As uncertainty and sensitivity analysis of complex models grows ever more important, the difficulty of their timely realizations highlights a need for more efficient numerical operations. Non-intrusive Polynomial Chaos methods are highly efficient and accurate methods of mapping input-output relationships to investigate complex models. There is substantial potential to increase the efficacy of the method regarding the selected sampling scheme. We examine state-of-the-art sampling schemes categorized in space-filling-optimal designs such as Latin Hypercube sampling and L1-optimal sampling and compare their empirical performance against standard random sampling. The analysis was performed in the context of L1 minimization using the least-angle regression algorithm to fit the GPCE regression models. Due to the random nature of the sampling schemes, we compared different sampling approaches using statistical stability measures and evaluated the success rates to construct a surrogate model with relative errors of $ < 0.1 $%, $ < 1 $%, and $ < 10 $%, respectively. The sampling schemes are thoroughly investigated by evaluating the y of surrogate models constructed for various distinct test cases, which represent different problem classes covering low, medium and high dimensional problems. Finally, the sampling schemes are tested on an application example to estimate the sensitivity of the self-impedance of a probe that is used to measure the impedance of biological tissues at different frequencies. We observed strong differences in the convergence properties of the methods between the analyzed test functions.
Citation: Konstantin Weise, Erik Müller, Lucas Poßner, Thomas R. Knösche. Comparison of the performance and reliability between improved sampling strategies for polynomial chaos expansion[J]. Mathematical Biosciences and Engineering, 2022, 19(8): 7425-7480. doi: 10.3934/mbe.2022351
As uncertainty and sensitivity analysis of complex models grows ever more important, the difficulty of their timely realizations highlights a need for more efficient numerical operations. Non-intrusive Polynomial Chaos methods are highly efficient and accurate methods of mapping input-output relationships to investigate complex models. There is substantial potential to increase the efficacy of the method regarding the selected sampling scheme. We examine state-of-the-art sampling schemes categorized in space-filling-optimal designs such as Latin Hypercube sampling and L1-optimal sampling and compare their empirical performance against standard random sampling. The analysis was performed in the context of L1 minimization using the least-angle regression algorithm to fit the GPCE regression models. Due to the random nature of the sampling schemes, we compared different sampling approaches using statistical stability measures and evaluated the success rates to construct a surrogate model with relative errors of $ < 0.1 $%, $ < 1 $%, and $ < 10 $%, respectively. The sampling schemes are thoroughly investigated by evaluating the y of surrogate models constructed for various distinct test cases, which represent different problem classes covering low, medium and high dimensional problems. Finally, the sampling schemes are tested on an application example to estimate the sensitivity of the self-impedance of a probe that is used to measure the impedance of biological tissues at different frequencies. We observed strong differences in the convergence properties of the methods between the analyzed test functions.
| [1] |
