A Framework for Distributed Approximation of Moments with Higher-Order Derivatives Through Automatic Differentiation

摘要

We present a framework for the distributed approximation of moments, enabling the evaluation of the uncertainty in a dynamical system. The first and second moment, mean, and variance are computed with up to third-order Taylor series expansion. The required derivatives for the expansion are generated automatically by automatic differentiation and propagated through an implicit time stepper. The computational kernels are the accumulation of the derivatives (Jacobian, Hessian, tensor) and the covariance matrix. We apply distributed parallelism to the Hessian or third-order tensor, and the user merely has to provide a function for the differential equation, thus achieving similar ease of use as Monte Carlo-based methods. We demonstrate our approach using with benchmarks on Theta, a KNL-based system at the Argonne Leadership Computing Facility.