An algorithm-based error detection scheme for the multigrid method

Algorithm-based fault tolerance (ABFT) is a technique to provide system level error detection and correction on array processors as well as multiprocessors at a low cost. Since the early 1980s the technique has been extensively applied to several linear algebraic algorithms, e.g., matrix multiplication, Gaussian elimination, QR factorization, and singular value decompositions, etc. An important class of problems in numerical linear algebra dealing with the iterative solution of linear algebraic equations arising due to the finite difference discretization or the finite element discretization of a partial differential equation, however, has been overlooked. The only exception is the recent application of algorithm based error detection (ABED) encodings to the successive overrelaxation algorithm for Laplace's equation. In this paper, ABED is applied to a multigrid algorithm for the iterative solution of a Poisson equation in two dimensions. Invariants are created to implement checking in the relaxation, the restriction, and the interpolation operators. Modifications to invariants due to roundoff errors accumulated within the operators, which often lead to a situation known as false alarms, have been addressed by deriving the expressions for the roundoff errors in the algebraic processes in the operators and correcting the invariants accordingly. The ABED encoded multigrid algorithm is shown to be insensitive to the size and the range of the input data besides providing excellent error coverage at a low latency for floating-point, integer, and memory errors.