Reliability approximations via asymptotic distribution S Adhikari Department of Aerospace Engineering, University of Bristol, Bristol, U.K. Email: S.Adhikari@bristol.ac.uk URL: http://www.aer.bris.ac.uk/contact/academic/adhikari/home.html ICOSSAR 2005, Rome Asymptotic distribution in high dimensions – p.1/26
Outline of the presentation Introduction to structural reliability analysis Limitation of FORM/SORM in high dimensions Asymptotic distribution of quadratic forms Strict asymptotic formulation Weak asymptotic formulation Numerical results Conclusions & discussions ICOSSAR 2005, Rome Asymptotic distribution in high dimensions – p.2/26
Structural reliability analysis Probability of failure � e − x T x / 2 d x P f = (2 π ) − n/ 2 g ( x ) ≤ 0 x ∈ R n : Gaussian parameter vector g ( x ) : failure surface Maximum contribution comes from the neighborhood where x T x / 2 is minimum subject to g ( x ) ≤ 0 . The design point x ∗ : x ∗ : min { ( x T x ) / 2 } subject to g ( x ) = 0 . ICOSSAR 2005, Rome Asymptotic distribution in high dimensions – p.3/26
Graphical explanation x 2 ✻ SORM approximation y n = β + y T Ay ✏ ✏ ✏ ✏ ✏ ✏ ✮ ❅ FORM approximation ❅ ✘ ✘ y n = β ✘ ✘ ❅ ✘ ✘ ✘ Failure domain ✾ ✘ ❅ g ( x ) ≤ 0 y n ❅ ✒ � ❅ � ❅ • x ∗ � ❅ Actual failure surface � ❅ g ( x ) = 0 � β ❅ � ✲ ❅ x 1 ❅ O β = − ∇ g x ∗ | ∇ g | = α ∗ ICOSSAR 2005, Rome Asymptotic distribution in high dimensions – p.4/26
FORM/SORM approximations y n ≥ β + y T Ay � � P f ≈ Prob = Prob [ y n ≥ β + U ] (1) where U : R n − 1 �→ R = y T Ay , is a quadratic form in Gaussian random variable. The eigenvalues of A , say a j , can be related to the principal curvatures of the surface κ j as a j = κ j / 2 . Considering A = O in Eq. (1), we have the FORM: P f ≈ Φ( − β ) ICOSSAR 2005, Rome Asymptotic distribution in high dimensions – p.5/26
SORM approximations Breitung’s asymptotic formula (1984): P f → Φ( − β ) � I n − 1 + 2 β A � − 1 / 2 when β → ∞ Hohenbichler and Rackwitz’s improved formula (1988): − 1 / 2 � � � I n − 1 + 2 ϕ ( β ) � � P f ≈ Φ( − β ) Φ( − β ) A � � � ICOSSAR 2005, Rome Asymptotic distribution in high dimensions – p.6/26
Numerical example Consider a problem for which the failure surface is exactly parabolic: g = − y n + β + y T Ay We choose n and the value of Trace ( A ) When Trace ( A ) = 0 the failure surface is effectively linear. Therefore, the more the value of Trace ( A ) , the more non-linear the failure surface becomes. It is assumed that the eigenvalues of A are uniform random numbers. ICOSSAR 2005, Rome Asymptotic distribution in high dimensions – p.7/26
P f for small n 10 0 Asymptotic: β → ∞ (Breitung, 84) Hohenbichler & Rackwitz, 88 Exact (MCS) P f / Φ (− β ) 10 −1 10 −2 0 1 2 3 4 5 6 β Failure probability for n − 1 = 3 , Trace ( A ) = 1 ICOSSAR 2005, Rome Asymptotic distribution in high dimensions – p.8/26
P f for large n 10 0 Asymptotic: β → ∞ (Breitung, 84) Hohenbichler & Rackwitz, 88 Exact (MCS) 10 −1 P f / Φ (− β ) 10 −2 10 −3 0 1 2 3 4 5 6 β Failure probability for n − 1 = 100 , Trace ( A ) = 1 ICOSSAR 2005, Rome Asymptotic distribution in high dimensions – p.9/26
The problem with high-dimension If n , i.e. the dimension is large, the computation time to obtain P f using any tools will be high. ICOSSAR 2005, Rome Asymptotic distribution in high dimensions – p.10/26
The problem with high-dimension If n , i.e. the dimension is large, the computation time to obtain P f using any tools will be high. Question 1: Suppose we have followed the ‘normal route’ and obtained x ∗ , β and A . Why the results from classical FORM/SORM is not satisfactory in a high dimensional problem? ICOSSAR 2005, Rome Asymptotic distribution in high dimensions – p.10/26
The problem with high-dimension If n , i.e. the dimension is large, the computation time to obtain P f using any tools will be high. Question 1: Suppose we have followed the ‘normal route’ and obtained x ∗ , β and A . Why the results from classical FORM/SORM is not satisfactory in a high dimensional problem? Question 2: What is a ‘high dimension’? ICOSSAR 2005, Rome Asymptotic distribution in high dimensions – p.10/26
