Global sensitivity analysis using polynomial chaos expansions

doi:10.1016/J.RESS.2007.04.002

Home
/
Papers
/
Global sensitivity analysis using polynomial chaos expansions

Journal Article•DOI•

Global sensitivity analysis using polynomial chaos expansions

Bruno Sudret¹•Institutions (1)

Électricité de France¹

01 Jul 2008-Reliability Engineering & System Safety (Elsevier)-Vol. 93, Iss: 7, pp 964-979

TL;DR: In this article, generalized polynomial chaos expansions (PCE) are used to build surrogate models that allow one to compute the Sobol' indices analytically as a post-processing of the PCE coefficients.

read less

About: This article is published in Reliability Engineering & System Safety.The article was published on 2008-07-01. It has received 1934 citations till now. The article focuses on the topics: Sobol sequence & Polynomial chaos.

...read moreread less

Citations

PDF

Open Access

More filters

Reliability Engineering and System Safety

[...]

Sharif Rahman

01 Jan 2011

TL;DR: In this paper, a polynomial dimensional decomposition (PDD) method for global sensitivity analysis of stochastic systems subject to independent random input following arbitrary probability distributions is presented.

...read moreread less

Abstract: This paper presents a polynomial dimensional decomposition (PDD) method for global sensitivity analysis of stochastic systems subject to independent random input following arbitrary probability distributions. The method involves Fourier-polynomial expansions of lower-variate component functions of a stochastic response by measure-consistent orthonormal polynomial bases, analytical formulae for calculating the global sensitivity indices in terms of the expansion coefficients, and dimension-reduction integration for estimating the expansion coefficients. Due to identical dimensional structures of PDD and analysis-of-variance decomposition, the proposed method facilitates simple and direct calculation of the global sensitivity indices. Numerical results of the global sensitivity indices computed for smooth systems reveal significantly higher convergence rates of the PDD approximation than those from existing methods, including polynomial chaos expansion, random balance design, state-dependent parameter, improved Sobol’s method, and sampling-based methods. However, for non-smooth functions, the convergence properties of the PDD solution deteriorate to a great extent, warranting further improvements. The computational complexity of the PDD method is polynomial, as opposed to exponential, thereby alleviating the curse of dimensionality to some extent. Mathematical modeling of complex systems often requires sensitivity analysis to determine how an output variable of interest is influenced by individual or subsets of input variables. A traditional local sensitivity analysis entails gradients or derivatives, often invoked in design optimization, describing changes in the model response due to the local variation of input. Depending on the model output, obtaining gradients or derivatives, if they exist, can be simple or difficult. In contrast, a global sensitivity analysis (GSA), increasingly becoming mainstream, characterizes how the global variation of input, due to its uncertainty, impacts the overall uncertain behavior of the model. In other words, GSA constitutes the study of how the output uncertainty from a mathematical model is divvied up, qualitatively or quantitatively, to distinct sources of input variation in the model [1].

...read moreread less

1,296 citations

Journal Article•DOI•

Adaptive sparse polynomial chaos expansion based on least angle regression

[...]

Géraud Blatman¹, Bruno Sudret¹•Institutions (1)

International Facility Management Association¹

01 Mar 2011-Journal of Computational Physics

TL;DR: A non intrusive method that builds a sparse PC expansion, which may be obtained at a reduced computational cost compared to the classical ''full'' PC approximation.

...read moreread less

1,112 citations

Cites background or methods from "Global sensitivity analysis using p..."

...It is worth mentioning that an algorithm has been devised in [12] to select a minimum number of roots of orthogonal polynomials in the design....
[...]
...Two approaches based on a PC representation of the response are usually distinguished: – the projection approach: each PC coefficient is recast as a multidimensional integral [7, 8] which can be computed either by simulation or quadrature; – the regression approach [9, 10, 11, 12]: the PC coefficients are estimated by minimizing the mean square error of the response approximation....
[...]
...It is also possible to derive inexpensively the global sensitivity indices of the response to the input variables [12]....
[...]

Journal Article•DOI•

Sensitivity analysis of environmental models

[...]

Francesca Pianosi¹, Keith Beven², Jim Freer¹, Jim W. Hall³, Jonathan Rougier¹, David B. Stephenson⁴, Thorsten Wagener¹ - Show less +3 more•Institutions (4)

University of Bristol¹, Lancaster University², Environmental Change Institute³, University of Exeter⁴

01 May 2016-Environmental Modelling and Software

TL;DR: This paper presents an overview of SA and its link to uncertainty analysis, model calibration and evaluation, robust decision-making, and provides practical guidelines by developing a workflow for the application of SA.

...read moreread less

Abstract: Sensitivity Analysis (SA) investigates how the variation in the output of a numerical model can be attributed to variations of its input factors. SA is increasingly being used in environmental modelling for a variety of purposes, including uncertainty assessment, model calibration and diagnostic evaluation, dominant control analysis and robust decision-making. In this paper we review the SA literature with the goal of providing: (i) a comprehensive view of SA approaches also in relation to other methodologies for model identification and application; (ii) a systematic classification of the most commonly used SA methods; (iii) practical guidelines for the application of SA. The paper aims at delivering an introduction to SA for non-specialist readers, as well as practical advice with best practice examples from the literature; and at stimulating the discussion within the community of SA developers and users regarding the setting of good practices and on defining priorities for future research. We present an overview of SA and its link to uncertainty analysis, model calibration and evaluation, robust decision-making.We provide a systematic review of existing approaches, which can support users in the choice of an SA method.We provide practical guidelines by developing a workflow for the application of SA and discuss critical choices.We give best practice examples from the literature and highlight trends and gaps for future research.

...read moreread less

888 citations

Cites methods from "Global sensitivity analysis using p..."