N. Wiener, The Homogeneous Chaos, Am. J. Math., 60 (1938), 897. http://dx.doi.org/10.2307/2371268 doi: 10.2307/2371268
|
| [2] |
D. Xiu, G. E. Karniadakis, The Wiener–Askey Polynomial Chaos for Stochastic Differential Equations, SIAM J. Sci. Comput., 24 (2002), 619–644. http://dx.doi.org/10.1137/S1064827501387826 doi: 10.1137/S1064827501387826
|
| [3] | O. P. Le Maître, O. M. Knio, Spectral methods for uncertainty quantification: With applications to computational fluid dynamics, Scientific computation. Springer, Dordrecht, ISBN 978-90-481-3519-6 (2010), http://dx.doi.org/10.1007/978-90-481-3520-2 |
| [4] | M. Eldred, C. Webster, P. Constantine, Evaluation of Non-Intrusive Approaches for Wiener-Askey Generalized Polynomial Chaos, In 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, American Institute of Aeronautics and Astronautics, Reston, Virigina (2008). http://dx.doi.org/10.2514/6.2008-1892 |
| [5] |
I. M. Sobol', Y. Levitan, On the use of variance reducing multipliers in Monte Carlo computations of a global sensitivity index, Comput. Phys. Commun., 117 (1999), 52–61. http://dx.doi.org/10.1016/s0010-4655(98)00156-8 doi: 10.1016/s0010-4655(98)00156-8
|
| [6] | D. Xiu, Fast Numerical Methods for Stochastic Computations: A Review, Commun. Comput. Phys., 5 (2009), 242–272. |
| [7] | R. G. Ghanem, P. D. Spanos, Stochastic finite elements: A spectral approach. rev. ed. edition, Dover Publ, Mineola, NY, ISBN 9780486428185, (2003). |
| [8] |
K. Weise, M. Carlstedt, M. Ziolkowski, H. Brauer, Uncertainty analysis in Lorentz force eddy current testing. IEEE Trans. Magn., 52 (2016), 1–4. http://dx.doi.org/10.1109/TMAG.2015.2480046 doi: 10.1109/TMAG.2015.2480046
|
| [9] |
L. Codecasa, L. Di Rienzo, K. Weise, S. Gross, J. Haueisen, Fast MOR-Based Approach to Uncertainty Quantification in Transcranial Magnetic Stimulation, IEEE Trans. Magn., 52 (2016), 1–4. http://dx.doi.org/10.1109/TMAG.2015.2475120 doi: 10.1109/TMAG.2015.2475120
|
| [10] |
K. Weise, L. Di Rienzo, H. Brauer, J. Haueisen, H. Toepfer, Uncertainty analysis in transcranial magnetic stimulation using nonintrusive polynomial chaos expansion, IEEE Trans. Magn., 51 (2015), 1–8. http://dx.doi.org/10.1109/TMAG.2015.2390593 doi: 10.1109/TMAG.2015.2390593
|
| [11] |
G. B. Saturnino, A. Thielscher, K. H. Madsen, T. R. Knösche, K. Weise, A principled approach to conductivity uncertainty analysis in electric field calculations, NeuroImage, 188 (2019), 821–834. http://dx.doi.org/10.1016/j.neuroimage.2018.12.053 doi: 10.1016/j.neuroimage.2018.12.053
|
| [12] |
K. Weise, O. Numssen, A. Thielscher, G. Hartwigsen, T. R. Knösche, A novel approach to localize cortical TMS effects, NeuroImage, 209 (2020), 116486. http://dx.doi.org/10.1016/j.neuroimage.2019.116486 doi: 10.1016/j.neuroimage.2019.116486
|
| [13] |
K. Sepahvand, S. Marburg, H.-J. Hardtke, Stochastic free vibration of orthotropic plates using generalized polynomial chaos expansion, J. Sound Vib., 331 (2012), 167–179. http://dx.doi.org/10.1016/j.jsv.2011.08.012 doi: 10.1016/j.jsv.2011.08.012
|
| [14] | S. Hosder, R. Walters, M. Balch, Efficient Sampling for Non-Intrusive Polynomial Chaos Applications with Multiple Uncertain Input Variables, In 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, American Institute of Aeronautics and Astronautics, Reston, Virigina, ISBN 978-1-62410-013-0 (2007). http://dx.doi.org/10.2514/6.2007-1939 |
| [15] |
A. Kaintura, T. Dhaene, D. Spina, Review of Polynomial Chaos-Based Methods for Uncertainty Quantification in Modern Integrated Circuits, Electronics, 7 (2018), 30. http://dx.doi.org/10.3390/electronics7030030 doi: 10.3390/electronics7030030
