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Introduction to Sensitivity Analysis

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• Bertrand Iooss 4 , 5 &
• Andrea Saltelli 6 , 7  

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Sensitivity analysis provides users of mathematical and simulation models with tools to appreciate the dependency of the model output from model input and to investigate how important is each model input in determining its output. All application areas are concerned, from theoretical physics to engineering and socio-economics. This introductory paper provides the sensitivity analysis aims and objectives in order to explain the composition of the overall “Sensitivity Analysis” chapter of the Springer Handbook. It also describes the basic principles of sensitivity analysis, some classification grids to understand the application ranges of each method, a useful software package, and the notations used in the chapter papers. This section also offers a succinct description of sensitivity auditing, a new discipline that tests the entire inferential chain including model development, implicit assumptions, and normative issues and which is recommended when the inference provided by the model needs to feed into a regulatory or policy process. For the “Sensitivity Analysis” chapter, in addition to this introduction, eight papers have been written by around twenty practitioners from different fields of application. They cover the most widely used methods for this subject: the deterministic methods as the local sensitivity analysis, the experimental design strategies, the sampling-based and variance-based methods developed from the 1980s, and the new importance measures and metamodel-based techniques established and studied since the 2000s. In each paper, toy examples or industrial applications illustrate their relevance and usefulness.

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Industrial Risk Management Department, EDF R&D, Chatou, France

Bertrand Iooss

Institut de Mathématiques de Toulouse, Université Paul Sabatier, Toulouse, France

Centre for the Study of the Sciences and the Humanities (SVT), University of Bergen (UIB), Bergen, Norway

Andrea Saltelli

Institut de Ciència i Tecnologia Ambientals (ICTA), Universitat Autonoma de Barcelona (UAB), Barcelona, Spain

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Los Alamos National Laboratory, Los Alamos, New Mexico, USA

David Higdon

California Institute of Technology , Pasadena, California, USA

Houman Owhadi

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Iooss, B., Saltelli, A. (2015). Introduction to Sensitivity Analysis. In: Ghanem, R., Higdon, D., Owhadi, H. (eds) Handbook of Uncertainty Quantification. Springer, Cham. https://doi.org/10.1007/978-3-319-11259-6_31-1

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DOI : https://doi.org/10.1007/978-3-319-11259-6_31-1

Received : 20 December 2014

Accepted : 24 June 2015

Published : 26 March 2016

Publisher Name : Springer, Cham

Online ISBN : 978-3-319-11259-6

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Wan, Din Wan Ibrahim. "Sensitivity analysis intolerance allocation." Thesis, Queen's University Belfast, 2014. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.675480.

Poveda, David. "Sensitivity analysis of capital projects." Thesis, University of British Columbia, 1988. http://hdl.handle.net/2429/27990.

Faria, Jairo Rocha de. "Second order topological sensitivity analysis." Laboratório Nacional de Computação Científica, 2008. http://www.lncc.br/tdmc/tde_busca/arquivo.php?codArquivo=141.

Witzgall, Zachary F. "Parametric sensitivity analysis of microscrews." Morgantown, W. Va. : [West Virginia University Libraries], 2006. https://eidr.wvu.edu/etd/documentdata.eTD?documentid=4892.

鄧國良 and Kwok-leong Tang. "Sensitivity analysis of bootstrap methods." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1993. http://hub.hku.hk/bib/B31977479.

Fang, Xinding S. M. Massachusetts Institute of Technology. "Sensitivity analysis of fracture scattering." Thesis, Massachusetts Institute of Technology, 2010. http://hdl.handle.net/1721.1/59789.

Masinde, Brian. "Birds' Flight Range. : Sensitivity Analysis." Thesis, Linköpings universitet, Institutionen för datavetenskap, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-166248.

Tang, Kwok-leong. "Sensitivity analysis of bootstrap methods." [Hong Kong] : University of Hong Kong, 1993. http://sunzi.lib.hku.hk/hkuto/record.jsp?B13793792.

Munster, Drayton William. "Sensitivity Enhanced Model Reduction." Thesis, Virginia Tech, 2013. http://hdl.handle.net/10919/23169.

Konarski, Roman. "Sensitivity analysis for structural equation models." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp04/nq22893.pdf.

Sulieman, Hana. "Parametric sensitivity analysis in nonlinear regression." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk2/tape15/PQDD_0004/NQ27858.pdf.

