High-order shock tracking

Unsteady optimization with high-order discretizations

Model reduction for multiscale problems

PDE-constrained optimization using adaptive reduced-order models

Local model reduction

Miscellaneous

PDE-constrained optimization using adaptive reduced-order models

An adaptive approach to using reduced-order models as surrogates in PDE-constrained optimization is introduced that breaks the traditional offline-online framework of model order reduction. A sequence of optimization problems constrained by a given Reduced-Order Model (ROM) is defined with the goal of converging to the solution of a given PDE-constrained optimization problem. For each reduced optimization problem, the constraining ROM is trained from sampling the High-Dimensional Model (HDM) at the solution of previous problems in the sequence. The reduced optimization problems are equipped with a nonlinear trust-region based on a residual error indicator to keep the optimization trajectory in a region of the parameter space where the ROM is accurate.


Adaptive model reduction framework for PDE-constrained optimization globalized with trust regions. Top left: Flow chart, bottom left: breakdown of computational time, and right: parameter space.

Aerodynamic shape optimization with ROM optimization framework: 2D airfoil. Left: optimal/target configuration (RAE2822), center: shape and pressure coefficient distribution, and right: convergence history.

A methodology is proposed for incorporating reduced-order models in a nonlinear topology optimization framework, which reduces the time required to obtain a topologically optimal structural design. The approach constructs two low-dimensional bases: one to reduce the dimension of the structural displacements and governing equations, and one to reduce the dimension of the parameter space, i.e. element volume fractions. The displacement basis is constructed adaptively during the reduced optimization problem to ensure accuracy is maintained along the optimization trajectory. The parameter basis is constructed apriori by lumping elements into macro-elements. As the reduced optimal solution will not coincide with the optimal topology, in general, the reduced optimal solution is used to warm-start a standard topology optimization method. Compared to a popular topology optimization method, without reduction or warm-starting, the proposed method achieves an order of magnitude speedup when applied to compliance minimization of a 2D cantilever and 3D trestle.

2D cantilever: compliance minimization. Left: optimal solution, center: solution using ROM framework (N seconds), right: olution using HDM after N seconds.

3D trestle: compliance minimization. Left: optimal solution, center: solution using ROM framework (N seconds), right: solution using HDM after N seconds.

Journal papers

  • M. J. Zahr and C. Farhat, “Progressive construction of a parametric reduced-order model for PDE-constrained optimization,” International Journal for Numerical Methods in Engineering, vol. 102, no. 5, pp. 1111--1135, 2015. [ bib | DOI | link | arxiv ]

  • D. Amsallem, M. J. Zahr, Y. Choi, and C. Farhat, “Design optimization using hyper-reduced-order models,” Structural and Multidisciplinary Optimization, pp. 1--22, 2014. [ bib | DOI | link ]

Conference papers

  • M. J. Zahr, D. Amsallem, and C. Farhat, “Construction of parametrically-robust CFD-based reduced-order models for PDE-constrained optimization,” in Proc. of the 21st AIAA Computational Fluid Dynamics Conference, vol. AIAA-2013-2685, American Institute of Aeronautics and Astronautics, 6/24/2013 -- 6/27/2013. [ bib | link | paper ]

Talks

  • M. J. Zahr, “Efficient PDE-constrained optimization under uncertainty using adaptive model reduction and sparse grids,” in 2017 West Coast ROM Workshop, (Berkeley, California), Lawrence Berkeley National Laboratory, 11/17/2017. [ bib | slides ]

  • M. J. Zahr, “Optimization-based computational physics and high-order methods: from optimized analysis to design and data assimilation,” in LBNL CRD Postdoc Seminar Series, (Berkeley, California), Lawrence Berkeley National Laboratory, 9/18/2017. [ bib | slides ]

  • M. J. Zahr, K. Carlberg, and D. P. Kouri, “Efficient PDE-constrained optimization under uncertainty using adaptive model reduction and sparse grids,” in SIAM Conference on Computational Science and Engineering, (Atlanta, Georgia), 2/27/2017 -- 3/3/2017. [ bib | slides ]

  • M. J. Zahr, “Efficient PDE-constrained optimization under uncertainty using adaptive model reduction and sparse grids,” in BIRS Workshop: Data-Driven Methods for ROMs and Stochastic PDEs, (Banff, Alberta, Canada), Banff International Reseach Station, 1/30/2017 -- 2/3/2017. [ bib | slides ]

  • M. J. Zahr, “Adaptive model reduction to accelerate optimization problems governed by partial differential equations,” in Farhat Research Group Seminar, (Stanford, California), Stanford University, 1/10/2017. [ bib | slides ]

  • M. J. Zahr, “Adaptive model reduction to accelerate optimization problems governed by partial differential equations,” in LBNL Postdoc Seminar Series, (Berkeley, California), Lawrence Berkeley National Laboratory, 1/9/2017. [ bib | slides ]

