Bibliography
Thomas Bendokat, Ralf Zimmermann, and P. -A. Absil. A grassmann manifold handbook: basic geometry and computational aspects. 2020. arXiv:2011.13699.
P.-A. Absil, R. Mahony, and R. Sepulchre. Riemannian geometry of grassmann manifolds with a view on algorithmic computation. Acta Applicandae Mathematica, 80(2):199–220, January 2004. URL: https://doi.org/10.1023/b:acap.0000013855.14971.91, doi:10.1023/b:acap.0000013855.14971.91.
Ronald R. Coifman and Stéphane Lafon. Diffusion maps. Applied and Computational Harmonic Analysis, 21(1):5–30, 2006. Special Issue: Diffusion Maps and Wavelets. URL: https://www.sciencedirect.com/science/article/pii/S1063520306000546, doi:https://doi.org/10.1016/j.acha.2006.04.006.
Carmeline J Dsilva, Ronen Talmon, Ronald R Coifman, and Ioannis G Kevrekidis. Parsimonious representation of nonlinear dynamical systems through manifold learning: a chemotaxis case study. Applied and Computational Harmonic Analysis, 44(3):759–773, 2018.
Julien Weiss. A Tutorial on the Proper Orthogonal Decomposition, chapter, pages. American Institute of Aeronautics and Astronautics, 2019. URL: https://arc.aiaa.org/doi/abs/10.2514/6.2019-3333, arXiv:https://arc.aiaa.org/doi/pdf/10.2514/6.2019-3333, doi:10.2514/6.2019-3333.
Lawrence Sirovich. Turbulence and the dynamics of coherent structures. i. coherent structures. Quarterly of applied Mathematics, 45(3):561–571, 1987. URL: https://doi.org/10.1090/qam/910462, doi:10.1090/qam/910462.
D.G. Giovanis and M.D. Shields. Variance-based simplex stochastic collocation with model order reduction for high-dimensional systems. International Journal for Numerical Methods in Engineering, 117(11):1079–1116, 2019. URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.5992, arXiv:https://onlinelibrary.wiley.com/doi/pdf/10.1002/nme.5992, doi:https://doi.org/10.1002/nme.5992.
Lieven De Lathauwer, Bart De Moor, and Joos Vandewalle. A multilinear singular value decomposition. SIAM Journal on Matrix Analysis and Applications, 21(4):1253–1278, January 2000. URL: https://doi.org/10.1137/s0895479896305696, doi:10.1137/s0895479896305696.
Pauli Virtanen, Ralf Gommers, Travis E. Oliphant, Matt Haberland, Tyler Reddy, David Cournapeau, Evgeni Burovski, Pearu Peterson, Warren Weckesser, Jonathan Bright, Stéfan J. van der Walt, Matthew Brett, Joshua Wilson, K. Jarrod Millman, Nikolay Mayorov, Andrew R. J. Nelson, Eric Jones, Robert Kern, Eric Larson, C J Carey, İlhan Polat, Yu Feng, Eric W. Moore, Jake VanderPlas, Denis Laxalde, Josef Perktold, Robert Cimrman, Ian Henriksen, E. A. Quintero, Charles R. Harris, Anne M. Archibald, Antônio H. Ribeiro, Fabian Pedregosa, Paul van Mulbregt, and SciPy 1.0 Contributors. SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python. Nature Methods, 17:261–272, 2020. doi:10.1038/s41592-019-0686-2.
Ralph Smith. Uncertainty quantification : Theory, Implementation, and Applications. Society for Industrial and Applied Mathematics, Philadelphia, 2013. ISBN 978-1-611973-21-1.
Kenneth P. Burnham and David R. Anderson, editors. Model Selection and Multimodel Inference. Springer New York, 2004. URL: https://doi.org/10.1007/b97636, doi:10.1007/b97636.
Adrian Raftery, Michael Newton, Jaya Satagopan, and Pavel Krivitsky. Estimating the integrated likelihood via posterior simulation using the harmonic mean identity. Bayesian statistics, 8:, 01 2007.
Siu-Kui Au and James L. Beck. Estimation of small failure probabilities in high dimensions by subset simulation. Probabilistic Engineering Mechanics, 16(4):263–277, 2001. URL: https://www.sciencedirect.com/science/article/pii/S0266892001000194, doi:https://doi.org/10.1016/S0266-8920(01)00019-4.