Yu, Jianbin. "Flexible reinforced pavement structure-sensitivity analysis." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2000. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape3/PQDD_0015/MQ52682.pdf.

Kolen, A. W. J., Kan A. H. G. Rinnooy, Hoesel C. P. M. Van, and Albert Wagelmans. "Sensitivity Analysis of List Scheduling Heuristics." Massachusetts Institute of Technology, Operations Research Center, 1990. http://hdl.handle.net/1721.1/5268.

Maginot, Jeremy. "Sensitivity analysis for multidisciplinary design optmization." Thesis, Cranfield University, 2007. http://dspace.lib.cranfield.ac.uk/handle/1826/5667.

North, Simon John. "High sensitivity mass spectrometric glycoprotein analysis." Thesis, Imperial College London, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.404993.

Maginot, Jeremy. "Sensitivity analysis for multidisciplinary design optimization." Thesis, Cranfield University, 2007. http://dspace.lib.cranfield.ac.uk/handle/1826/5667.

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Santos, Miguel Duque. "UK pension funds : liability sensitivity analysis." Master's thesis, Instituto Superior de Economia e Gestão, 2019. http://hdl.handle.net/10400.5/19509.

Sen, Sharma Pradeep Kumar. "Sensitivity analysis of ship longitudinal strength." Thesis, Virginia Tech, 1988. http://hdl.handle.net/10919/45183.

DeBrunner, Victor Earl. "Sensitivity analysis of digital filter structures." Thesis, Virginia Polytechnic Institute and State University, 1986. http://hdl.handle.net/10919/104319.

Wycoff, Nathan Benjamin. "Gradient-Based Sensitivity Analysis with Kernels." Diss., Virginia Tech, 2021. http://hdl.handle.net/10919/104683.

Kern, Simon. "Sensitivity Analysis in 3D Turbine CFD." Thesis, KTH, Mekanik, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-210821.

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Kennedy, Christopher Brandon. "GPT-Free Sensitivity Analysis for Reactor Depletion and Analysis." Thesis, North Carolina State University, 2015. http://pqdtopen.proquest.com/#viewpdf?dispub=3710730.

Introduced in this dissertation is a novel approach that forms a reduced-order model (ROM), based on subspace methods, that allows for the generation of response sensitivity profiles without the need to set up or solve the generalized inhomogeneous perturbation theory (GPT) equations. The new approach, denoted hereinafter as the generalized perturbation theory free (GPT-Free) approach, computes response sensitivity profiles in a manner that is independent of the number or type of responses, allowing for an efficient computation of sensitivities when many responses are required. Moreover, the reduction error associated with the ROM is quantified by means of a Wilks’ order statistics error metric denoted by the κ-metric.

Traditional GPT has been recognized as the most computationally efficient approach for performing sensitivity analyses of models with many input parameters, e.g. when forward sensitivity analyses are computationally overwhelming. However, most neutronics codes that can solve the fundamental (homogenous) adjoint eigenvalue problem do not have GPT (inhomogenous) capabilities unless envisioned during code development. Additionally, codes that use a stochastic algorithm, i.e. Monte Carlo methods, may have difficult or undefined GPT equations. When GPT calculations are available through software, the aforementioned efficiency gained from the GPT approach diminishes when the model has both many output responses and many input parameters. The GPT-Free approach addresses these limitations, first by only requiring the ability to compute the fundamental adjoint from perturbation theory, and second by constructing a ROM from fundamental adjoint calculations, constraining input parameters to a subspace. This approach bypasses the requirement to perform GPT calculations while simultaneously reducing the number of simulations required.

In addition to the reduction of simulations, a major benefit of the GPT-Free approach is explicit control of the reduced order model (ROM) error. When building a subspace using the GPT-Free approach, the reduction error can be selected based on an error tolerance for generic flux response-integrals. The GPT-Free approach then solves the fundamental adjoint equation with randomly generated sets of input parameters. Using properties from linear algebra, the fundamental k-eigenvalue sensitivities, spanned by the various randomly generated models, can be related to response sensitivity profiles by a change of basis. These sensitivity profiles are the first-order derivatives of responses to input parameters. The quality of the basis is evaluated using the κ-metric, developed from Wilks’ order statistics, on the user-defined response functionals that involve the flux state-space. Because the κ-metric is formed from Wilks’ order statistics, a probability-confidence interval can be established around the reduction error based on user-defined responses such as fuel-flux, max-flux error, or other generic inner products requiring the flux. In general, The GPT-Free approach will produce a ROM with a quantifiable, user-specified reduction error.