The problem with high-dimension If n , i.e. the dimension is large, the computation time to obtain P f using any tools will be high. Question 1: Suppose we have followed the ‘normal route’ and obtained x ∗ , β and A . Why the results from classical FORM/SORM is not satisfactory in a high dimensional problem? Question 2: What is a ‘high dimension’? Only simulation methods (Au & Beck, 2003; Koutsourelakis et al., 2004) are available at present for problems with high dimension. ICOSSAR 2005, Rome Asymptotic distribution in high dimensions – p.10/26
Asymptotic distribution of quadratic forms Moment generating function: n − 1 M U ( s ) = � I n − 1 − 2 s A � − 1 / 2 = � (1 − 2 sa k ) − 1 / 2 k =1 Now construct a sequence of new random variables q = U/ √ n . The moment generating function of q : n − 1 M q ( s ) = M U ( s/ √ n ) = 1 − 2 sa k / √ n � − 1 / 2 � � k =1 ICOSSAR 2005, Rome Asymptotic distribution in high dimensions – p.11/26
Asymptotic distribution Truncating the Taylor series expansion: ln ( M q ( s )) ≈ Trace ( A ) s/ √ n + A 2 �� s 2 / 2 n � � 2 Trace We assume n is large such that the following conditions hold 2 A 2 � � n Trace < ∞ 2 r n r/ 2 r Trace ( A r ) → 0 , ∀ r ≥ 3 and ICOSSAR 2005, Rome Asymptotic distribution in high dimensions – p.12/26
Asymptotic distribution Therefore, the moment generating function of U = q √ n can be approximated by: 2 �� � � A Trace ( A ) s + s 2 / 2 2 Trace M U ( s ) ≈ e From the uniqueness of the Laplace Transform pair it follows that U asymptotically approaches a Gaussian random variable with mean Trace ( A ) and A 2 � � variance 2Trace , that is A 2 �� � � U ≃ N 1 Trace ( A ) , 2 Trace when n → ∞ ICOSSAR 2005, Rome Asymptotic distribution in high dimensions – p.13/26
Minimum number of random variables The error in neglecting higher order terms: � 2 s � r 1 Trace ( A r ) , for r ≥ 3 . √ n r Using s = β and assuming there exist a small real number ǫ (the error) we have (2 β ) r 4 β 2 1 � 2 � n r/ 2 Trace ( A r ) < ǫ or n > � Trace ( A r ) √ r r r r 2 ǫ 2 ICOSSAR 2005, Rome Asymptotic distribution in high dimensions – p.14/26
Strict asymptotic formulation We rewrite (1): P f ≈ Prob [ y n ≥ β + U ] = Prob [ y n − U ≥ β ] Since U is asymptotically Gaussian, the vari- able z = y n − U is also Gaussian with mean A 2 � � ( − Trace ( A )) and variance (1 + 2 Trace ) . Thus, β +Trace ( A ) P f Strict → Φ ( − β 1 ) , β 1 = 2 � , n → ∞ � � A 1+2 Trace ICOSSAR 2005, Rome Asymptotic distribution in high dimensions – p.15/26
� � Graphical explanation r � � . Failure surface: y n − U ≥ β . Using the standard- izing transformation Y = ( U − m ) /σ , modified m = Trace ( A ) , σ 2 = 2Trace A 2 y n Y failure surface β + m + ≥ 1 . − β + m σ y n tan θ σ √ √ From △ AOB, sin θ = 1+tan 2 θ = 1+ σ 2 . ✻ ✔ Failure ( β + m ) A Therefore, from △ OB y ∗ : SORM approximation ✔ domain β +Trace ( A ) y n = β + y T Ay β 1 = β + m β + m √ sin θ = 1+ σ 2 = ✔ σ A 2 1+2 Trace ✔ If n is small, m, σ will be small. When m, σ → 0 , ✔ modified ✔ ② ❳ AB rotates clockwise and eventually becomes • ❳ ❳ ❳❳❳ design point original ✔ ③ parallel to the Y-axis with a shift of + β . In this sit- y ∗ ❜ β • design point x ∗ uation y ∗ → x ∗ in the y n -axis and β 1 → β as ex- ✔ ❜ β 1 ❜ ✔ ✛ ❜ pected. This explains why classical F/SORM ap- B θ Y ✔ proximations based on the original design point ( β + m ) /σ O x ∗ do not work well when a large number of ran- dom variables are considered. ICOSSAR 2005, Rome Asymptotic distribution in high dimensions – p.16/26
Weak asymptotic formulation P f ≈ Prob [ y n ≥ β + U ] �� ∞ � � = ϕ ( y n ) dy n p U ( u ) du = E [Φ( − β − U )] R β + u Noticing that u ∈ R + as A is positive definite we rewrite � + e ln[Φ( − β − u )]+ln[ p U ( u )] du P f ≈ R ICOSSAR 2005, Rome Asymptotic distribution in high dimensions – p.17/26
Weak asymptotic formulation For the maxima of the integrand (say at point u ∗ ) ∂ ∂u { ln [Φ( − β − u )] + ln [ p U ( u )] } = 0 Recalling that p U ( u ) = (2 π ) − 1 / 2 σ − 1 e − ( u − m ) 2 / (2 σ 2 ) we have Φ( − ( β + u )) = m − u ϕ ( β + u ) σ 2 ICOSSAR 2005, Rome Asymptotic distribution in high dimensions – p.18/26
Weak asymptotic formulation After some simplifications, the failure probability using weak asymptotic formulation: 2 � � � A � − β 2 Trace ( A ) 2 β 2 − 2 Trace P f Weak → Φ ( − β 2 ) e , � � I n − 1 + 2 β 2 A � β + Trace ( A ) A 2 � when n → ∞ where β 2 = � 1 + 2 Trace For the small n case, neglecting the ‘trace effect’ it can be seen that P f Weak approaches to Breitung’s formula. ICOSSAR 2005, Rome Asymptotic distribution in high dimensions – p.19/26
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