Reliability Analysis in High Dimensions S Adhikari Department of - PowerPoint PPT Presentation

Reliability Analysis in High Dimensions 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 PMC 2004 Reliability analysis in high dimensions – p.1/30

Outline of the presentation Introduction to structural reliability analysis Limitation of current methods in high dimension Asymptotic distribution of quadratic forms Strict asymptotic formulation Weak asymptotic formulation Numerical result Open problems & discussions PMC 2004 Reliability analysis in high dimensions – p.2/30

Reliability analysis: basics 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 . PMC 2004 Reliability analysis in high dimensions – p.3/30

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 ∗ PMC 2004 Reliability analysis in high dimensions – p.4/30

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 ) PMC 2004 Reliability analysis in high dimensions – p.5/30

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 � � � PMC 2004 Reliability analysis in high dimensions – p.6/30

The curse of dimensionality If n , i.e. the dimension is large, the computation time to obtain P f using any tools will be high (no magic is possible!) PMC 2004 Reliability analysis in high dimensions – p.7/30

The curse of dimensionality If n , i.e. the dimension is large, the computation time to obtain P f using any tools will be high (no magic is possible!) Question 1: What is a ‘high dimension’? PMC 2004 Reliability analysis in high dimensions – p.7/30

The curse of dimensionality If n , i.e. the dimension is large, the computation time to obtain P f using any tools will be high (no magic is possible!) Question 1: What is a ‘high dimension’? Question 2: Suppose we have followed the ‘normal route’ and did all the calculations (i.e., x ∗ , β and A ). Can we still trust the results from classical FORM/SORM in high dimension? PMC 2004 Reliability analysis in high dimensions – p.7/30

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. PMC 2004 Reliability analysis in high dimensions – p.8/30

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 PMC 2004 Reliability analysis in high dimensions – p.9/30

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 PMC 2004 Reliability analysis in high dimensions – p.10/30

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 PMC 2004 Reliability analysis in high dimensions – p.11/30

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 PMC 2004 Reliability analysis in high dimensions – p.12/30

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 → ∞ PMC 2004 Reliability analysis in high dimensions – p.13/30

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 PMC 2004 Reliability analysis in high dimensions – p.14/30

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 PMC 2004 Reliability analysis in high dimensions – p.15/30

� � 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. PMC 2004 Reliability analysis in high dimensions – p.16/30

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 PMC 2004 Reliability analysis in high dimensions – p.17/30

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 PMC 2004 Reliability analysis in high dimensions – p.18/30

Weak asymptotic formulation Because this relationship holds at the optimal point u ∗ , define a constant η as Φ( − ( β + u ∗ )) = m − u ∗ ϕ ( β + u ∗ ) η = σ 2 Taking a first-order Taylor series expansion of ln [Φ( − β − u )] about u = u ∗ : ϕ ( β + u ∗ ) Φ( − β − u ) ≈ e ln[Φ( − ( β + u ∗ ))] − Φ( − ( β + u ∗ )) ( u − u ∗ ) PMC 2004 Reliability analysis in high dimensions – p.19/30

Weak asymptotic formulation Using η we have Φ( − β − u ) ≈ Φ( − β 2 ) e ηu ∗ e − ηu (1) where the modified reliability index β 2 = β + u ∗ Taking the expectation of (1) and using the expression of the moment generating function: P f ≈ E [Φ( − β − U )] = Φ( − β 2 ) e ηu ∗ � I n − 1 + 2 η A � − 1 / 2 PMC 2004 Reliability analysis in high dimensions – p.20/30

Weak asymptotic formulation Considering the asymptotic expansion of the ratio Φ( − ( β + u ∗ )) ≈ ( β + u ∗ ) = β 2 ≈ m − u ∗ ϕ ( β + u ∗ ) η = σ 2 We obtain u ∗ ≈ m − βσ 2 1 + σ 2 , β 2 = β + u ∗ ≈ β + m β + Trace ( A ) 1 + σ 2 = A 2 � � 1 + 2 Trace Since η ≈ β 2 , u ∗ can be expressed in terms of β 2 as u ∗ ≈ − β 2 σ 2 − m A 2 � � � � � � = − 2 β 2 Trace − Trace ( A ) PMC 2004 Reliability analysis in high dimensions – p.21/30