This dissertation demonstrates the GPT-Free approach for steady state and depletion reactor calculations modeled by SCALE6, an analysis tool developed by Oak Ridge National Laboratory. Future work includes the development of GPT-Free for new Monte Carlo methods where the fundamental adjoint is available. Additionally, the approach in this dissertation examines only the first derivatives of responses, the response sensitivity profile; extension and/or generalization of the GPT-Free approach to higher order response sensitivity profiles is natural area for future research.

Guo, Jia. "Uncertainty analysis and sensitivity analysis for multidisciplinary systems design." Diss., Rolla, Mo. : Missouri University of Science and Technology, 2008. http://scholarsmine.mst.edu/thesis/pdf/Guo_09007dcc8066e905.pdf.

Ekberg, Marie. "Sensitivity analysis of optimization : Examining sensitivity of bottleneck optimization to input data models." Thesis, Högskolan i Skövde, Institutionen för ingenjörsvetenskap, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:his:diva-12624.

Reineke, Jan. "Caches in WCET analysis : predictability, competitiveness, sensitivity /." Berlin : epubli, 2008. http://www.epubli.de/shop/showshopelement?pubId=882.

Chaban, Habib Fady Ruben. "A numerical sensitivity analysis of streamline simulation." Texas A&M University, 2004. http://hdl.handle.net/1969.1/1541.

Gajev, Ivan. "Sensitivity and Uncertainty Analysis of BWR Stability." Licentiate thesis, KTH, Kärnkraftsäkerhet, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-26387.

Nahum, Carole. "Second order sensitivity analysis in mathematical programming." Thesis, McGill University, 1989. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=74349.

Wu, QiongLi. "Sensitivity Analysis for Functional Structural Plant Modelling." Phd thesis, Ecole Centrale Paris, 2012. http://tel.archives-ouvertes.fr/tel-00719935.

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Rios, Insua David. "Sensitivity analysis in multi-objective decision making." Thesis, University of Leeds, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.236870.

Yin, Peng. "Local sensitivity analysis and bias model selection." Thesis, University of Newcastle upon Tyne, 2014. http://hdl.handle.net/10443/2385.

Zhu, Yitao. "Sensitivity Analysis and Optimization of Multibody Systems." Diss., Virginia Tech, 2015. http://hdl.handle.net/10919/71649.

Svenson, Joshua. "Computer Experiments: Multiobjective Optimization and Sensitivity Analysis." The Ohio State University, 2011. http://rave.ohiolink.edu/etdc/view?acc_num=osu1306361734.

Hakami, Amir. "Direct sensitivity analysis in air quality models." Diss., Available online, Georgia Institute of Technology, 2004:, 2003. http://etd.gatech.edu/theses/available/etd-04082004-180202/unrestricted/hakami%5Famir%5F200312%5Fphd.pdf.

Garmroodi, Doiran Mehdi. "Sensitivity Analysis for Future Grid Stability Studies." Thesis, The University of Sydney, 2016. http://hdl.handle.net/2123/15978.

Tiscareno-Lopez, Mario 1957. "Sensitivity analysis of the WEPP Watershed model." Thesis, The University of Arizona, 1991. http://hdl.handle.net/10150/292034.

Capozzi, Marco G. F. "FINITE ELEMENT ANALYSIS AND SENSITIVITY ANALYSIS FOR THE POTENTIAL EQUATION." MSSTATE, 2004. http://sun.library.msstate.edu/ETD-db/theses/available/etd-04222004-131403/.

Rapadamnaba, Robert. "Uncertainty analysis, sensitivity analysis, and machine learning in cardiovascular biomechanics." Thesis, Montpellier, 2020. http://www.theses.fr/2020MONTS058.

Wang, Mengchao. "Sensitivity analysis and evolutionary optimization for building design." Thesis, Loughborough University, 2014. https://dspace.lboro.ac.uk/2134/16282.

Van, Hoesel Stan, and Albert Wagelmans. "Sensitivity Analysis of the Economic Lot-Sizing Problem." Massachusetts Institute of Technology, Operations Research Center, 1990. http://hdl.handle.net/1721.1/5146.

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