A bibliography would be useful

[mbaudin47]

• The NASA Langley Multidisciplinary Uncertainty Quantification Challenge, Luis G. Crespo, Sean P. Kenny and Daniel P. Giesy. NASA Langley Research Center. https://uqtools.larc.nasa.gov/files/2013/02/NASA_LaRC_MUQ.pdf
• Waarts, P.-H. (2000). Structural reliability using finite element methods: an appraisal of DARS: Directional Adaptive Response Surface Sampling. Ph. D. thesis, Technical University of Delft, The Netherlands. Pages 58, 69, 160.
• Thèse Vincent Dubourg 2011, Méta-modèles adaptatifs pour l’analyse de fiabilité et l’optimisation sous contrainte fiabiliste, section "A two-dimensional four-branch serial system", page 182
• On the Ishigami function. Ishigami, T., & Homma, T. (1990, December). An importance quantification technique in uncertainty analysis for computer models. In Uncertainty Modeling and Analysis, 1990. Proceedings., First International Symposium on (pp. 398-403). IEEE.
• On the G-Sobol function. A. Saltelli and I.M. Sobol. About the use of rank transformation in sensitivity analysis of model output. Reliab. Eng. Syst. Safety, 50(3):225{239, 1995.

[mbaudin47]

There are interesting benchmark problems in the following references:

• "sensobol: an R package to compute variance-based sensitivity indices", Arnald Puy, Samuele Lo Piano, Andrea Saltelli, Simon A. Levin. https://arxiv.org/abs/2101.10103

• Here is a benchmark problem from Saltelli:

  x1 in (0.0, 1.0)
  x2 in (0.0, 1.0)
  y1 = x1 + 0.5 * x2
  y2 = x1 + 0.5 * x2 + x1 * x2
  y1, x1: SI1 = 4/5, STI1 = 4/5
  y1, x2: SI2 = 1/5, STI2 = 1/5
  y2, x1: SI1 = (9 / 4) * (12 / 42) ~= 0.643, 
      STI1 = (7.0 / 36.0) / (40.0 / 144.0) = 0.7
  y2, x2: SI2 = 12 / 42 ~= 0.286, 
      STI2 = (13.0 / 144.0) / (40.0 / 144.0) ~= 0.325

• Chan, Karen, Andrea Saltelli, and Stefano Tarantola. "Sensitivity analysis of model output: variance-based methods make the difference." Proceedings of the 29th conference on Winter simulation. 1997. There are 11 benchmark problems from p.34 to p.44.

• Borgonovo, Emanuele. "Sensitivity analysis." An Introduction for the Management Scientist. International Series in Operations Research and Management Science. Cham, Switzerland: Springer (2017). There is the following example p.18 and p.157:

  y = x1 * x2 + x3
  x1, x2, x3 ~ U(0,1)
  V = 7/36
  V1 = 1/12
  V2 = 0
  V3 = 1/12