Rüdiger Rackwitz and Bernd Flessler. Structural reliability under combined random load sequences. Computers & Structures, 9(5):489–494, 1978. URL: https://www.sciencedirect.com/science/article/pii/0045794978900469, doi:https://doi.org/10.1016/0045-7949(78)90046-9.
Xiaoping Du. Probabilistic engineering design, chapter 7, first order and second reliability methods. https://pdesign.sitehost.iu.edu/me360/ch7.pdf, 2005.
K. Breitung and M. Hohenbichler. Asymptotic approximations for multivariate integrals with an application to multinormal probabilities. Journal of Multivariate Analysis, 30(1):80–97, 1989. URL: https://www.sciencedirect.com/science/article/pii/0047259X89900894, doi:https://doi.org/10.1016/0047-259X(89)90089-4.
Audrey Olivier, Dimitris G. Giovanis, B.S. Aakash, Mohit Chauhan, Lohit Vandanapu, and Michael D. Shields. Uqpy: a general purpose python package and development environment for uncertainty quantification. Journal of Computational Science, 47:101204, 2020. URL: https://www.sciencedirect.com/science/article/pii/S1877750320305056, doi:https://doi.org/10.1016/j.jocs.2020.101204.
K.D. Tocher. The Art of Simulation. Electrical engineering series. English Universities Press, 1963. URL: https://books.google.com/books?id=ccEvAAAAYAAJ.
Michael D. Shields, Kirubel Teferra, Adam Hapij, and Raymond P. Daddazio. Refined stratified sampling for efficient monte carlo based uncertainty quantification. Reliability Engineering & System Safety, 142:310–325, 2015. URL: https://www.sciencedirect.com/science/article/pii/S0951832015001726, doi:https://doi.org/10.1016/j.ress.2015.05.023.
Michael D. Shields. Adaptive monte carlo analysis for strongly nonlinear stochastic systems. Reliability Engineering & System Safety, 175:207–224, 2018. URL: https://www.sciencedirect.com/science/article/pii/S0951832017308827, doi:https://doi.org/10.1016/j.ress.2018.03.018.
Wouter Edeling, Richard Dwight, and Paola Cinnella. Simplex-stochastic collocation method with improved scalability. Journal of Computational Physics, 310:301–328, 04 2016. doi:10.1016/j.jcp.2015.12.034.
B. Echard, N. Gayton, and M. Lemaire. Ak-mcs: an active learning reliability method combining kriging and monte carlo simulation. Structural Safety, 33(2):145–154, 2011. URL: https://www.sciencedirect.com/science/article/pii/S0167473011000038, doi:https://doi.org/10.1016/j.strusafe.2011.01.002.
Donald R. Jones, Matthias Schonlau, and William J. Welch. Efficient global optimization of expensive black-box functions.” journal of global optimization. Journal of Global Optimization, 13(4):455–492, 1998. URL: https://doi.org/10.1023/a:1008306431147, doi:10.1023/a:1008306431147.
V. S. Sundar and Michael D. Shields. Reliability analysis using adaptive kriging surrogates with multimodel inference. ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering, 2019. doi:10.1061/AJRUA6.0001005.
B. J. Bichon, M. S. Eldred, L. P. Swiler, S. Mahadevan, and J. M. McFarland. Efficient global reliability analysis for nonlinear implicit performance functions. AIAA Journal, 46(10):2459–2468, 2008. doi:10.2514/1.34321.
Chen Quin Lam. Sequential Adaptive Designs in Computer Experiments for Response Surface Model Fit. Ohio State University, USA, 2008. ISBN 9780549716860. AAI3321369.
Andrew Gelman, John B. Carlin, Hal S. Stern, David B. Dunson, Aki Vehtari, and Donald B. Rubin. Bayesian Data Analysis. Chapman and Hall/CRC, November 2013. URL: https://doi.org/10.1201/b16018, doi:10.1201/b16018.
Heikki Haario, Marko Laine, Antonietta Mira, and Eero Saksman. DRAM: efficient adaptive MCMC. Statistics and Computing, 16(4):339–354, December 2006. URL: https://doi.org/10.1007/s11222-006-9438-0, doi:10.1007/s11222-006-9438-0.
Jasper A Vrugt, James M Hyman, Bruce A Robinson, Dave Higdon, Cajo J F Ter Braak, and Cees G H Diks. Accelerating markov chain monte carlo simulation by differential evolution with self-adaptive randomized subspace sampling. International Journal of Nonlinear Sciences and Numerical Simulation, 1 2008. URL: https://www.osti.gov/biblio/960766.