|
| [16] |
P. Diaz, A. Doostan, J. Hampton, Sparse polynomial chaos expansions via compressed sensing and D-optimal design, Comput. Methods Appl. Mech. Eng., 336 (2018), 640–666. http://dx.doi.org/10.1016/j.cma.2018.03.020 doi: 10.1016/j.cma.2018.03.020
|
| [17] |
H. N. Najm, Uncertainty Quantification and Polynomial Chaos Techniques in Computational Fluid Dynamics, Annu. Rev. Fluid Mech., 41 (2009), 35–52. http://dx.doi.org/10.1146/ANNUREV.FLUID.010908.165248 doi: 10.1146/ANNUREV.FLUID.010908.165248
|
| [18] |
K. M. Burk, A. Narayan, J. A. Orr, Efficient sampling for polynomial chaos-based uncertainty quantification and sensitivity analysis using weighted approximate Fekete points, Int. J. Numer. Method. Biomed. Eng., 36 (2020). http://dx.doi.org/10.1002/cnm.3395 doi: 10.1002/cnm.3395
|
| [19] |
M. Grignard, C. Geuzaine, C. Phillips, Shamo: A Tool for Electromagnetic Modeling, Simulation and Sensitivity Analysis of the Head, Neuroinformatics, (2022). http://dx.doi.org/10.1007/s12021-022-09574-7 doi: 10.1007/s12021-022-09574-7
|
| [20] |
Z. Hu, D. Du, Y. Du, Generalized polynomial chaos-based uncertainty quantification and propagation in multi-scale modeling of cardiac electrophysiology. Comput. Biol. Med., 102 (2018), 57–74. http://dx.doi.org/10.1016/j.compbiomed.2018.09.006 doi: 10.1016/j.compbiomed.2018.09.006
|
| [21] |
E. C. Massoud, Emulation of environmental models using polynomial chaos expansion, Environ. Model. Softw., 111 (2019), 421–431. http://dx.doi.org/10.1016/j.envsoft.2018.10.008 doi: 10.1016/j.envsoft.2018.10.008
|
| [22] |
N. Pepper, L. Gerardo-Giorda, F. Montomoli, Meta-modeling on detailed geography for accurate prediction of invasive alien species dispersal, Sci. Rep., 9 (2019), 16237. http://dx.doi.org/10.1038/s41598-019-52763-9 doi: 10.1038/s41598-019-52763-9
|
| [23] |
J. Son, D. Du, Y. Du, Modified Polynomial Chaos Expansion for Efficient Uncertainty Quantification in Biological Systems, Appl. Mech., 1 (2020), 153–173. http://dx.doi.org/10.3390/applmech1030011 doi: 10.3390/applmech1030011
|
| [24] |
S. K. Sachdeva, P. B. Nair, A. J. Keane, Hybridization of stochastic reduced basis methods with polynomial chaos expansions, Probab. Eng. Mech., 21 (2006), 182–192. http://dx.doi.org/10.1016/j.probengmech.2005.09.003 doi: 10.1016/j.probengmech.2005.09.003
|
| [25] |
X. Wan, G. E. Karniadakis, Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures, SIAM J. Sci. Comput., 28 (2006), 901–928. http://dx.doi.org/10.1137/050627630 doi: 10.1137/050627630
|
| [26] |
G. Blatman, B. Sudret, Adaptive sparse polynomial chaos expansion based on least angle regression, J. Comput. Phys., 230 (2011), 2345–2367. http://dx.doi.org/10.1016/j.jcp.2010.12.021 doi: 10.1016/j.jcp.2010.12.021
|
| [27] |
L. Novák, M. Vořechovský, V. Sadílek, M. D. Shields, Variance-based adaptive sequential sampling for Polynomial Chaos Expansion, Comput. Methods Appl. Mech. Eng., 386 (2021), 114105. http://dx.doi.org/10.1016/j.cma.2021.114105 doi: 10.1016/j.cma.2021.114105
|
| [28] |
M. D. McKay, R. J. Beckman, W. J. Conover, Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code, Technometrics, 21 (1979), 239–245. http://dx.doi.org/10.1080/00401706.1979.10489755 doi: 10.1080/00401706.1979.10489755
|
| [29] |
J. C. Helton, F. J. Davis, Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems, Reliab. Eng. Syst. Saf., 81 (2003), 23–69. http://dx.doi.org/10.1016/S0951-8320(03)00058-9 doi: 10.1016/S0951-8320(03)00058-9
