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arxiv: 2606.17233 · v1 · pith:LFG74WJInew · submitted 2026-06-15 · 💻 cs.LG · stat.ML

Uncertainty Quantification of Engineering Structures by Polynomial Chaos Expansion and Multivariate Active Learning

Qitian Lu , Jafar Jafari-Asl , Panagiotis Spyridis , Lukas Novak This is my paper

Pith reviewed 2026-06-27 03:23 UTC · model grok-4.3

classification 💻 cs.LG stat.ML
keywords polynomial chaos expansionactive learningsurrogate modelinguncertainty quantificationmultivariate outputsadaptive samplingengineering structures
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The pith

A single adaptive sampling strategy for polynomial chaos expansions can approximate multiple engineering outputs simultaneously by aggregating their local variance contributions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that an existing sequential sampling method for building polynomial chaos expansion surrogates can be extended to cases where one model produces several quantities of interest at once. Instead of building separate experimental designs for each output, the approach selects new points from a candidate pool by measuring each point's contribution to total output variance across all quantities, while also maintaining distance-based coverage of the input space. This avoids the cost of repeated sampling and preserves correlations among outputs. Numerical tests on engineering problems show the resulting surrogates achieve higher accuracy and more stable estimates of means and variances than non-adaptive Latin hypercube designs of comparable size. The central object is therefore the aggregated-variance selection rule that turns a univariate active-learning procedure into a multivariate one.

Core claim

The adaptive sequential sampling procedure is generalized to vector-valued quantities of interest by replacing per-output variance indicators with a single scalar that sums the local variance contributions of every output; new samples are then drawn from a candidate pool to maximize this aggregate while enforcing a minimum-distance exploration term, yielding experimental designs that simultaneously improve surrogate accuracy, stability, and second-moment reliability for all outputs.

What carries the argument

The aggregated local-variance selection criterion, which sums each candidate point's estimated contribution to output variance across every quantity of interest and balances it against a distance-to-existing-points term.

If this is right

  • Fewer total model evaluations are needed to reach a target accuracy level when multiple quantities must be approximated from the same high-fidelity code.
  • Estimated means and variances of all outputs become more stable across repeated surrogate constructions.
  • Correlations among outputs are automatically respected because all outputs share the same set of training points.
  • The method scales to any number of outputs without a combinatorial increase in sampling effort.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same aggregation idea could be tested on other surrogate families such as Gaussian processes or neural networks to check whether the benefit is specific to polynomial chaos expansions.
  • If the outputs are known to be strongly correlated, the variance aggregation could be weighted by a correlation matrix estimated from an initial pilot sample.
  • The approach might be combined with dimension-reduction techniques when the input space itself is high-dimensional.

Load-bearing premise

That a single experimental design chosen from the summed variance contributions of all outputs will remain adequate for every individual output even when the outputs respond to the inputs with markedly different sensitivities.

What would settle it

A numerical experiment on a model whose outputs have strongly opposing sensitivities (one output most sensitive to input A, another to input B) in which the aggregated design produces visibly larger error on one output than an output-specific design of the same total size.

Figures

Figures reproduced from arXiv: 2606.17233 by Jafar Jafari-Asl, Lukas Novak, Panagiotis Spyridis, Qitian Lu.

