1 Introduction to Spectral Methods for Uncertainty Quantification
33
Therefore, SM are usually used, since their exponential rate of convergence
compensates the complexity. However, if the parametrisation of the sources of
uncertainty uses too many random variables, SM can perform poorly, given the
larger complexity of the problem. In this case, MC methods may be better suited,
given that its convergence does not depend on the number of independent variables
in the parametrisation.
One of the main goals of this chapter was to demonstrate the versatility of a
surrogate to find QoI. We demonstrated how this surrogate can be obtained by
SM. Providing that we can parametrise the conductivity field with a small enough
number of random variables, NISP or Galerkin methods are efficient ways to obtain
a PCE of the solution of a model, such as Eq. (1.5). Once making the effort of
computing this surrogate, the QoI is obtained almost “for free” using the coefficients
of the PCE. We illustrate this by computing the Sobol indices w.r.t. to each random
variable of the germ.
References
1. D.S. Sivia, Data Analysis: A Bayesian Tutorial, 1st edn. (Oxford University Press, Oxford,
2006)
2. S. Salsa, Partial Differential Equations in Action: From Modelling to Theory. UNITEXT - La
Matematica per il 3+2 (Springer, Cham, 2012)
3. F. Saleri, A. Quarteroni, R. Sacco, Numerical Mathematics. Texts in Applied Mathematics, 1st
edn. (Springer, New York, 2000)
4. A. Valli, A. Quarteroni, Numerical Analysis of Partial Differential Equations. Springer Series
in Computational Mathematics, 1st edn. (Springer, Berlin 2008)
5. L.C. Evans, Partial Differential Equations (American Mathematical Society, Providence,
2010)
6. M. Kac, A.J.F. Siegert, An explicit representation of a stationary Gaussian process. Ann. Math.
Stat. 18(3), 438–442 (1947)
7. K. Karhunen, Über lineare Methoden in der Wahrscheinlichkeitsrechnung. Number
ARRAY(0x4deaf30) in Suomalaisen Tiedeakatemian toimituksia. Suomalainen
Tiedeakatemia, Helsinki, 1947
8. M. Loeve, Fonctions alatoires du second ordre, in Processus Stochastique et Mouvement
Brownien, ed. by P. Lévy (Gauthier Villars, Paris, 1948)
9. O.P. Le Matre, O.M. Knio, Spectral Methods for Uncertainty Quantification: With Applications
to Computational Fluid Dynamics. Scientific Computation (Springer, Dordrecht, 2010)
10. R.G. Bartle, The Elements of Integration and Lebesgue Measure (Wiley, New York, 1995)
11. T. Sullivan, Introduction to Uncertainty Quantification, vol. 63 (Springer, Berlin, 2015)
12. D. Xiu, Numerical Methods for Stochastic Computations: A Spectral Method Approach
(Princeton University Press, Princeton, 2010)
13. D. Gamerman, Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference.
Texts in Statistical Science, 1st edn. (Chapman & Hall/CRC, Boca Raton, 1997)
14. G. Blatman, Adaptive sparse polynomial chaos expansions for uncertainty propagation and
sensitivity analysis. PhD thesis, Université Blaise Pascal BLAISE - Clermont II, Institut
Français de Mécanique Avancée et Université Blaise Pascal, Oct 2009
15. I.M. Sobol, Global sensitivity indices for nonlinear mathematical models and their Monte
Carlo estimates. Math. Comput. Simul. 55(1–3), 271–280 (2001)
33
Therefore, SM are usually used, since their exponential rate of convergence
compensates the complexity. However, if the parametrisation of the sources of
uncertainty uses too many random variables, SM can perform poorly, given the
larger complexity of the problem. In this case, MC methods may be better suited,
given that its convergence does not depend on the number of independent variables
in the parametrisation.
One of the main goals of this chapter was to demonstrate the versatility of a
surrogate to find QoI. We demonstrated how this surrogate can be obtained by
SM. Providing that we can parametrise the conductivity field with a small enough
number of random variables, NISP or Galerkin methods are efficient ways to obtain
a PCE of the solution of a model, such as Eq. (1.5). Once making the effort of
computing this surrogate, the QoI is obtained almost “for free” using the coefficients
of the PCE. We illustrate this by computing the Sobol indices w.r.t. to each random
variable of the germ.
References
1. D.S. Sivia, Data Analysis: A Bayesian Tutorial, 1st edn. (Oxford University Press, Oxford,
2006)
2. S. Salsa, Partial Differential Equations in Action: From Modelling to Theory. UNITEXT - La
Matematica per il 3+2 (Springer, Cham, 2012)
3. F. Saleri, A. Quarteroni, R. Sacco, Numerical Mathematics. Texts in Applied Mathematics, 1st
edn. (Springer, New York, 2000)
4. A. Valli, A. Quarteroni, Numerical Analysis of Partial Differential Equations. Springer Series
in Computational Mathematics, 1st edn. (Springer, Berlin 2008)
5. L.C. Evans, Partial Differential Equations (American Mathematical Society, Providence,
2010)
6. M. Kac, A.J.F. Siegert, An explicit representation of a stationary Gaussian process. Ann. Math.
Stat. 18(3), 438–442 (1947)
7. K. Karhunen, Über lineare Methoden in der Wahrscheinlichkeitsrechnung. Number
ARRAY(0x4deaf30) in Suomalaisen Tiedeakatemian toimituksia. Suomalainen
Tiedeakatemia, Helsinki, 1947
8. M. Loeve, Fonctions alatoires du second ordre, in Processus Stochastique et Mouvement
Brownien, ed. by P. Lévy (Gauthier Villars, Paris, 1948)
9. O.P. Le Matre, O.M. Knio, Spectral Methods for Uncertainty Quantification: With Applications
to Computational Fluid Dynamics. Scientific Computation (Springer, Dordrecht, 2010)
10. R.G. Bartle, The Elements of Integration and Lebesgue Measure (Wiley, New York, 1995)
11. T. Sullivan, Introduction to Uncertainty Quantification, vol. 63 (Springer, Berlin, 2015)
12. D. Xiu, Numerical Methods for Stochastic Computations: A Spectral Method Approach
(Princeton University Press, Princeton, 2010)
13. D. Gamerman, Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference.
Texts in Statistical Science, 1st edn. (Chapman & Hall/CRC, Boca Raton, 1997)
14. G. Blatman, Adaptive sparse polynomial chaos expansions for uncertainty propagation and
sensitivity analysis. PhD thesis, Université Blaise Pascal BLAISE - Clermont II, Institut
Français de Mécanique Avancée et Université Blaise Pascal, Oct 2009
15. I.M. Sobol, Global sensitivity indices for nonlinear mathematical models and their Monte
Carlo estimates. Math. Comput. Simul. 55(1–3), 271–280 (2001)