Grassmannian diffusion maps based surrogate modeling via&nbsp;geometric harmonics

dos Santos, Ketson R. M.; Giovanis, Dimitris G.; Kontolati, Katiana; Loukrezis, Dimitrios; Shields, Michael D.

doi:10.1002/nme.6977

Grassmannian diffusion maps based surrogate modeling via geometric harmonics

Journal Article · Mon Apr 11 00:00:00 EDT 2022 · International Journal for Numerical Methods in Engineering

DOI:https://doi.org/10.1002/nme.6977· OSTI ID:1862433

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Earthquake Engineering and Structural Dynamics Laboratory École Polytechnique Fédérale de Lausanne Lausanne Switzerland
Department of Civil &, Systems Engineering Johns Hopkins University Baltimore Maryland USA
Institute for Accelerator Science and Electromagnetic Fields (TEMF) Technische Universität Darmstadt Darmstadt Germany, Centre for Computational Engineering Technische Universität Darmstadt Darmstadt Germany

Abstract

A novel surrogate model based on the Grassmannian diffusion maps (GDMaps) and utilizing geometric harmonics (GH) is developed for predicting the response of complex physical phenomena. The method utilizes GDMaps to obtain a low‐dimensional representation of the underlying behavior of physical/mathematical systems with respect to uncertain input parameters. Using this representation, GH, an out‐of‐sample extension technique, is employed to create a global map from the input parameter space to a Grassmannian diffusion manifold. GH is further employed to locally map points on the diffusion manifold onto the tangent space of a Grassmann manifold. The exponential map is then used to project the points in the tangent space onto the Grassmann manifold, where reconstruction of the full solution is performed. The performance of the proposed surrogate model is verified with three examples. The first problem is a toy example used to illustrate the technique. In the second example, errors associated with the various mappings are assessed by studying response predictions of the electric potential of a dielectric cylinder in a homogeneous electric field. The last example applies the method for uncertainty prediction in the strain field evolution in a model amorphous material using the shear transformation zone theory of plasticity.

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Sponsoring Organization:: USDOE

Grant/Contract Number:: SC0020428

OSTI ID:: 1862433

Alternate ID(s):: OSTI ID: 1862434
OSTI ID: 2562791
OSTI ID: 1906446

Journal Information:: International Journal for Numerical Methods in Engineering, Journal Name: International Journal for Numerical Methods in Engineering Journal Issue: 15 Vol. 123; ISSN 0029-5981

Publisher:: Wiley Blackwell (John Wiley & Sons)Copyright Statement

Country of Publication:: United Kingdom

Language:: English

References (50)