|
| [30] |
D. L. Donoho, Compressed sensing, IEEE Trans. Inf. Theory, 52 (2006), 1289–1306. http://dx.doi.org/10.1109/tit.2006.871582 doi: 10.1109/tit.2006.871582
|
| [31] |
H. Rauhut, R. Ward, Sparse Legendre expansions via l1-minimization, J. Approximation Theory, 164 (2012), 517–533. http://dx.doi.org/10.1016/j.jat.2012.01.008 doi: 10.1016/j.jat.2012.01.008
|
| [32] |
J. Hampton, A. Doostan, Compressive sampling of polynomial chaos expansions: Convergence analysis and sampling strategies, J. Comput. Phys., 280 (2015), 363–386. http://dx.doi.org/10.1016/j.jcp.2014.09.019 doi: 10.1016/j.jcp.2014.09.019
|
| [33] |
Z. Shu, P. Jirutitijaroen, Latin Hypercube Sampling Techniques for Power Systems Reliability Analysis With Renewable Energy Sources, IEEE Trans. Power Syst., 26 (2011), 2066–2073. http://dx.doi.org/10.1109/TPWRS.2011.2113380 doi: 10.1109/TPWRS.2011.2113380
|
| [34] | D. Robinson, C. Atcitty, 40th Structures, Structural Dynamics, and Materials Conference and Exhibit. Reston, Virigina (04121999), http://dx.doi.org/10.2514/MSDM99 |
| [35] | R. Jin, W. Chen, A. Sudjianto, {An Efficient Algorithm for Constructing Optimal Design of Computer Experiments}, In Volume 2: 29th Design Automation Conference, Parts A and B, 545–554. ASMEDC, ISBN 0-7918-3700-9 (09022003). http://dx.doi.org/10.1115/detc2003/dac-48760 |
| [36] |
M. E. Johnson, L. M. Moore, D. Ylvisaker, Minimax and maximin distance designs, J. Stat. Plann. Inference, 26 (1990), 131–148. http://dx.doi.org/10.1016/0378-3758(90)90122-b doi: 10.1016/0378-3758(90)90122-b
|
| [37] |
E. J. Candes, M. B. Wakin, An Introduction To Compressive Sampling, IEEE Signal Process Mag., 25 (2008), 21–30. http://dx.doi.org/10.1109/MSP.2007.914731 doi: 10.1109/MSP.2007.914731
|
| [38] |
E. J. Candès, J. K. Romberg, T. Tao, Stable signal recovery from incomplete and inaccurate measurements, Commun. Pure Appl. Math., 59 (2006), 1207–1223. http://dx.doi.org/10.1002/cpa.20124 doi: 10.1002/cpa.20124
|
| [39] |
M. Elad, Optimized Projections for Compressed Sensing, IEEE Trans. Signal Process., 55 (2007), 5695–5702. http://dx.doi.org/10.1109/tsp.2007.900760 doi: 10.1109/tsp.2007.900760
|
| [40] | M. Lustig, J. M. Santos, J.-H. Lee, D. L. Donoho, J. M. Pauly, Application of "Compressed Sensing" for Rapid MR Imaging (2005), Available from: https://www.researchgate.net/profile/david_donoho/publication/249926097_application_of_compressed_sensing_for_rapid_mr_imaging |
| [41] |
J. L. Paredes, G. R. Arce, Z. Wang, Ultra-Wideband Compressed Sensing: Channel Estimation, IEEE J. Sel. Top. Signal Process., 1 (2007), 383–395. http://dx.doi.org/10.1109/JSTSP.2007.906657 doi: 10.1109/JSTSP.2007.906657
|
| [42] |
J. H. Ender, On compressive sensing applied to radar, Signal Process., 90 (2010), 1402–1414. http://dx.doi.org/10.1016/j.sigpro.2009.11.009 doi: 10.1016/j.sigpro.2009.11.009
|
| [43] |
G. Karagiannis, G. Lin, Selection of polynomial chaos bases via Bayesian model uncertainty methods with applications to sparse approximation of PDEs with stochastic inputs, J. Comput. Phys., 259 (2014), 114–134. http://dx.doi.org/10.1016/j.jcp.2013.11.016 doi: 10.1016/j.jcp.2013.11.016
|
| [44] |
J. D. Jakeman, M. S. Eldred, K. Sargsyan, Enhancing $\ell$1 -minimization estimates of polynomial chaos expansions using basis selection, J. Comput. Phys., 289 (2015), 18–34. http://dx.doi.org/10.1016/j.jcp.2015.02.025 doi: 10.1016/j.jcp.2015.02.025
|
| [45] |