Figure 1
Figure 1. Figure 1: Sample distributions for the 2D mirror-line singularity problem: (a) samples generated by LHS and (b) initial and [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: 2D mirror line singularities: comparison of [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Geometry and loading configuration of the three-span reinforced concrete beam [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Reinforced concrete beam: comparison of Θ criterion and LHS for PCE surrogate model construction: mean and standard deviation of maximum absolute error [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Reinforced concrete beam: comparison of Θ criterion and LHS for PCE surrogate model construction: mean and standard deviation of MAE. -4 -3 -2 -1 0 l o g 1 0 ( ε v a r ) Output 1 Theta LHS Output 2 Output 3 Output 4 Output 5 -4 -3 -2 -1 0 l o g 1 0 ( ε v a r ) Output 6 Output 7 Output 8 Output 9 Output 10 100 200 300 400 500 nsample -4 -3 -2 -1 0 l o g 1 0 ( ε v a r ) Output 11 100 200 300 400 500 nsample … view at source ↗
Figure 6
Figure 6. Figure 6: Reinforced concrete beam: comparison of Θ criterion and LHS for PCE surrogate model construction: mean and standard deviation of relative variance error. 12 [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Reinforced concrete beam: comparison of Θ criterion and LHS for PCE surrogate model construction: mean and standard deviation of LOO error. 200 400 600 800 1000 nsample 2 4 6 log10 (max A E) Output 1 Theta LHS σ 2 ref 200 400 600 800 1000 nsample 2 4 6 Output 2 200 400 600 800 1000 nsample 1 2 3 4 log10 (M A E) Output 1 200 400 600 800 1000 nsample 2 3 4 Output 2 200 400 600 800 1000 nsample 3 6 9 12 log10… view at source ↗
Figure 8
Figure 8. Figure 8: OWT under wind excitation: comparison of [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Schematic of multiphysics environmental interactions affecting the offshore wind turbine [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Multiphysics OWT: comparison of Θ criterion and LHS for PCE surrogate model construction: mean and standard deviation of maximum absolute error. such as LAR are employed for PCE coefficient estimation, the iterative selection of basis functions can introduce additional computational overhead. While LAR has advantages for handling large candidate basis sets in high￾dimensional cases, its cost may become no… view at source ↗
read the original abstract

In many engineering applications, a single high-fidelity model produces multiple quantities of interest (QoIs) under the same input parameters, e.g. finite element models of complex physical systems. To alleviate the high computational cost of direct model evaluations, surrogate models are widely used to construct efficient approximations of model responses. Naturally, the accuracy of surrogates strongly depends on the quality of the experimental design (ED). However, a single ED may not provide an adequate representation for all outputs simultaneously, especially when different outputs exhibit varying sensitivities to the input variables. A straightforward solution is to perform separate sampling for each output, but this results in increased sampling complexity and computational cost. From a statistical perspective, such an approach also ignores potential correlations among all outputs and may compromise data consistency. To address this issue, an adaptive sequential sampling method for constructing polynomial chaos expansion surrogate models is generalized for vector valued QoIs. The method sequentially selects new samples from a candidate pool based on their local contribution to the output variance, while balancing distance-based exploration of the input space and exploitation of aggregated variance information across all outputs. Its performance is compared with non-sequential Latin Hypercube Sampling through several numerical examples from engineering problems. Numerical results demonstrate that the proposed strategy improves both surrogate accuracy and stability, and provides a more reliable estimation of second-order statistics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript generalizes an adaptive sequential sampling method for polynomial chaos expansion (PCE) surrogates to vector-valued quantities of interest (QoIs). New samples are chosen from a candidate pool by maximizing an acquisition function that aggregates local variance contributions across outputs while balancing distance-based exploration; performance is compared to non-sequential Latin Hypercube Sampling on several engineering numerical examples, with the claim that the strategy improves surrogate accuracy, stability, and second-order statistics estimation.

Significance. If the empirical claims are substantiated with per-output quantitative validation, the method could reduce sampling cost for multi-output engineering UQ problems while preserving output correlations, offering a practical alternative to separate per-QoI designs.

major comments (2)
  1. [Abstract and numerical examples] Abstract and numerical examples section: the central claim that the aggregated-variance design improves accuracy for every QoI rests on the assumption that high-variance regions overlap sufficiently across outputs. No per-output error tables, comparisons against output-specific adaptive designs, or worst-case analysis are supplied to verify that the single experimental design does not systematically under-sample regions critical to the most sensitive output.
  2. [Method description] Method description: the acquisition function aggregates local variance contributions, yet no convergence-rate bound or sensitivity-misalignment test is provided to guarantee that the resulting PCE still converges at the expected rate for each marginal output when sensitivities differ.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address the major comments point by point below, indicating planned revisions to the manuscript.

read point-by-point responses

  1. Referee: [Abstract and numerical examples] Abstract and numerical examples section: the central claim that the aggregated-variance design improves accuracy for every QoI rests on the assumption that high-variance regions overlap sufficiently across outputs. No per-output error tables, comparisons against output-specific adaptive designs, or worst-case analysis are supplied to verify that the single experimental design does not systematically under-sample regions critical to the most sensitive output.