Riemannian center of mass and mollifier smoothing Karcher, H. Communications on Pure and Applied Mathematics, Vol. 30, Issue 5 https://doi.org/10.1002/cpa.3160300502	journal	September 1977
Diffusion maps‐based surrogate modeling: An alternative machine learning approach Kalogeris, I.; Papadopoulos, V. International Journal for Numerical Methods in Engineering, Vol. 121, Issue 4 https://doi.org/10.1002/nme.6236	journal	October 2019
Expanding the Family of Grassmannian Kernels: An Embedding Perspective Harandi, Mehrtash T.; Salzmann, Mathieu; Jayasumana, Sadeep Computer Vision – ECCV 2014 https://doi.org/10.1007/978-3-319-10584-0_27	book	January 2014
Gaussian Processes in Machine Learning Rasmussen, Carl Edward Advanced Lectures on Machine Learning https://doi.org/10.1007/978-3-540-28650-9_4	book	January 2004
On principal angles between subspaces in Rn Miao, Jianming; Ben-Israel, Adi Linear Algebra and its Applications, Vol. 171 https://doi.org/10.1016/0024-3795(92)90251-5	journal	July 1992
Modeling uncertainty in flow simulations via generalized polynomial chaos Xiu, Dongbin; Karniadakis, George Em Journal of Computational Physics, Vol. 187, Issue 1 https://doi.org/10.1016/S0021-9991(03)00092-5	journal	May 2003
Diffusion maps, spectral clustering and reaction coordinates of dynamical systems Nadler, Boaz; Lafon, Stéphane; Coifman, Ronald R. Applied and Computational Harmonic Analysis, Vol. 21, Issue 1, p. 113-127 https://doi.org/10.1016/j.acha.2005.07.004	journal	July 2006
Geometric harmonics: A novel tool for multiscale out-of-sample extension of empirical functions Coifman, Ronald R.; Lafon, Stéphane Applied and Computational Harmonic Analysis, Vol. 21, Issue 1 https://doi.org/10.1016/j.acha.2005.07.005	journal	July 2006
Diffusion maps Coifman, Ronald R.; Lafon, Stéphane Applied and Computational Harmonic Analysis, Vol. 21, Issue 1 https://doi.org/10.1016/j.acha.2006.04.006	journal	July 2006
Multiscale data sampling and function extension Bermanis, Amit; Averbuch, Amir; Coifman, Ronald R. Applied and Computational Harmonic Analysis, Vol. 34, Issue 1 https://doi.org/10.1016/j.acha.2012.03.002	journal	January 2013
Parsimonious representation of nonlinear dynamical systems through manifold learning: A chemotaxis case study Dsilva, Carmeline J.; Talmon, Ronen; Coifman, Ronald R. Applied and Computational Harmonic Analysis, Vol. 44, Issue 3 https://doi.org/10.1016/j.acha.2015.06.008	journal	May 2018
Manifold learning for coarse-graining atomistic simulations: Application to amorphous solids Kontolati, Katiana; Alix-Williams, Darius; Boffi, Nicholas M. Acta Materialia, Vol. 215 https://doi.org/10.1016/j.actamat.2021.117008	journal	August 2021
Radial basis function approximations: comparison and applications Majdisova, Zuzana; Skala, Vaclav Applied Mathematical Modelling, Vol. 51 https://doi.org/10.1016/j.apm.2017.07.033	journal	November 2017
Data-driven surrogates for high dimensional models using Gaussian process regression on the Grassmann manifold Giovanis, D. G.; Shields, M. D. Computer Methods in Applied Mechanics and Engineering, Vol. 370 https://doi.org/10.1016/j.cma.2020.113269	journal	October 2020
Diffusion maps-aided Neural Networks for the solution of parametrized PDEs Kalogeris, Ioannis; Papadopoulos, Vissarion Computer Methods in Applied Mechanics and Engineering, Vol. 376 https://doi.org/10.1016/j.cma.2020.113568	journal	April 2021
Parallel three-dimensional simulations of quasi-static elastoplastic solids Boffi, Nicholas M.; Rycroft, Chris H. Computer Physics Communications https://doi.org/10.1016/j.cpc.2020.107254	journal	March 2020
Data-driven probability concentration and sampling on manifold Soize, C.; Ghanem, R. Journal of Computational Physics, Vol. 321 https://doi.org/10.1016/j.jcp.2016.05.044	journal	September 2016
Uncertainty quantification for complex systems with very high dimensional response using Grassmann manifold variations Giovanis, D. G.; Shields, M. D. Journal of Computational Physics, Vol. 364 https://doi.org/10.1016/j.jcp.2018.03.009	journal	July 2018
UQpy: A general purpose Python package and development environment for uncertainty quantification Olivier, Audrey; Giovanis, Dimitris G.; Aakash, B. S. Journal of Computational Science, Vol. 47 https://doi.org/10.1016/j.jocs.2020.101204	journal	November 2020
Refined Stratified Sampling for efficient Monte Carlo based uncertainty quantification Shields, Michael D.; Teferra, Kirubel; Hapij, Adam Reliability Engineering & System Safety, Vol. 142 https://doi.org/10.1016/j.ress.2015.05.023	journal	October 2015
Adaptive Monte Carlo analysis for strongly nonlinear stochastic systems Shields, Michael D. Reliability Engineering & System Safety, Vol. 175 https://doi.org/10.1016/j.ress.2018.03.018	journal	July 2018
AK-MCS: An active learning reliability method combining Kriging and Monte Carlo Simulation Echard, B.; Gayton, N.; Lemaire, M. Structural Safety, Vol. 33, Issue 2 https://doi.org/10.1016/j.strusafe.2011.01.002	journal	March 2011
Spectral identification of nonlinear multi-degree-of-freedom structural systems with fractional derivative terms based on incomplete non-stationary data dos Santos, Ketson R. M.; Brudastova, Olga; Kougioumtzoglou, Ioannis A. Structural Safety, Vol. 86 https://doi.org/10.1016/j.strusafe.2020.101975	journal	September 2020
Monte Carlo and quasi-Monte Carlo methods Caflisch, Russel E. Acta Numerica, Vol. 7 https://doi.org/10.1017/S0962492900002804	journal	January 1998