G. Deman, K. Konakli, B. Sudret, J. Kerrou, P. Perrochet, H. Benabderrahmane, Using sparse polynomial chaos expansions for the global sensitivity analysis of groundwater lifetime expectancy in a multi-layered hydrogeological model, Reliab. Eng. Syst. Saf., 147 (2016), 156–169. http://dx.doi.org/10.1016/j.ress.2015.11.005 doi: 10.1016/j.ress.2015.11.005
|
| [46] |
R. Tibshirani, I. Johnstone, T. Hastie, B. Efron, Least angle regression, Ann. Stat., 32 (2004), 407–499. http://dx.doi.org/10.1214/009053604000000067 doi: 10.1214/009053604000000067
|
| [47] | Y. C. Pati, R. Rezaiifar, P. S. Krishnaprasad, Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition, In Proceedings of 27th Asilomar Conference on Signals, Systems and Computers, 40–44. IEEE Comput. Soc. Press, ISBN 0-8186-4120-7 (1-3 Nov, 1993). http://dx.doi.org/10.1109/ACSSC.1993.342465 |
| [48] |
E. J. Candès, The restricted isometry property and its implications for compressed sensing, C.R. Math., 346 (2008), 589–592. http://dx.doi.org/10.1016/j.crma.2008.03.014 doi: 10.1016/j.crma.2008.03.014
|
| [49] |
N. Alemazkoor, H. Meidani, A near-optimal sampling strategy for sparse recovery of polynomial chaos expansions, J. Comput. Phys., 371 (2018), 137–151. http://dx.doi.org/10.1016/j.jcp.2018.05.025 doi: 10.1016/j.jcp.2018.05.025
|
| [50] |
K. Weise, L. Poßner, E. Müller, R. Gast, T. R. Knösche, Pygpc: A sensitivity and uncertainty analysis toolbox for Python, SoftwareX, 11 (2020), 100450. http://dx.doi.org/10.1016/j.softx.2020.100450 doi: 10.1016/j.softx.2020.100450
|
| [51] |
R. Askey, J. A. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Memoirs of the American Mathematical Society, 54 (1985). http://dx.doi.org/10.1090/memo/0319 doi: 10.1090/memo/0319
|
| [52] |
J. Hampton, A. Doostan, Coherence motivated sampling and convergence analysis of least squares polynomial Chaos regression, Comput. Methods Appl. Mech. Eng., 290 (2015), 73–97. http://dx.doi.org/10.1016/j.cma.2015.02.006 doi: 10.1016/j.cma.2015.02.006
|
| [53] |
A. M. Bruckstein, D. L. Donoho, M. Elad, From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images, SIAM Rev., 51 (2009), 34–81. http://dx.doi.org/10.1137/060657704 doi: 10.1137/060657704
|
| [54] |
S. S. Chen, D. L. Donoho, M. A. Saunders, Atomic Decomposition by Basis Pursuit, SIAM Rev., 43 (2001), 129–159. http://dx.doi.org/10.1137/S003614450037906X doi: 10.1137/S003614450037906X
|
| [55] |
M. Hadigol, A. Doostan, Least squares polynomial chaos expansion: A review of sampling strategies, Comput. Methods Appl. Mech. Eng., 332 (2018), 382–407. http://dx.doi.org/10.1016/j.cma.2017.12.019 doi: 10.1016/j.cma.2017.12.019
|
| [56] | W. K. Hastings, Monte Carlo sampling methods using Markov chains and their applications, 50 (1970). http://dx.doi.org/10.2307/2334940 |
| [57] | F. Pukelsheim, Optimal design of experiments, SIAM, ISBN 0898716047 (2006). |
| [58] | A. C. Atkinson, Optimum experimental designs, with SAS, volume 34 of Oxford statistical science series, Oxford Univ. Press, Oxford, ISBN 9780199296606 (2007). |
| [59] |
M. D. Morris, T. J. Mitchell, Exploratory designs for computational experiments, J. Stat. Plann. Inference, 43 (1995), 381–402. http://dx.doi.org/10.1016/0378-3758(94)00035-t doi: 10.1016/0378-3758(94)00035-t
|
| [60] | P. Audze, V. Eglais, New approach for planning out of experiments, Prob. Dyn. Str., (1977), 104–107. |
| [61] | S. Bates, J. Sienz, V. Toropov, Formulation of the Optimal Latin Hypercube Design of Experiments Using a Permutation Genetic Algorithm, In 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, American Institute of Aeronautics and Astronautics, Palm Springs, California, ISBN 978-1-62410-079-6 (2004). http://dx.doi.org/10.2514/6.2004-2011 |