    Authors: The manuscript reports aggregate accuracy and stability metrics over the vector QoI, consistent with the goal of a single design that respects output correlations. We acknowledge the absence of per-output error tables and direct comparisons to output-specific designs. To address this, the revised manuscript will include per-output error metrics for the numerical examples and a brief discussion of cases where output sensitivities may differ substantially. revision: yes

  2. Referee: [Method description] Method description: the acquisition function aggregates local variance contributions, yet no convergence-rate bound or sensitivity-misalignment test is provided to guarantee that the resulting PCE still converges at the expected rate for each marginal output when sensitivities differ.

    Authors: The method is presented as a practical heuristic that aggregates variance information while incorporating distance-based exploration; no theoretical convergence-rate analysis is derived. The numerical examples demonstrate reliable performance on the chosen engineering test cases. A general guarantee for arbitrary sensitivity misalignment would require substantial additional theoretical development beyond the scope of this applied contribution. We will add a short limitations paragraph noting this point. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation remains self-contained

full rationale

The paper describes an adaptive sequential sampling strategy for multivariate PCE that aggregates local variance contributions across outputs to select new design points while balancing exploration. No equations, derivations, or performance claims reduce the reported accuracy/stability improvements to a quantity defined by construction from the method's own fitted inputs or self-citations. Numerical examples are presented as external empirical checks rather than tautological predictions. The central premise relies on standard PCE theory and active-learning heuristics without load-bearing self-referential loops.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the method description does not introduce new mathematical objects or fitting constants beyond the standard polynomial chaos framework.

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Reference graph

Works this paper leans on

147 extracted references · 78 canonical work pages

  1. [1]

    2022 , issn =

    On distribution-based global sensitivity analysis by polynomial chaos expansion , journal =. 2022 , issn =. doi:https://doi.org/10.1016/j.compstruc.2022.106808 , author =

    work page doi:10.1016/j.compstruc.2022.106808 2022

  2. [2]

    Papadopoulos and Dimitris Syvridis and Elias N

    Aristeides D. Papadopoulos and Dimitris Syvridis and Elias N. Glytsis , journal =. Sparse polynomial chaos algorithm with a variance-adaptive design domain for the uncertainty quantification and optimization of grating structures , volume =. 2025 , doi =

    2025

  3. [3]

    2025 , issn =

    Multivariate sensitivity-adaptive polynomial chaos expansion for high-dimensional surrogate modeling and uncertainty quantification , journal =. 2025 , issn =. doi:https://doi.org/10.1016/j.apm.2024.115746 , author =

    work page doi:10.1016/j.apm.2024.115746 2025

  4. [4]

    2026 , issn =

    Polynomial chaos expansion for operator learning , journal =. 2026 , issn =. doi:https://doi.org/10.1016/j.cma.2026.118796 , author =

    work page doi:10.1016/j.cma.2026.118796 2026

  5. [5]

    2024 , issn =

    Physics-constrained polynomial chaos expansion for scientific machine learning and uncertainty quantification , journal =. 2024 , issn =. doi:https://doi.org/10.1016/j.cma.2024.117314 , author =

    work page doi:10.1016/j.cma.2024.117314 2024

  6. [6]

    , year =

    Physics-informed polynomial chaos expansions , journal =. 2024 , issn =. doi:https://doi.org/10.1016/j.jcp.2024.112926 , author =

    work page doi:10.1016/j.jcp.2024.112926 2024

  7. [7]

    AIP Advances , volume =

    Ayoob, Keenjhar and Zafar, Tayyab and Hamza, Amir and Wang, Zhonglai , title =. AIP Advances , volume =. 2025 , month =

    2025

  8. [8]

    Structural Concrete , volume =

    Slowik, Ondřej and Lehký, David and Novák, Drahomír , title =. Structural Concrete , volume =. doi:https://doi.org/10.1002/suco.202000455 , year =

    work page doi:10.1002/suco.202000455

  9. [9]

