Here, we formulate a new projection-based reduced-order modeling technique for non-linear dynamical systems. The proposed technique, which we refer to as the Adjoint Petrov–Galerkin (APG) method, is derived by decomposing the generalized coordinates of a dynamical system into a resolved coarse-scale set and an unresolved fine-scale set. A Markovian finite memory assumption within the Mori–Zwanzig formalism is then used to develop a reduced-order representation of the coarse scales. This procedure leads to a closed reduced-order model that displays commonalities with the adjoint stabilization method used in finite elements. The formulation is shown to be equivalent to a Petrov–Galerkin method with a non-linear, time-varying test basis, thus sharing some similarities with the Least-Squares Petrov–Galerkin method. Theoretical analysis examining a priori error bounds and computational cost is presented. Numerical experiments on the compressible Navier–Stokes equations demonstrate that the proposed method can lead to improvements in numerical accuracy, robustness, and computational efficiency over the Galerkin method on problems of practical interest. Improvements in numerical accuracy and computational efficiency over the Least-Squares Petrov–Galerkin method are observed in most cases.
Parish, Eric J., et al. "The Adjoint Petrov–Galerkin method for non-linear model reduction." Computer Methods in Applied Mechanics and Engineering, vol. 365, no. C, Mar. 2020. https://doi.org/10.1016/j.cma.2020.112991
Parish, Eric J., Wentland, Christopher R., & Duraisamy, Karthik (2020). The Adjoint Petrov–Galerkin method for non-linear model reduction. Computer Methods in Applied Mechanics and Engineering, 365(C). https://doi.org/10.1016/j.cma.2020.112991
Parish, Eric J., Wentland, Christopher R., and Duraisamy, Karthik, "The Adjoint Petrov–Galerkin method for non-linear model reduction," Computer Methods in Applied Mechanics and Engineering 365, no. C (2020), https://doi.org/10.1016/j.cma.2020.112991
@article{osti_1618094,
author = {Parish, Eric J. and Wentland, Christopher R. and Duraisamy, Karthik},
title = {The Adjoint Petrov–Galerkin method for non-linear model reduction},
annote = {Here, we formulate a new projection-based reduced-order modeling technique for non-linear dynamical systems. The proposed technique, which we refer to as the Adjoint Petrov–Galerkin (APG) method, is derived by decomposing the generalized coordinates of a dynamical system into a resolved coarse-scale set and an unresolved fine-scale set. A Markovian finite memory assumption within the Mori–Zwanzig formalism is then used to develop a reduced-order representation of the coarse scales. This procedure leads to a closed reduced-order model that displays commonalities with the adjoint stabilization method used in finite elements. The formulation is shown to be equivalent to a Petrov–Galerkin method with a non-linear, time-varying test basis, thus sharing some similarities with the Least-Squares Petrov–Galerkin method. Theoretical analysis examining a priori error bounds and computational cost is presented. Numerical experiments on the compressible Navier–Stokes equations demonstrate that the proposed method can lead to improvements in numerical accuracy, robustness, and computational efficiency over the Galerkin method on problems of practical interest. Improvements in numerical accuracy and computational efficiency over the Least-Squares Petrov–Galerkin method are observed in most cases.},
doi = {10.1016/j.cma.2020.112991},
url = {https://www.osti.gov/biblio/1618094},
journal = {Computer Methods in Applied Mechanics and Engineering},
issn = {ISSN 0045-7825},
number = {C},
volume = {365},
place = {United States},
publisher = {Elsevier},
year = {2020},
month = {03}}
Sandia National Laboratories (SNL-CA), Livermore, CA (United States)
Sponsoring Organization:
USDOE National Nuclear Security Administration (NNSA); US Air Force Office of Scientific Research (AFOSR)
Grant/Contract Number:
AC04-94AL85000; NA0003525
OSTI ID:
1618094
Alternate ID(s):
OSTI ID: 1776272
Report Number(s):
SAND--2019-1114J; 672058
Journal Information:
Computer Methods in Applied Mechanics and Engineering, Journal Name: Computer Methods in Applied Mechanics and Engineering Journal Issue: C Vol. 365; ISSN 0045-7825