| [62] |
S.-K. Choi, R. V. Grandhi, R. A. Canfield, C. L. Pettit, Polynomial chaos expansion with Latin hypercube sampling for estimating response variability, AIAA J., 42 (2004), 1191–1198. http://dx.doi.org/10.2514/1.2220 doi: 10.2514/1.2220
|
| [63] |
M. Vořechovský, J. Eliáš, Modification of the Maximin and $\varphi_p$ Criteria to Achieve Statistically Uniform Distribution of Sampling Points, Technometrics, 62 (2020), 371–386. http://dx.doi.org/10.1080/00401706.2019.1639550 doi: 10.1080/00401706.2019.1639550
|
| [64] |
J. Eliáš, M. Vořechovský, V. Sadílek, Periodic version of the minimax distance criterion for Monte Carlo integration, Adv. Eng. Software, 149 (2020), 102900. http://dx.doi.org/10.1016/j.advengsoft.2020.102900 doi: 10.1016/j.advengsoft.2020.102900
|
| [65] |
M. Vořechovský, J. Mašek, Distance-based optimal sampling in a hypercube: Energy potentials for high-dimensional and low-saturation designs, Adv. Eng. Software, 149 (2020), 102880. http://dx.doi.org/10.1016/j.advengsoft.2020.102880 doi: 10.1016/j.advengsoft.2020.102880
|
| [66] |
L. Gang, Z. Zhu, D. Yang, L. Chang, H. Bai, On Projection Matrix Optimization for Compressive Sensing Systems, IEEE Trans. Signal Process., 61 (2013), 2887–2898. http://dx.doi.org/10.1109/tsp.2013.2253776 doi: 10.1109/tsp.2013.2253776
|
| [67] |
P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, et al., SciPy 1.0: fundamental algorithms for scientific computing in Python, Nat. Methods, 17 (2020), 261–272. http://dx.doi.org/10.1038/s41592-019-0686-2 doi: 10.1038/s41592-019-0686-2
|
| [68] | T. Ishigami, T. Homma, An importance quantification technique in uncertainty analysis for computer models, In Uncertainty Modeling and Analysis, First International Symposium on (ISUMA '90) (ed. B. M. Ayyub), IEEE Computer Society Press, Los Alamitos, ISBN 0818621079 (1990). http://dx.doi.org/10.1109/isuma.1990.151285 |
| [69] |
T. Crestaux, O. Le Maitre, J.-M. Martinez, Polynomial chaos expansion for sensitivity analysis. Reliab. Eng. Syst. Saf., 94 (2009), 1161–1172. http://dx.doi.org/10.1016/j.ress.2008.10.008 doi: 10.1016/j.ress.2008.10.008
|
| [70] |
A. Marrel, B. Iooss, B. Laurent, O. Roustant, Calculations of Sobol indices for the Gaussian process metamodel, Reliab. Eng. Syst. Saf., 94 (2009), 742–751. http://dx.doi.org/10.1016/j.ress.2008.07.008 doi: 10.1016/j.ress.2008.07.008
|
| [71] | L. C. W. Dixon, G. P. Szego, The Global Optimization Problem: An Introduction, In Towards Global Optimisation 2, North-Holland Pub. Co, Amsterdam (1978). |
| [72] | M. Molga, C. Smutnicki, Test functions for optimization needs, (2015). http://new.zsd.iiar.pwr.wroc.pl/files/docs/functions.pdf |
| [73] |
V. Picheny, T. Wagner, D. Ginsbourger, A benchmark of kriging-based infill criteria for noisy optimization, Struct. Multidiscip. Optim., 48 (2013), 607–626. http://dx.doi.org/10.1007/s00158-013-0919-4 doi: 10.1007/s00158-013-0919-4
|
| [74] | W. Li, Optimal Designs Using CP Algorithms, In Proceedings of the 2nd World Conference of the International Association for statistical computing, 130139 (1997). |
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Konstantin Weise, Erik Müller, Lucas Poßner, Thomas R. Knösche. Comparison of the performance and reliability between improved sampling strategies for polynomial chaos expansion[J]. Mathematical Biosciences and Engineering, 2022, 19(8): 7425-7480. doi: 10.3934/mbe.2022351
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