  • M. J. Zahr, “Adaptive model reduction to accelerate optimization problems governed by partial differential equations,” in Thesis Defense, (Stanford, California), Stanford University, 8/3/2016. [ bib | slides ]

  • M. J. Zahr, K. Carlberg, and D. P. Kouri, “Efficient PDE-constrained optimization under uncertainty using adaptive model reduction and sparse grids,” in SIAM Annual Meeting, (Boston, Massachusetts), 7/11/2016 -- 7/15/2016. [ bib | slides ]

  • M. J. Zahr, “Efficient PDE-constrained optimization under uncertainty using adaptive model reduction and sparse grids,” in CME 500 Seminar, (Stanford, California), Stanford University, 4/11/2016. [ bib | slides ]

  • M. J. Zahr, K. Carlberg, and D. P. Kouri, “Adaptive stochastic collocation for PDE-constrained optimization under uncertainty using sparse grids and model reduction,” in SIAM Conference on Uncertainty Quantification, (Lausanne, Switzerland), Ecole Polytechnique Federale de Lausanne, 4/5/2016 -- 4/8/2016. [ bib | slides ]

  • M. J. Zahr, “Accelerating PDE-constrained optimization problems using adaptive reduced-order models,” in University of Notre Dame Aerospace and Mechanical Engineering Seminar (Host: Gretar Tryggvason), (South Bend, Indiana), University of Notre Dame, 3/3/2016 -- 3/4/2016. [ bib | slides ]

  • M. J. Zahr, “Accelerating PDE-constrained optimization problems using adaptive reduced-order models,” in University of Southern California Aerospace and Mechanical Engineering Seminar (Host: Geoff Spedding), (Los Angeles, California), University of Southern California, 2/25/2016 -- 2/26/2017. [ bib | slides ]

  • M. J. Zahr, “Accelerating PDE-constrained optimization problems using adaptive reduced-order models,” in Luis W. Alvarez Fellowship Seminar (Host: Jonathan Carterl), (Berkeley, California), Lawrence Berkeley National Laboratory, 2/9/2016. [ bib | slides ]

  • M. J. Zahr, “Accelerating PDE-constrained optimization problems using adaptive reduced-order models,” in J. H. Wilkinson Fellowship Seminar (Host: Sven Leyffer), (Argonne, Illinois), Argonne National Laboratory, 1/15/2016. [ bib | slides ]

  • M. J. Zahr, “Accelerating PDE-constrained optimization problems using adaptive reduced-order models,” in John von Neumann Postdoctoral Fellowship Seminar (Host: Denis Ridzal), (Albuquerque, New Mexico), Sandia National Laboratories, 1/11/2016. [ bib | slides ]

  • M. J. Zahr, “Accelerating PDE-constrained optimization problems using adaptive reduced-order models,” in Sidney Fernbach Postdoctoral Fellowship Seminar (Host: Jeffrey A. F. Hittinger), (Livermore, California), Lawrence Livermore National Laboratory, 12/9/2015. [ bib | slides ]

  • M. J. Zahr, “High-order methods for optimization and control of conservation laws on deforming domains,” in Farhat Research Group Seminar, (Stanford, California), Stanford University, 12/8/2015. [ bib | slides ]

  • M. J. Zahr and C. Farhat, “A nonlinear trust-region framework for PDE-constrained optimization using adaptive model reduction,” in West Coast ROM Workshop, (Livermore, California), Sandia National Laboratories, 11/19/2015. [ bib | slides ]

  • M. J. Zahr and C. Farhat, “Accelerating PDE-constrained optimization using adaptive reduced-order models,” in Seminar at Sandia National Laboratories (Host: Drew Kouri), (Albuquerque, New Mexico), 7/8/2015. [ bib | slides ]

  • M. J. Zahr, “Accelerating PDE-constrained optimization using adaptive reduced-order models: application to topology optimization,” in Robert J. Melosh Medal Competition, (Durham, North Carolina), Duke University, 4/24/2015. [ bib | slides ]

  • M. J. Zahr and C. Farhat, “A nonlinear trust-region framework for PDE-constrained optimization using progressively constructed reduced-order models,” in 2015 SIAM Conference on Computational Science and Engineering (CSE15), (Salt Lake City, Utah), 3/14/2015 -- 3/18/2015. [ bib | slides ]

  • M. J. Zahr and C. Farhat, “Accelerating PDE-constrained optimization using progressively constructed reduced-order models,” in Bay Area ROM Workshop, (Livermore, California), Sandia National Laboratories, 8/8/2014. [ bib | slides ]

  • M. J. Zahr and C. Farhat, “PDE-constrained optimization using progressively constructed reduced-order models,” in World Congress on Computational Mechanics XI (WCCM XI), (Barcelona, Spain), 7/20/2014 -- 7/25/2014. [ bib | slides ]

