Using automatic differentiation for compressive sensing in uncertainty quantification

Abstract

This paper employs automatic differentiation (AD) in the compressive sensing-based generalized polynomial chaos (gPC) expansion, which computes a sparse approximation of the Quantity of Interest (QoI) using orthogonal polynomials as basis functions. An earlier approach without AD relies on an iterative procedure to refine the solution by approximating the gradient of the QoI. With AD, the gradient can be accurately evaluated, and a set of basis functions of the gPC expansion associated with new random variables can be efficiently identified. The computational complexity of the algorithm using AD is independent of the number of basis functions, whereas an earlier algorithm had complexity proportional to the square of this number. Our test problems include synthetic problems and a high-dimensional stochastic partial differential equation. With the new basis, the coefficient vector in the gPC expansion is sparser than the original basis. We demonstrate that introducing AD can greatly improve the performance by computing solutions 2 to 10 times faster than an earlier approach. The accuracy of the gPC expansion is also improved; sparse gpC expansions are obtained without iterative refinement, even for high dimensions when an earlier approach fails.