checksum stream. Thus, our proposal can be used in fault-generating processor hardware or safety-critical applications, where high reliability is required without the cost of ABFT or modular redundancy

Fault Tolerant Integer Data Computations: Algorithms and Applications

As computing units move to higher transistor integration densities and computing clusters become highly heterogeneous, studies begin to predict that, rather than being exceptions, data corruptions in memory and processor failures are likely to become more prevalent. It has therefore become imperative to improve the reliability of systems in the face of increasing soft error probabilities in memory and computing logic units of silicon CMOS integrated chips. This thesis introduces a new class of algorithms for fault tolerance in compute-intensive linear and sesquilinear (“one-and-half-linear”) data computations on integer data inputs within high-performance computing systems. The key difference between the proposed algorithms and existing fault tolerance methods is the elimination of the traditional requirement for additional hardware resources for system reliability. The first contribution of this thesis is in the detection of hardware-induced errors in integer matrix products. The proposed method of numerical packing for detecting a single error within a quadruple of matrix outputs is described in Chapter 2. The chapter includes analytic calculations of the proposed method’s computational complexity and reliability. Experimental results show that the proposed algorithm incurs comparable execution time overhead to existing algorithms for the detection and correction of a limited number of errors within generic matrix multiplication (GEMM) outputs. On the other hand, numerical packing becomes substantially more efficient in the mitigation of multiple errors. The achieved execution time gain of numerical packing is further analyzed with respect to its energy saving equivalent, thus paving the way for a new class of silent data corruption (SDC) mitigation method for integer matrix products that are fast, energy efficient, and highly reliable. A further advancement of the proposed numerical packing approach for the mitigation of core/processor failures in computing clusters (a.k.a., failstop failures) is described in Chapter 3 . The key advantage of this new packing approach is the ability to tolerate processor failures for all classes of sum-of-product computations. Because multimedia applications running on cloud computing platforms are now required to mitigate an increasing number of failures and outages at runtime, we analyze the efficiency of numerical packing within an image retrieval framework deployed over a cluster of AWS EC2 spot (i.e., low-cost albeit terminable) instances. Our results show that more than 70% reduction of cost can be achieved in comparison to conventional failure-intolerant processing based on AWS EC2 on-demand (i.e., higher-cost albeit guaranteed) instances. Finally, beyond numerical packing, we present a second approach for reliability in the case of linear and sesquilinear integer data computations by generalizing the recently-proposed concept of numerical entanglement. The proposed approach is capable of recovering from multiple fail-stop failures in a parallel/distributed computing environment. We present theoretical analysis of the computational and bit-width requirements of the proposed method in comparison to existing methods of checksum generation and processing. Our experiments with integer matrix products show that the proposed approach incurs 1.72% − 37.23% reduction in processing throughput in comparison to failure-intolerant processing while allowing for the mitigation of multiple fail-stop failures without the use of additional computing resources