    2023 , issn =

    Active learning-based domain adaptive localized polynomial chaos expansion , journal =. 2023 , issn =. doi:https://doi.org/10.1016/j.ymssp.2023.110728 , author =

    work page doi:10.1016/j.ymssp.2023.110728 2023

  10. [10]

    SIAM Journal on Scientific Computing , volume =

    Wan, Xiaoliang and Karniadakis, George Em , title =. SIAM Journal on Scientific Computing , volume =. 2006 , doi =

    2006

  11. [11]

    Kiefer and J

    J. Kiefer and J. Wolfowitz , title =. doi:10.4153/cjm-1960-030-4 , year =

    work page doi:10.4153/cjm-1960-030-4 1960

  12. [12]

    Journal of Engineering Mechanics , volume =

    Armen Der Kiureghian and Pei‐Ling Liu , title =. Journal of Engineering Mechanics , volume =

  13. [13]

    Distance-based optimal sampling in a hypercube:

    Miroslav Vo. Distance-based optimal sampling in a hypercube:. Advances in Engineering Software , volume =. 2019 , issn =

    2019

  14. [14]

    Distance-based optimal sampling in a hypercube:

    Miroslav Vo. Distance-based optimal sampling in a hypercube:. Advances in Engineering Software , volume =. 2020 , issn =

    2020

  15. [15]

    A literature survey of benchmark functions for global optimisation problems , author=. Int. J. Math. Model. Numer. Optimisation , year=

  16. [16]

    2019 , issn =

    Polynomial chaos expansions for dependent random variables , journal =. 2019 , issn =. doi:https://doi.org/10.1016/j.cma.2019.03.049 , author =

    work page doi:10.1016/j.cma.2019.03.049 2019

  17. [17]

    2010 , issn =

    Efficient computation of global sensitivity indices using sparse polynomial chaos expansions , journal =. 2010 , issn =. doi:10.1016/j.ress.2010.06.015 , author =

    work page doi:10.1016/j.ress.2010.06.015 2010

  18. [18]

    SIAM/ASA Journal on Uncertainty Quantification , volume =

    Lüthen, Nora and Marelli, Stefano and Sudret, Bruno , title =. SIAM/ASA Journal on Uncertainty Quantification , volume =. 2021 , doi =

    2021

  19. [19]

    2012 , note =

    Global sensitivity analysis using sparse grid interpolation and polynomial chaos , journal =. 2012 , note =. doi:10.1016/j.ress.2011.07.011 , author =

    work page doi:10.1016/j.ress.2011.07.011 2012

  20. [20]

    Narayan and J

    Akil C. Narayan and J. Jakeman and Tao Zhou , journal=. A. 2017 , volume=

    2017

  21. [21]

    2014 , issn =

    Grid and basis adaptive polynomial chaos techniques for sensitivity and uncertainty analysis , journal =. 2014 , issn =. doi:10.1016/j.jcp.2013.12.025 , author =

    work page doi:10.1016/j.jcp.2013.12.025 2014

  22. [22]

    The SMAI journal of computational mathematics , pages =

    Albert Cohen and Giovanni Migliorati , title =. The SMAI journal of computational mathematics , pages =. 2017 , doi =

    2017

  23. [23]

    2019 , issn =

    An efficient and robust adaptive sampling method for polynomial chaos expansion in sparse Bayesian learning framework , journal =. 2019 , issn =. doi:10.1016/j.cma.2019.04.046 , author =

    work page doi:10.1016/j.cma.2019.04.046 2019

  24. [24]

    SIAM Journal on Numerical Analysis , volume =

    Migliorati, Giovanni , title =. SIAM Journal on Numerical Analysis , volume =. 2019 , doi =

    2019

  25. [25]

    Journal of Computational Physics , volume =

    Basis adaptive sample efficient polynomial chaos. Journal of Computational Physics , volume =. 2018 , issn =. doi:10.1016/j.jcp.2018.03.035 , author =

    work page doi:10.1016/j.jcp.2018.03.035 2018

  26. [26]

    doi:10.1016/j.cma.2017.12.019 , year =

    Mohammad Hadigol and Alireza Doostan , title =. doi:10.1016/j.cma.2017.12.019 , year =

    work page doi:10.1016/j.cma.2017.12.019 2017

  27. [27]

    and Canfield, Robert A

    Choi, Seung-Kyum and Grandhi, Ramana V. and Canfield, Robert A. and Pettit, Chris L. , title =. AIAA Journal , volume =. 2004 , doi =