Jasper A. Vrugt. Markov chain monte carlo simulation using the dream software package: theory, concepts, and matlab implementation. Environmental Modelling & Software, 75:273–316, 2016. URL: https://www.sciencedirect.com/science/article/pii/S1364815215300396, doi:https://doi.org/10.1016/j.envsoft.2015.08.013.
Jonathan Goodman and Jonathan Weare. Ensemble samplers with affine invariance. Communications in Applied Mathematics and Computational Science, 5(1):65–80, January 2010. URL: https://doi.org/10.2140/camcos.2010.5.65, doi:10.2140/camcos.2010.5.65.
Daniel Foreman-Mackey, David W. Hogg, Dustin Lang, and Jonathan Goodman. Emcee: the mcmc hammer. Publications of the Astronomical Society of the Pacific, 125(925):306–312, Mar 2013. URL: http://dx.doi.org/10.1086/670067, doi:10.1086/670067.
David J. Earl and Michael W. Deem. Parallel tempering: theory, applications, and new perspectives. Phys. Chem. Chem. Phys., 7:3910–3916, 2005. URL: http://dx.doi.org/10.1039/B509983H, doi:10.1039/B509983H.
Paul M. Goggans and Ying Chi. Using thermodynamic integration to calculate the posterior probability in bayesian model selection problems. In AIP Conference Proceedings 707, 59. 2004. doi:https://doi.org/10.1063/1.1751356.
Jianye Ching and Yi-Chu Chen. Transitional markov chain monte carlo method for bayesian model updating, model class selection, and model averaging. Journal of Engineering Mechanics, 133(7):816–832, 2007. URL: https://ascelibrary.org/doi/abs/10.1061/%28ASCE%290733-9399%282007%29133%3A7%28816%29, arXiv:https://ascelibrary.org/doi/pdf/10.1061/%28ASCE%290733-9399%282007%29133%3A7%28816%29, doi:10.1061/(ASCE)0733-9399(2007)133:7(816).
Han Xiao, Kashif Rasul, and Roland Vollgraf. Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. 2017. arXiv:cs.LG/1708.07747.
Solomon Kullback and Richard A Leibler. On information and sufficiency. The annals of mathematical statistics, 22(1):79–86, 1951.
Ponkrshnan Thiagarajan and Susanta Ghosh. Jensen-shannon divergence based novel loss functions for bayesian neural networks. arXiv preprint arXiv:2209.11366, 2022.
Jacob Deasy, Nikola Simidjievski, and Pietro Liò. Constraining variational inference with geometric jensen-shannon divergence. Advances in Neural Information Processing Systems, 33:10647–10658, 2020.
Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differential equations. arXiv preprint arXiv:2010.08895, 2021. URL: https://arxiv.org/abs/2010.08895.
Yarin Gal and Zoubin Ghahramani. Dropout as a bayesian approximation: representing model uncertainty in deep learning. In international conference on machine learning, 1050–1059. PMLR, 2016.
Lu Lu, Pengzhan Jin, Guofei Pang, Zhongqiang Zhang, and George Em Karniadakis. Learning nonlinear operators via deeponet based on the universal approximation theorem of operators. Nature Machine Intelligence, 3:218–229, 2021. URL: https://doi.org/10.1038/s42256-021-00302-5.
Somdatta Goswami, Katiana Kontolati, Michael D Shields, and George Em Karniadakis. Deep transfer operator learning for partial differential equations under conditional shift. Nature Machine Intelligence, 4(12):1155–1164, 2022.
Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: convolutional networks for biomedical image segmentation. In Medical image computing and computer-assisted intervention–MICCAI 2015: 18th international conference, Munich, Germany, October 5-9, 2015, proceedings, part III 18, 234–241. Springer, 2015.
Charles Blundell, Julien Cornebise, Koray Kavukcuoglu, and Daan Wierstra. Weight uncertainty in neural network. In International conference on machine learning, 1613–1622. PMLR, 2015.
Steve Brooks, Andrew Gelman, Galin Jones, and Xiao-Li Meng. Handbook of Markov Chain Monte Carlo. Chapman and Hall/CRC, May 2011. ISBN 9780429138508. URL: http://dx.doi.org/10.1201/b10905, doi:10.1201/b10905.
Sourav Chatterjee. A new coefficient of correlation. Journal of the American Statistical Association, 116(536):2009–2022, 2021. URL: https://doi.org/10.1080/01621459.2020.1758115, arXiv:https://doi.org/10.1080/01621459.2020.1758115, doi:10.1080/01621459.2020.1758115.
