The Stochastic Preconditioned Conjugate Gradient method

Abstract

This paper presents the Stochastic Preconditioned Conjugate Gradient method (SPCG), an iterative equation solver that can greatly reduce the computational effort associated with the repeated calculations required in probabilistic finite element analysis. The method is implemented in a Monte Carlo simulation code and demonstrated for two example problems — a simple cantilever beam and a 3-D space truss. Excellent convergence properties are demonstrated in both cases. For the 3-D space truss problem (99 members, 72 DOF (degrees of freedom)), the method converged in only 3–4 iterations, on average. The method is also compared with the well-known Neumann expansion technique and shown to possess several advantageous properties with regard to speed of convergence. It is also shown that the computational effort required at each iteration of the SPCG method is effectively the same as the effort for each iteration (or expansion term) in Neumann expansion. Finally, it is pointed out that the method is also well-suited for parallel processing implementation. For parallel implementation, solution strategies are necessary that require minimal storage, can be vectorized, and can take advantage of concurrent processing. The SPCG method meets these criteria. Thus, the method promises to contribute toward making probabilistic finite element analysis practical for large, complex structures.