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Computer tutorial on global sensitivity analysisAlen Alexanderian, Pierre Gremaud, and Ralph SmithDepartment of Mathematics, North Carolina State UniversityJune 8, 2019Alexanderian, Gremaud, Smith (NCSU)GSA TutorialJune 8, 20191 / 10

Model problem: SIR dynamics with time varyingtransmission rateGoverning equations:dS β(t)SIwithdtdI β(t)SI p3 IdtdR p3 Idtβ(t) p1 p2 sin(t)Uncertain parameter vector: p1p p2 p3Alexanderian, Gremaud, Smith (NCSU)GSA TutorialJune 8, 20192 / 10

DEMO 1: Nominal system parametersp 0.002, p 0.001, p 0) 499 I (0) 1R(0) 0MATLAB: demo nominal system.mAlexanderian, Gremaud, Smith (NCSU)GSA TutorialJune 8, 20193 / 10

DEMO 2: A scalar quantity of interest (QoI)Parameterize uncertainty in p using x R3 with xi U( 1, 1)QoI: infected population at a fixed timef (x) I (T ; x)with T 00QoIMATLAB: demo sample.mAlexanderian, Gremaud, Smith (NCSU)GSA TutorialJune 8, 20194 / 10

DEMO 3: Sobol’ indices1total Sobol index0.80.60.40.20p1p2p3MATLAB: demo sobol scalar.mAlexanderian, Gremaud, Smith (NCSU)GSA TutorialJune 8, 20195 / 10

DEMO 4: Derivative based GSAInequality:cνi (f )DSitot (f ) withn f 2 o4νi (f ) Ec 2 xiπD Var {f }10 0StotiBound10 -210 -410 -6p1p2p3MATLAB: demo dgsm.mAlexanderian, Gremaud, Smith (NCSU)GSA TutorialJune 8, 20196 / 10

DEMO 5: Fixing parameters0.0150.020.035full modelmodel with p 2 fixedfull modelmodel with p 1 p 2 fixed0.025PDFPDF0.01PDFfull modelmodel with p 3 00150200250050QoIRecall the model100150200QoIdS β(t)SIwithdtdI β(t)SI p3 IdtdR p3 Idt250050100150200250QoIβ(t) p1 p2 sin(t)MATLAB: demo fixing var.mAlexanderian, Gremaud, Smith (NCSU)GSA TutorialJune 8, 20197 / 10

DEMO 6: Sobol’ indices over timeSobol’ indices for I (t; x) over time1p1p2total Sobol indices0.8p30.60.40.20010203040timeNote:Sitot (t) total contribution of ith input to variance at time tMATLAB demo sobol time.mAlexanderian, Gremaud, Smith (NCSU)GSA TutorialJune 8, 20198 / 10

DEMO 6: Sobol’ indices over timeSobol’ indices for I (t; x) over time1p1p2total Sobol indices0.8p30.60.40.20010203040timeSitot (t)Note: total contribution of ith input to variance at time tvariance changes over time pointwise-in-time Sobol’ indices have limited use8050040060std deviationinfected populationmean2 std dev300200402010000010203040010203040timetimeMATLAB demo sobol time.mAlexanderian, Gremaud, Smith (NCSU)GSA TutorialJune 8, 20198 / 10

Some suggested computational experimentsGenerate system trajectory with different sets of nominal parametersCompute Sobol’ indices / DGSMs with different sample sizesChange ranges of uncertainty on parametersConsider different scalar QoIs (redefine sir scalar qoi.m)Think more about the time-dependent case (look atdemo sobol time generalized.m)Consider different time-dependent QoIs (redefine sir timedep qoi.m).Alexanderian, Gremaud, Smith (NCSU)GSA TutorialJune 8, 20199 / 10

Further readingBooksRalph Smith, Uncertainty Quantification: Theory, Implementation, and Applications, 2014.A. Saltelli, M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, M. Saisana. and S. Tarantola,Global sensitivity analysis: the primer. John Wiley & Sons. 2008.Le Maitre, Olivier, and Omar M. Knio. Spectral methods for uncertainty quantification: withapplications to computational fluid dynamics. 2010.Some relevant papersI.M. Sobol’. Global sensitivity indices for nonlinear mathematical models and their monte carloestimates. Mathematics and computers in simulation, 55(1):271–280, 2001.I.M. Sobol’ and S. Kucherenko. Derivative based global sensitivity measures and their link with globalsensitivity indices. Mathematics and Computers in Simulation, 79(10), 2009.I.M. Sobol’, S. Tarantola, D. Gatelli, S. Kucherenko, and W. Mauntz. Estimating the ap- proximationerror when fixing unessential factors in global sensitivity analysis. Reliability Engineering & SystemSafety, 92(7), 2007.S. Kucherenko and B. Iooss, Derivative-based global sensitivity measures. Handbook of uncertaintyquantification, 2017.J. Hart and P. Gremaud. An approximation theoretic perspective of Sobol’ indices with dependentvariables. International Journal for Uncertainty Quantification. 2018.https://arxiv.org/pdf/1801.01359.pdf.J. Hart, A. Alexanderian, P. Gremaud Efficient computation of Sobol’indices for stochastic models.SIAM Journal on Scientific Computing, 39(4), 2017.A. Alexanderian, P.A. Gremaud, and R.C. Smith. Variance-based sensitivity analysis for time-dependentprocesses. In revision. https://arxiv.org/abs/1711.08030. 2019.Alexanderian, Gremaud, Smith (NCSU)GSA TutorialJune 8, 201910 / 10

Global sensitivity analysis: the primer. John Wiley & Sons. 2008. Le Maitre, Olivier, and Omar M. Knio. Spectral methods for uncertainty quanti cation: with applications to computational uid dynamics. 2010. Some relevant papers I.M. Sobol'. Global sensitivity indices for nonlinear mathematical models and their monte carlo estimates.