...For example Sudret (2008) presents an approach where generalized polynomial chaos expansions (PCE) are used as emulators and variance-based sensitivity indices (see Section 3.5) are computed analytically as a post-processing of the PCE coefficients....
[...]

Journal Article•DOI•

An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis

[...]

Géraud Blatman¹, Bruno Sudret¹•Institutions (1)

International Facility Management Association¹

01 Apr 2010-Probabilistic Engineering Mechanics

TL;DR: A non-intrusive method that builds a sparse PC expansion and an adaptive regression-based algorithm is proposed for automatically detecting the significant coefficients of the PC expansion in a suitable polynomial chaos basis.

...read moreread less

710 citations

Cites background or methods from "Global sensitivity analysis using p..."

...Two approaches based on a PC representation of the response are usually distinguished: – the projection approach: each PC coefficient is recast as a multidimensional integral [6, 7] which can be computed either by simulation or quadrature; – the regression approach [8, 9, 10, 11]: the PC coefficients are estimated by minimizing the mean square error of the response approximation in the L2 sense....
[...]
...Indeed, the iterative procedure outlined herein help improve the PC-based GSA method detailed in [11], as shown in [38]....
[...]
...It is however our belief that in most applications, the number of significant terms in the PC expansion is relatively small, because of the two following points: – high order interaction effects are usually negligible compared to main effects and low order interaction effects (this property is referred to as a low effective dimension in applications); – the input variables might have a different impact on the model response, as shown from global sensitivity analysis [11]....
[...]
...It was shown in [10, 11] that a PC approximation of degree p = 2 usually provides satisfactory results for moment and sensitivity analyses, whereas a degree p = 3 is often necessary when performing a reliability analysis....
[...]

Journal Article•DOI•

Flood inundation modelling

[...]

Jin Teng¹, Anthony Jakeman¹, Jai Vaze², Barry Croke¹, Dushmanta Dutta², Shaun Kim² - Show less +2 more•Institutions (2)

Australian National University¹, Commonwealth Scientific and Industrial Research Organisation²

01 Apr 2017-Environmental Modelling and Software

TL;DR: A review of state-of-the-art empirical, hydrodynamic and simple conceptual models for determining flood inundation is presented in this paper, where guidance is provided for selecting the most suitable method/model for solving practical flood related problems, taking into account the specific outputs required for the modelling purpose, the data available and computational demands.

...read moreread less

Abstract: This paper reviews state-of-the-art empirical, hydrodynamic and simple conceptual models for determining flood inundation. It explores their advantages and limitations, highlights the most recent advances and discusses future directions. It addresses how uncertainty is analysed in this field with the various approaches and identifies opportunities for handling it better. The aim is to inform scientists new to the field, and help emergency response agencies, water resources managers, insurance companies and other decision makers keep up-to-date with the latest developments. Guidance is provided for selecting the most suitable method/model for solving practical flood related problems, taking into account the specific outputs required for the modelling purpose, the data available and computational demands. Multi-model, multi-discipline approaches are recommended in order to further advance this research field. This paper reviews state-of-the-art flood inundation models.It explores their advantages and limitations.It highlights the most recent advances and discusses future directions.It addresses how uncertainty is analysed and identifies opportunities for handling it better.

...read moreread less

694 citations

1
2
3
4
…
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200

Collapse

References

PDF

Open Access

More filters

Journal Article•DOI•

Handbook of Mathematical Functions

[...]

Milton Abramowitz, Irene A. Stegun, Donald A. McQuarrie

01 Feb 1966-American Journal of Physics

46,339 citations

Journal Article•DOI•

Numerical recipes

[...]

William H. Press, Saul A. Teukolsky

01 Oct 1990-Computers in Physics

17,845 citations

Handbook of Mathematical Functions

[...]

Milton Abramowitz, Irene A. Stegun

01 Jan 1972

9,153 citations

Book•

Stochastic Finite Elements: A Spectral Approach

[...]

Roger Ghanem¹, Pol D. Spanos²•Institutions (2)

State University of New York System¹, Rice University²

20 Dec 1990

TL;DR: In this article, a representation of stochastic processes and response statistics are represented by finite element method and response representation, respectively, and numerical examples are provided for each of them.

...read moreread less

Abstract: Representation of stochastic processes stochastic finite element method - response representation stochastic finite element method - response statistics numerical examples.

...read moreread less

5,495 citations

"Global sensitivity analysis using p..." refers background or methods in this paper

...1 Introduction Polynomial chaos expansions developed back in the 30’s by Wiener (1938) have been brought back in the field of engineering mechanics by Ghanem and Spanos (1991). The usual Hermite chaos is made of the multivariate Hermite polynomials {Ψj, j ∈N} in standard normal variables {ξn, n ∈ N} (called basic variables in the sequel) and allows to represent any random variable with finite variance as follows:...
[...]
...The so-called Spectral stochastic finite element method, named after the pioneering work by (Ghanem and Spanos, 1991) has been applied to many engineering problems (see a review by Sudret and Der Kiureghian (2000) and many recent papers, among...
[...]

Journal Article•DOI•

The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations

[...]

Dongbin Xiu¹, George Em Karniadakis¹•Institutions (1)

Brown University¹

01 Feb 2002-SIAM Journal on Scientific Computing

TL;DR: This work represents the stochastic processes with an optimum trial basis from the Askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error.

...read moreread less

Abstract: We present a new method for solving stochastic differential equations based on Galerkin projections and extensions of Wiener's polynomial chaos Specifically, we represent the stochastic processes with an optimum trial basis from the Askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error Several continuous and discrete processes are treated, and numerical examples show substantial speed-up compared to Monte Carlo simulations for low dimensional stochastic inputs

...read moreread less

4,473 citations