    2004

  28. [28]

    Jerrad Hampton and Alireza Doostan , title =. 2015. doi:10.1016/j.cma.2015.02.006

    work page doi:10.1016/j.cma.2015.02.006 2015

  29. [29]

    , TITLE =

    Conover, W.J. , TITLE =. 1975 , url =

    1975

  30. [30]

    Jerrad Hampton and Alireza Doostan , title =. 2015. doi:10.1016/j.jcp.2014.09.019

    work page doi:10.1016/j.jcp.2014.09.019 2015

  31. [31]

    doi:10.1137/16m1103488 , year =

    Noura Fajraoui and Stefano Marelli and Bruno Sudret , title =. doi:10.1137/16m1103488 , year =

    work page doi:10.1137/16m1103488

  32. [32]

    Technometrics , volume =

    Otto Dykstra , title =. Technometrics , volume =. 1971 , publisher =

    1971

  33. [33]

    Sparse polynomial chaos expansions via compressed sensing and D -optimal design

    Paul Diaz and Alireza Doostan and Jerrad Hampton. Sparse polynomial chaos expansions via compressed sensing and D -optimal design. 2018. doi:10.1016/j.cma.2018.03.020

    work page doi:10.1016/j.cma.2018.03.020 2018

  34. [34]

    Journal of the Royal Statistical Society

    Algorithm. Journal of the Royal Statistical Society. Series C (Applied Statistics) , number =. doi:10.2307/2986264 , author =

    work page doi:10.2307/2986264

  35. [35]

    Shields , title =

    Michael D. Shields , title =. Reliability Engineering. doi:10.1016/j.ress.2018.03.018 , year =

    work page doi:10.1016/j.ress.2018.03.018 2018

  36. [36]

    Comparison of Design Optimality Criteria of Reduced Models for Response Surface Designs in the Hypercube , journal =

    John J Borkowski and Elsie S Valeroso , publisher =. Comparison of Design Optimality Criteria of Reduced Models for Response Surface Designs in the Hypercube , journal =. doi:10.1198/00401700152672564 , year =

    work page doi:10.1198/00401700152672564

  37. [37]

    Shields , journal=

    M. Shields , journal=. Refined Latinized Stratified Sampling: a Robust Sequential Sample Size Extension Methodology for High-Dimensional. 2016 , volume=

    2016

  38. [38]

    2006 , issn =

    Refinement strategies for stratified sampling methods , journal =. 2006 , issn =. doi:10.1016/j.ress.2005.11.027 , author =

    work page doi:10.1016/j.ress.2005.11.027 2006

  39. [39]

    2017 , issn =

    Efficient space-filling and near-orthogonality sequential. 2017 , issn =. doi:10.1016/j.cma.2017.05.020 , author =

    work page doi:10.1016/j.cma.2017.05.020 2017

  40. [40]

    Tropp and Anna C

    Joel A. Tropp and Anna C. Gilbert , title =. doi:10.1109/tit.2007.909108 , year =

    work page doi:10.1109/tit.2007.909108 2007

  41. [41]

    2008 , issn =

    Spectral stochastic finite element analysis for laminated composite plates , journal = CMAME, volume =. 2008 , issn =. doi:10.1016/j.cma.2008.07.003 , author =

    work page doi:10.1016/j.cma.2008.07.003 2008

  42. [42]

    2019 , issn =

    A painless intrusive polynomial chaos method with RANS-based applications , journal = CMAME, volume =. 2019 , issn =. doi:10.1016/j.cma.2019.01.018 , author =

    work page doi:10.1016/j.cma.2019.01.018 2019

  43. [43]

    An innovating analysis of the

    R. An innovating analysis of the. Probabilistic Engineering Mechanics , volume =. 2009 , doi =

    2009

  44. [44]

