A systematic study of efficient sampling methods to quantify uncertainty in crack propagation and the Burgers equation

Abstract

We present a systematic study of efficient sampling methods to quantify parametric uncertainty in stage II crack propagation and in a Burgers equation with random viscosity. The samplingmethods employed are Monte Carlo and quasi-Monte Carlo methods enhanced through variance reduction techniques including stratified sampling. We also present a recent variance reduction technique called sensitivity derivative enhanced sampling that utilizes sensitivity derivatives to make more efficient use of random samples. The crack propagation problem is a benchmark problem proposed by the Society of Automotive Engineers (SAE); this problem was selected because its large variance can hinder the already slow convergence of typical Monte Carlo methods and thus provides an ideal test for efficient sampling methods. We also examine a steady-state Burgers equation, a model equation of perennial interest for its connections with the equations of fluid mechanics, with uncertain viscosity. We demonstrate that sensitivity derivative enhanced sampling (SDES) can reduce the required number of samples by an order of magnitude to achieve the same accuracy as a traditionalMonte Carlo method. When SDES is coupled with other sampling techniques such as stratified sampling, the number of samples required can be reduced by as much as two orders of magnitude. In addition, we introduce an analogous approach that utilizes sensitivity derivatives which, under appropriate conditions, results in a more efficient quasi-Monte Carlo method.