Algorithm based techniques are based on checking for the preservation of certain properties possessed by global data following a set of computations. This often involves the introduction of a check variable which is updated in such a manner that, in the absence of roundoff errors, it equals the value of some function which involves all the data elements participating in the algorithm. However, roundoff errors accumulate in different ways in the updates involving the check variables and the computations involving data elements; this makes it highly unlikely that the equality is preserved exactly for an implementation of the algorithm on a real computer. Thus, the check step involves verifying the preservation of the equality to within a tolerance value. We propose a method for determination of the tolerance based on error analysis techniques. We present results on three numerical algorithms which show the effectiveness of our approach for data sets of varying sizes and data ranges.

Algorithm-based fault-tolerance (ABFT) is an inexpensive method of incorporating fault-tolerance into existing applications. Applications are modified to operate on encoded data and produce encoded results which may then be checked for correctness. An attractive feature of the scheme is that it requires little or no modification to the underlying hardware or system software. Previous algorithm-based methods for developing reliable versions of numerical programs for general-purpose multicomputers have mostly concerned themselves with error detection. A truly fault-tolerant algorithm, however, needs to locate errors and recover from them once they are located. In a parallel processing environment, this corresponds to locating the faulty processors and recovering the data corrupted by the faulty processors. In this paper, we first present a general scheme for performing fault-location and recovery under the ABFT framework. Our fault model assumes that a faulty processor can corrupt all the data it possesses. The fault-location scheme is an application of system-level diagnosis theory to the ABFT framework, while the fault-recovery scheme uses ideas from coding theory to maintain redundant data and uses this to recover corrupted data in the event of processor failures. Results are presented on implementations of three numerical algorithms on a 16-processor Intel iPSC/2 hypercube multicomputer, which demonstrate acceptably low overheads for the single and double fault location and recovery cases.

Algorithm-based fault tolerance is an inexpensive method of achieving fault tolerance without requiring any hardware modifications. For numerical applications involving the iterative solution of linear systems arising from discretization of various PDEs, there exist almost no fault-tolerant algorithms in the literature. We describe an error-detecting version of a parallel algorithm for iteratively solving the Laplace equation over a rectangular grid. This error-detecting algorithm is based on the popular successive overrelaxation scheme with red-black ordering. We use the Laplace equation merely as a vehicle for discussion; we show how to modify the algorithm to devise error-detecting iterative schemes for solving linear systems arising from discretizations of other PDEs, such as the Poisson equation and a variant of the Laplace equation with a mixed derivative term. We also discuss a modification of the basic scheme to handle situations where the underlying solution domain is not rectangular. We then discuss a somewhat different error-detecting algorithm for iterative solution of PDEs which can be expected to yield better error coverage. We also present a new way of dealing with the roundoff errors which complicate the check phase of algorithm-based schemes. Our approach is based on error analysis incorporating some simplifications and gives high fault coverage and no false alarms for a large variety of data sets. We report experimental results on the error coverage and performance overhead of our algorithm-based error-detection schemes on an Intel iPSC/2 hypercube multiprocessor.

Algorithm based fault tolerance is an inexpensive method of achieving fault tolerance without requiring any hardware modifications. Algorithm-based schemes have been proposed for a wide variety of numerical applications. However, for a particular class of numerical applications, namely those involving the iterative solution of linear systems, there exist almost no fault-tolerant algorithms in the literature. In this paper, we describe a fault-tolerant version of a parallel algorithm for iteratively solving the Laplace equation over a grid.