    A generalization of the

    R. A generalization of the. doi:10.1016/j.probengmech.2008.05.001 , year =

    work page doi:10.1016/j.probengmech.2008.05.001 2008

  45. [45]

    Reliability Engineering

    Fan Wang and Heng Li , title =. Reliability Engineering. doi:10.1016/j.ress.2017.12.018 , year =

    work page doi:10.1016/j.ress.2017.12.018 2017

  46. [46]

    doi:10.1002/nme.5505 , year =

    Computation of higher-order moments of generalized polynomial chaos expansions , journal =. doi:10.1002/nme.5505 , year =

    work page doi:10.1002/nme.5505

  47. [47]

    Journal of Computational and Applied Mathematics , doi =

    On linearization and connection coefficients for generalized. Journal of Computational and Applied Mathematics , doi =. 2011 , volume =

    2011

  48. [48]

    Expressing a Probability Density Function in Terms of another

    M. Expressing a Probability Density Function in Terms of another. Journal of Mathematical Chemistry , doi =. 2006 , volume =

    2006

  49. [49]

    Canadian Journal of Mathematics , doi =

    George Gasper , title =. Canadian Journal of Mathematics , doi =. 1970 , publisher =

    1970

  50. [50]

    R. Do. Probabilistic Engineering Mechanics , year =. doi:10.1016/j.probengmech.2009.04.006 , pages =

    work page doi:10.1016/j.probengmech.2009.04.006 2009

  51. [51]

    Journal of Engineering Mechanics , year =

    Armen Der Kiureghian and Pei‐Ling Liu , title =. Journal of Engineering Mechanics , year =. doi:10.1061/(asce)0733-9399(1986)112:1(85) , pages =

    work page doi:10.1061/(asce)0733-9399(1986)112:1(85 1986

  52. [52]

    Katafygiotis and K.M

    L.S. Katafygiotis and K.M. Zuev , title =. doi:10.1016/j.probengmech.2007.12.026 , year =

    work page doi:10.1016/j.probengmech.2007.12.026 2007

  53. [53]

    High-Dimensional p -Norms , booktitle =

    G. High-Dimensional p -Norms , booktitle =. doi:10.1007/978-3-319-12442-1_3 , year =

    work page doi:10.1007/978-3-319-12442-1_3

  54. [54]

    Borgonovo , title=

    E. Borgonovo , title=. Reliability Engineering. 2007 , doi =

    2007

  55. [55]

    Beton- und Stahlbetonbau , doi =

    Novak, Lukas and Novak, Drahomir , title =. Beton- und Stahlbetonbau , doi =. 2018 , volume =

    2018

  56. [56]

    Bayesian Compressive Sensing , year=

    Shihao Ji and Ya Xue and Lawrence Carin , journal=. Bayesian Compressive Sensing , year=

  57. [57]

    Chaospy: An open source tool for designing methods of uncertainty quantification

    Jonathan Feinberg and Hans Petter Langtangen. Chaospy: An open source tool for designing methods of uncertainty quantification. Journal of Computational Science , volume = "11", pages =. 2015. doi:10.1016/j.jocs.2015.08.008

    work page doi:10.1016/j.jocs.2015.08.008 2015

  58. [58]

    Sensitivity Analysis Based on

    Fabrice Gamboa and Thierry Klein and Agn. Sensitivity Analysis Based on. 2018 , volume =

    2018

  59. [59]

    Reliability calculation of time-consuming problems using a small-sample artificial neural network-based response surface method , journal=

    David Lehk. Reliability calculation of time-consuming problems using a small-sample artificial neural network-based response surface method , journal=. doi:10.1007/s00521-016-2485-3 , year =

    work page doi:10.1007/s00521-016-2485-3

  60. [60]

    Ibrahim , title =

    Lingjian Shi and Beibei Sun and Dauda Sh. Ibrahim , title =. doi:10.1007/s00158-019-02210-0 , year =

    work page doi:10.1007/s00158-019-02210-0

  61. [61]

    doi:10.1016/j.strusafe.2018.06.003 , year =

    Stefano Marelli and Bruno Sudret , title =. doi:10.1016/j.strusafe.2018.06.003 , year =

    work page doi:10.1016/j.strusafe.2018.06.003 2018

  62. [62]