  • M. J. Zahr and C. Farhat, “Rapid nonlinear topology optimization using precomputed reduced-order models,” in 17th US National Congress on Theoretical and Applied Mechanics (USNCTAM), (East Lansing, Michigan), 6/15/2014 -- 6/20/2014. [ bib | slides ]

  • M. J. Zahr and C. Farhat, “PDE-constrained optimization using hyper-reduced models,” in SIAM Conference on Optimization, (San Diego, California), 5/19/2014 -- 5/22/2014. [ bib | slides ]

  • M. J. Zahr, “Rapid topology optimization using reduced-order models,” in 2013 Berkeley/Stanford Computational Mechanics Festival (CompFest), (Berkeley, California), University of California, Berkeley, 10/19/2013. [ bib | slides ]

  • M. J. Zahr and C. Farhat, “Rapid nonlinear topology optimization using reduced-order models,” in 12th U.S. National Congress on Computational Mechanics (USNCCM12), (Raleigh, North Carolina), 7/22/2013 -- 7/25/2013. [ bib | slides ]

  • M. J. Zahr, D. Amsallem, and C. Farhat, “Construction of parametrically robust CFD-based reduced-order models for PDE-constrained optimization,” in 43rd AIAA Fluid Dynamics Conference and Exhibit, (San Diego, California), 6/24/2013 -- 6/27/2013. [ bib | slides | link ]

  • D. Amsallem, M. J. Zahr, Y. Choi, and C. Farhat, “Design optimization using hyper-reduced order models,” in 10th World Congress on Structural and Multidisciplinary Optimization (WCSMO10), (Orlando, Florida), 3/19/2013 -- 3/24/2013. [ bib ]

  • M. J. Zahr and C. Farhat, “Construction of parametrically robust reduced-order models for PDE-constrained optimization,” in 10th World Congress on Structural and Multidisciplinary Optimization (WCSMO10), (Orlando, Florida), 3/19/2013 -- 3/24/2013. [ bib ]

Posters

  • M. J. Zahr and P.-O. Persson, “Adjoint-based optimization, uncertainty quantification, and data assimilation of multiphysics systems using high-order numerical discretizations,” in DOE ASCR Applied Mathematics PI Meeting, (Washington D.C.), 9/11/2017 -- 9/12/2017. [ bib | poster ]

  • M. J. Zahr, “Efficient PDE-constrained optimization using adaptive model reduction,” in Institute for Mathematics and its Applications: Frontiers in PDE-Constrained Optimization, (Minneapolis, Minnesota), 6/6/2016 -- 6/10/2016. [ bib | poster ]

  • M. J. Zahr, “Efficient PDE-constrained optimization using adaptive model reduction,” in 2016 Stanford Computational Mathematics and Engineering Affiliates Meeting, (Stanford, California), 5/1/2016. [ bib | poster ]

  • M. J. Zahr, “Efficient PDE-constrained optimization using adaptive model reduction,” in 2016 Stanford Aerospace and Astronautics Affiliates Meeting, (Stanford, California), 4/26/2016. [ bib | poster ]

  • M. J. Zahr and C. Farhat, “Accelerating PDE-constrained optimization using adaptive reduced-order models,” in Army High Performance Computing Research Center (AHPCRC) Review Meeting, (Santa Cruz, California), 1/18/2016 -- 1/20/2016. [ bib | poster ]

  • M. J. Zahr and C. Farhat, “Accelerating PDE-constrained optimization using progressively-constructed reduced-order models,” in Army High Performance Computing Research Center (AHPCRC) Review Meeting, (Santa Cruz, California), 8/10/2015 -- 8/12/2016. [ bib | poster ]

  • M. J. Zahr and C. Farhat, “Progressive construction of a parametric reduced-order model for PDE-constrained optimization,” in 2014 DOE CSGF Annual Program Review, (Washington D.C.), 7/14/2014 -- 7/17/2014. [ bib | poster ]

  • M. J. Zahr, “PDE-constrained optimization using progressively constructed reduced-order models,” in 2014 Stanford Aerospace and Astronautics Affiliates Meeting, (Stanford, California), 4/28/2014. [ bib | poster ]

  • M. J. Zahr and C. Farhat, “Rapid topology optimization using reduced-order models,” in 2013 DOE CSGF Annual Program Review, (Washington D.C.), 7/25/2013 -- 7/27/2013. [ bib | poster ]

  • M. J. Zahr and C. Farhat, “Rapid structural shape optimization using progressively constructed reduced-order models,” in 12th U.S. National Congress on Computational Mechanics (USNCCM12), (Raleigh, North Carolina), 7/22/2013 -- 7/25/2013. [ bib | poster ]