Fabrice Gamboa, Pierre Gremaud, Thierry Klein, and Agnès Lagnoux. Global sensitivity analysis: a new generation of mighty estimators based on rank statistics. 2020. arXiv:2003.01772.
Fabrice Gamboa, Thierry Klein, and Agnès Lagnoux. Sensitivity analysis based on cramér–von mises distance. SIAM/ASA Journal on Uncertainty Quantification, 6(2):522–548, 2018. URL: https://doi.org/10.1137/15M1025621, arXiv:https://doi.org/10.1137/15M1025621, doi:10.1137/15M1025621.
Fabrice Gamboa, Alexandre Janon, Thierry Klein, and Agnès Lagnoux. Sensitivity analysis for multidimensional and functional outputs. Electronic Journal of Statistics, 8(1):575 – 603, 2014. URL: https://doi.org/10.1214/14-EJS895, doi:10.1214/14-EJS895.
Francesca Campolongo, Jessica Cariboni, and Andrea Saltelli. An effective screening design for sensitivity analysis of large models. Environmental Modelling & Software, 22(10):1509–1518, 2007. Modelling, computer-assisted simulations, and mapping of dangerous phenomena for hazard assessment. URL: https://www.sciencedirect.com/science/article/pii/S1364815206002805, doi:https://doi.org/10.1016/j.envsoft.2006.10.004.
A. Saltelli. Global sensitivity analysis: the primer. John Wiley, 2008. ISBN 9780470059975. URL: https://onlinelibrary.wiley.com/doi/book/10.1002/9780470725184.
Andrea Saltelli. Making best use of model evaluations to compute sensitivity indices. Computer Physics Communications, 145(2):280–297, 2002. URL: https://www.sciencedirect.com/science/article/pii/S0010465502002801, doi:https://doi.org/10.1016/S0010-4655(02)00280-1.
M. Shinozuka and C.-M. Jan. Digital simulation of random processes and its applications. Journal of Sound and Vibration, 25(1):111–128, 1972. URL: https://www.sciencedirect.com/science/article/pii/0022460X72906001, doi:https://doi.org/10.1016/0022-460X(72)90600-1.
Masanobu Shinozuka and George Deodatis. Simulation of Stochastic Processes by Spectral Representation. Applied Mechanics Reviews, 44(4):191–204, 04 1991. URL: https://doi.org/10.1115/1.3119501, arXiv:https://asmedigitalcollection.asme.org/appliedmechanicsreviews/article-pdf/44/4/191/5435905/191\_1.pdf, doi:10.1115/1.3119501.
Masanobu Shinozuka and George Deodatis. Simulation of Multi-Dimensional Gaussian Stochastic Fields by Spectral Representation. Applied Mechanics Reviews, 49(1):29–53, 01 1996. URL: https://doi.org/10.1115/1.3101883, arXiv:https://asmedigitalcollection.asme.org/appliedmechanicsreviews/article-pdf/49/1/29/5437403/29\_1.pdf, doi:10.1115/1.3101883.
George Deodatis. Simulation of ergodic multivariate stochastic processes. Journal of Engineering Mechanics, 122(8):778–787, 1996. doi:10.1061/(ASCE)0733-9399(1996)122:8(778).
S. P. Huang, S. T. Quek, and K. K. Phoon. Convergence study of the truncated karhunen–loeve expansion for simulation of stochastic processes. International Journal for Numerical Methods in Engineering, 52(9):1029–1043, 2001. URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.255, arXiv:https://onlinelibrary.wiley.com/doi/pdf/10.1002/nme.255, doi:https://doi.org/10.1002/nme.255.
K.K. Phoon, S.P. Huang, and S.T. Quek. Simulation of second-order processes using karhunen–loeve expansion. Computers & Structures, 80(12):1049–1060, 2002. URL: https://www.sciencedirect.com/science/article/pii/S0045794902000640, doi:https://doi.org/10.1016/S0045-7949(02)00064-0.
Wolfgang Betz, Iason Papaioannou, and Daniel Straub. Numerical methods for the discretization of random fields by means of the karhunen–loève expansion. Computer Methods in Applied Mechanics and Engineering, 271:109–129, 2014. URL: https://www.sciencedirect.com/science/article/pii/S0045782513003502, doi:https://doi.org/10.1016/j.cma.2013.12.010.
Michael D. Shields and Hwanpyo Kim. Simulation of higher-order stochastic processes by spectral representation. Probabilistic Engineering Mechanics, 47:1–15, 2017.
