Efficient Uncertainty Quantification using Automatic Differentiation and Taylor Series Expansion

  • Millwater, Harry (Univ. of Texas at San Antonio)
  • Rios, Aaron (Univ. of Texas at San Antonio)
  • Posso, Daniela (University of Texas at San Antonio)
  • Roberts, Sam (University of Texas at San Antonio)
  • Restrepo, David (University of Texas at San Antonio)
  • Montoya, Arturo (University of Texas at San Antonio)

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The implementation and exploitation of automatic differentiation (AD) to construct sufficiently accurate surrogate models is explained and demonstrated using multiple uncertainty quantification (UQ) methods. AD exists in multiple forms: software modification, adjoint methods, hypercomplex variable automatic differentiation (HYPAD) and its application to large-scale finite element models is increasing. As a result, exploiting the resultant Taylor series as a surrogate model for UQ can lead to significant efficiency gains. The recent development of a non-intrusive “residual” formulation using HYPAD provides an extremely efficient method to compute partial derivatives of nonlinear finite element models. For example, partial derivatives with respect to 22 parameters of a transient nonlinear additive manufacturing thermal simulation were obtained in only 0.8X additional computational time compared to the standard thermal analysis [1]. These derivatives were then used to construct a Taylor series expansion (TSE) for UQ analysis. In this work, multiple UQ methods will be combined with the Taylor series expansions developed using HYPAD from nonlinear finite element models to demonstrate, compare, and contrast the strengths and weaknesses of using TSE for UQ. Comparisons will be made with standard Monte Carlo sampling, polynomial chaos expansions, multiple fidelity Monte Carlo methods, and others. The methods will exploit the open source codes TSEUQLIB [2] and JetGP [3]. The UQ methods investigated include: • Statistical moments using high order expansions using TSEUQlib • Use of control variates combined with a Taylor series • Combining interval analysis with TSEUQlib • Use of directional derivatives for high dimensional problems • Derivative-enhanced Gaussian process models using JetGP The results will indicate the advantages of using TSE combined with AD for efficient UQ. REFERENCES [1] J.-S. Rincon-Tabares, M. Aristizabal, M. Balcer, A. Montoya, H. Millwater, and D. Restrepo, “Efficient sensitivity analysis of the thermal profile in powder bed fusion of metals using hypercomplex automatic differentiation finite element method,” Additive Manufacturing. (2024) Aug, 15:104488, https://doi.org/10.1016/j.addma.2024.104488 [2] https://github.com/lanl/TSEUQLib [3] https://github.com/Samm-Py/jetgp