Variable-free exploration of stochastic models: A gene regulatory network example Erban, Radek; Frewen, Thomas A.; Wang, Xiao The Journal of Chemical Physics, Vol. 126, Issue 15 https://doi.org/10.1063/1.2718529	journal	April 2007
Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps Coifman, R. R.; Lafon, S.; Lee, A. B. Proceedings of the National Academy of Sciences, Vol. 102, Issue 21 https://doi.org/10.1073/pnas.0500334102	journal	May 2005
Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data Donoho, D. L.; Grimes, C. Proceedings of the National Academy of Sciences, Vol. 100, Issue 10 https://doi.org/10.1073/pnas.1031596100	journal	April 2003
Differential Geometry of Grassmann Manifolds Wong, Yung-Chow Proceedings of the National Academy of Sciences, Vol. 57, Issue 3 https://doi.org/10.1073/pnas.57.3.589	journal	March 1967
LIII. On lines and planes of closest fit to systems of points in space Pearson, Karl The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Vol. 2, Issue 11 https://doi.org/10.1080/14786440109462720	journal	November 1901
Dynamics of viscoplastic deformation in amorphous solids Falk, M. L.; Langer, J. S. Physical Review E, Vol. 57, Issue 6 https://doi.org/10.1103/PhysRevE.57.7192	journal	June 1998
Nonequilibrium thermodynamics of driven amorphous materials. III. Shear-transformation-zone plasticity Bouchbinder, Eran; Langer, J. S. Physical Review E, Vol. 80, Issue 3 https://doi.org/10.1103/PhysRevE.80.031133	journal	September 2009
Coarse graining atomistic simulations of plastically deforming amorphous solids Hinkle, Adam R.; Rycroft, Chris H.; Shields, Michael D. Physical Review E, Vol. 95, Issue 5 https://doi.org/10.1103/PhysRevE.95.053001	journal	May 2017
The Nystrom Extension for Signals Defined on a Graph Heimowitz, Ayelet; Eldar, Yonina C. ICASSP 2018 - 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) https://doi.org/10.1109/ICASSP.2018.8462408	conference	April 2018
The Laplacian Pyramid as a Compact Image Code Burt, P.; Adelson, E. IEEE Transactions on Communications, Vol. 31, Issue 4 https://doi.org/10.1109/TCOM.1983.1095851	journal	April 1983
Face recognition using Laplacianfaces No authors listed IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 27, Issue 3 https://doi.org/10.1109/TPAMI.2005.55	journal	March 2005
A Global Geometric Framework for Nonlinear Dimensionality Reduction Tenenbaum, J. B. Science, Vol. 290, Issue 5500 https://doi.org/10.1126/science.290.5500.2319	journal	December 2000
Nonlinear Dimensionality Reduction by Locally Linear Embedding Roweis, S. T. Science, Vol. 290, Issue 5500 https://doi.org/10.1126/science.290.5500.2323	journal	December 2000
Diffusion Maps, Reduction Coordinates, and Low Dimensional Representation of Stochastic Systems Coifman, R. R.; Kevrekidis, I. G.; Lafon, S. Multiscale Modeling & Simulation, Vol. 7, Issue 2 https://doi.org/10.1137/070696325	journal	January 2008
Uncertainty Quantification: Theory, Implementation, and Applications Smith, Ralph C. Society for Industrial and Applied Mathematics https://doi.org/10.1137/1.9781611973228	book	January 2013
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data Babuška, Ivo; Nobile, Fabio; Tempone, Raúl SIAM Review, Vol. 52, Issue 2 https://doi.org/10.1137/100786356	journal	January 2010
Numerical Algorithms on the Affine Grassmannian Lim, Lek-Heng; Sze-Wai Wong, Ken; Ye, Ke SIAM Journal on Matrix Analysis and Applications, Vol. 40, Issue 2 https://doi.org/10.1137/18M1169321	journal	January 2019
The Geometry of Algorithms with Orthogonality Constraints Edelman, Alan; Arias, Tomás A.; Smith, Steven T. SIAM Journal on Matrix Analysis and Applications, Vol. 20, Issue 2 https://doi.org/10.1137/S0895479895290954	journal	January 1998
Grassmann discriminant analysis: a unifying view on subspace-based learning Hamm, Jihun; Lee, Daniel D. Proceedings of the 25th international conference on Machine learning - ICML '08 https://doi.org/10.1145/1390156.1390204	conference	January 2008
A benchmark study on intelligent sampling techniques in Monte Carlo simulation Santos, K. R. M. dos; Beck, A. T. Latin American Journal of Solids and Structures, Vol. 12, Issue 4 https://doi.org/10.1590/1679-78251245	journal	August 2015
Extending Classical Surrogate Modeling to high Dimensions Through Supervised Dimensionality Reduction: a Data-Driven Approach Lataniotis, Christos; Marelli, Stefano; Sudret, Bruno International Journal for Uncertainty Quantification, Vol. 10, Issue 1 https://doi.org/10.1615/Int.J.UncertaintyQuantification.2020031935	journal	January 2020
A Survey of Unsupervised Learning Methods for High-Dimensional Uncertainty Quantification in Black-Box-Type Problems Kontolati, Katiana; Loukrezis, Dimitrios; Giovanis, Dimitrios G. SSRN Electronic Journal https://doi.org/10.2139/ssrn.4041125	journal	January 2022
A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code McKay, M. D.; Beckman, R. J.; Conover, W. J. Technometrics, Vol. 21, Issue 2 https://doi.org/10.2307/1268522	journal	May 1979
The Homogeneous Chaos Wiener, Norbert American Journal of Mathematics, Vol. 60, Issue 4 https://doi.org/10.2307/2371268	journal	October 1938
Interpolation Method for Adapting Reduced-Order Models and Application to Aeroelasticity Amsallem, David; Farhat, Charbel AIAA Journal, Vol. 46, Issue 7 https://doi.org/10.2514/1.35374	journal	July 2008
Reduced Models in Chemical Kinetics via Nonlinear Data-Mining Chiavazzo, Eliodoro; Gear, Charles; Dsilva, Carmeline Processes, Vol. 2, Issue 1 https://doi.org/10.3390/pr2010112	journal	January 2014

Grassmannian diffusion maps based surrogate modeling via geometric harmonics

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