Lohit Vandanapu and Michael D. Shields. 3rd-order spectral representation method: simulation of multi-dimensional random fields and ergodic multi-variate random processes with fast fourier transform implementation. Probabilistic Engineering Mechanics, 64:103128, 2021. URL: https://www.sciencedirect.com/science/article/pii/S0266892021000126, doi:https://doi.org/10.1016/j.probengmech.2021.103128.
Lohit Vandanapu and Michael D. Shields. 3rd-order spectral representation method: part ii – ergodic multi-variate random processes with fast fourier transform. 2019. arXiv:1911.10251.
Mircea Grigoriu. Applied non-gaussian processes : examples, theory, simulation, linear random vibration, and matlab solutions. In Applied non-Gaussian processes : examples, theory, simulation, linear random vibration, and MATLAB solutions. 1995.
M.D. Shields and G. Deodatis. A simple and efficient methodology to approximate a general non-gaussian stationary stochastic vector process by a translation process with applications in wind velocity simulation. Probabilistic Engineering Mechanics, 31:19–29, 2013. URL: https://www.sciencedirect.com/science/article/pii/S0266892012000549, doi:https://doi.org/10.1016/j.probengmech.2012.10.003.
Hwanpyo Kim and Michael Shields. Modeling strongly non-gaussian non-stationary stochastic processes using the iterative translation approximation method and karhunen-loeve expansion. Computers & Structures, 161:31–42, 08 2015. doi:10.1016/j.compstruc.2015.08.010.
Zhibao Zheng and Hongzhe Dai. Simulation of multi-dimensional random fields by karhunen–loève expansion. Computer Methods in Applied Mechanics and Engineering, 324:221–247, 2017. URL: https://www.sciencedirect.com/science/article/pii/S0045782516318692, doi:https://doi.org/10.1016/j.cma.2017.05.022.
M. Grigoriu. Reduced order models for random functions. application to stochastic problems. Applied Mathematical Modelling, 33(1):161–175, 2009. URL: https://www.sciencedirect.com/science/article/pii/S0307904X07002818, doi:https://doi.org/10.1016/j.apm.2007.10.023.
Nora Lüthen, Stefano Marelli, and Bruno Sudret. Automatic selection of basis-adaptive sparse polynomial chaos expansions for engineering applications. 2021. arXiv:2009.04800.
Géraud Blatman and Bruno Sudret. Adaptive sparse polynomial chaos expansion based on least angle regression. Journal of Computational Physics, 230(6):2345–2367, 2011. doi:10.1016/j.jcp.2010.12.021.
Bradley Efron, Trevor Hastie, Iain Johnstone, and Robert Tibshirani. Least angle regression. The Annals of Statistics, 32(2):407–451, 2004. doi:10.2307/3448465.
A. Nataf. Détermination des distributions de probabilités dont les marges sont données. C. R. Acad. Sci., Paris, 255:42–43, 1962.
Régis Lebrun and Anne Dutfoy. An innovating analysis of the nataf transformation from the copula viewpoint. Probabilistic Engineering Mechanics, 24(3):312–320, 2009. URL: https://www.sciencedirect.com/science/article/pii/S0266892008000660, doi:https://doi.org/10.1016/j.probengmech.2008.08.001.
Gene H. Golub and Charles F. Van Loan. Matrix Computations. The Johns Hopkins University Press, third edition, 1996.
Ke Ye and Lek-Heng Lim. Schubert varieties and distances between subspaces of different dimensions. 2016. arXiv:1407.0900.
Jihun Hamm and Daniel D. Lee. Grassmann discriminant analysis: a unifying view on subspace-based learning. In Proceedings of the 25th International Conference on Machine Learning, ICML '08, 376–383. New York, NY, USA, 2008. Association for Computing Machinery. URL: https://doi.org/10.1145/1390156.1390204, doi:10.1145/1390156.1390204.
Mehrtash T. Harandi, Mathieu Salzmann, Sadeep Jayasumana, Richard Hartley, and Hongdong Li. Expanding the family of grassmannian kernels: an embedding perspective. 2014. arXiv:1407.1123.
George D Pasparakis, Lori Graham-Brady, and Michael D Shields. Bayesian neural networks for predicting uncertainty in full-field material response. arXiv preprint arXiv:2406.14838, 2024.
Anindya Bhaduri, Ashwini Gupta, and Lori Graham-Brady. Stress field prediction in fiber-reinforced composite materials using a deep learning approach. Composites Part B: Engineering, 238:109879, 2022.