    Reliability Engineering

    Yicheng Zhou and Zhenzhou Lu and Wanying Yun , title =. Reliability Engineering. doi:10.1016/j.ress.2020.107025 , year =

    work page doi:10.1016/j.ress.2020.107025 2020

  63. [63]

    doi:10.1002/nme.6351 , year =

    Kai Cheng and Zhenzhou Lu , title =. doi:10.1002/nme.6351 , year =

    work page doi:10.1002/nme.6351

  64. [64]

    doi:10.1002/nme.6495 , year =

    Xufeng Yang and Xin Cheng , title =. doi:10.1002/nme.6495 , year =

    work page doi:10.1002/nme.6495

  65. [65]

    Echard and N

    B. Echard and N. Gayton and M. Lemaire , title =. doi:10.1016/j.strusafe.2011.01.002 , year =

    work page doi:10.1016/j.strusafe.2011.01.002 2011

  66. [66]

    Support-vector networks

    Corinna Cortes and Vladimir Vapnik , title =. doi:10.1007/bf00994018 , year =

    work page doi:10.1007/bf00994018

  67. [67]

    and Novak, D

    Novak, L. and Novak, D. , title=. Proceedings of 2019 International Conference on Quality, Reliability, Risk, Maintenance, and Safety Engineering, QR2MSE 2019 , year=

    2019

  68. [68]

    An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis , journal =

    G. An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis , journal =. 2010 , doi =

    2010

  69. [69]

    Nataf , title =

    A. Nataf , title =. Comptes Rendus de l'Académie des Sciences , year =

  70. [70]

    Journal of Applied Mechanics , volume =

    Polynomial Chaos in Stochastic Finite Elements. Journal of Applied Mechanics , volume =. 1990 , doi =

    1990

  71. [71]

    Berveiller and Bruno Sudret and M

    M. Berveiller and Bruno Sudret and M. Lemaire , title =. Proc. 9th ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability , year =

  72. [72]

    Lukas Novak and Drahomir Novak , title =. Proc. of the Fib Symposium 2019: Concrete - Innovations in Materials, Design and Structures , year =

    2019

  73. [73]

    Robbins and S

    Rosenblatt, Murray , title =. The Annals of Mathematical Statistics , year =. doi:10.1214/aoms/1177729394 , volume =

    work page doi:10.1214/aoms/1177729394

  74. [74]

    Mulani and Robert W

    Mishal Thapa and Sameer B. Mulani and Robert W. Walters. Adaptive weighted least-squares polynomial chaos expansion with basis adaptivity and sequential adaptive sampling. 2020. doi:10.1016/j.cma.2019.112759

    work page doi:10.1016/j.cma.2019.112759 2020

  75. [75]

    Candes and Yaniv Plan , title =

    Emmanuel J. Candes and Yaniv Plan , title =. 2011 , doi =

    2011

  76. [76]

    Jakeman and Akil Narayan and Tao Zhou , title =

    John D. Jakeman and Akil Narayan and Tao Zhou , title =. 2017 , doi =

    2017

  77. [77]

    Vulnerability, Uncertainty, and Risk , pages =

    Stefano Marelli and Bruno Sudret , title =. Vulnerability, Uncertainty, and Risk , pages =. doi:10.1061/9780784413609.257 , year =

    work page doi:10.1061/9780784413609.257

  78. [78]

    OpenCossan 2.0: an efficient computational toolbox for risk, reliability and resilience analysis

    Edoardo Patelli and Silvia Tolo and Hindolo George-Williams and Jonathan Sadeghi and Roberto Rocchetta and de Angelis , Marco and Matteo Broggi. OpenCossan 2.0: an efficient computational toolbox for risk, reliability and resilience analysis. Proceedings of the joint ICVRAM ISUMA UNCERTAINTIES conference. 2018

    2018

  79. [79]

    Dongbin Xiu and George Em Karniadakis , journal =. The. 2002 , doi =

    2002

  80. [80]

    Bruno Sudret and Armen Der Kiureghian , title =

Showing first 80 references.