Uncertainty Quantification and Propagation in Structural Mechanics: A Random Matrix Approach Sondipon 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 Random Matrix Method – p.1/63 B E College, India, January 2007
Overview of Predictive Methods in Engineering There are five key steps: Physics (mechanics) model building Uncertainty Quantification (UQ) Uncertainty Propagation (UP) Model Verification & Validation (V & V) Prediction Tools are available for each of these steps. In this talk we will focus mainly on UQ and UP in linear dynamical systems. Random Matrix Method – p.2/63 B E College, India, January 2007
Bristol Aerospace Random Matrix Method – p.3/63 B E College, India, January 2007
Structural dynamics The equation of motion: M ¨ x ( t ) + C ˙ x ( t ) + Kx ( t ) = p ( t ) Due to the presence of uncertainty M , C and K become random matrices. The main objectives in the ‘forward problem’ are: to quantify uncertainties in the system matrices to predict the variability in the response vector x Random Matrix Method – p.4/63 B E College, India, January 2007
Current Methods Two different approaches are currently available Low frequency : Stochastic Finite Element Method (SFEM) - assumes that stochastic fields describing parametric uncertainties are known in details High frequency : Statistical Energy Analysis (SEA) - do not consider parametric uncertainties in details Random Matrix Method – p.5/63 B E College, India, January 2007
Random Matrix Method (RMM) The objective : To have an unified method which will work across the frequency range. The methodology : Derive the matrix variate probability density functions of M , C and K Propagate the uncertainty (using Monte Carlo simulation or analytical methods) to obtain the response statistics (or pdf) Random Matrix Method – p.6/63 B E College, India, January 2007
Outline of the presentation In what follows next, I will discuss: Introduction to Matrix variate distributions Matrix factorization approach Optimal Wishart distribution Some examples Open problems & discussions Random Matrix Method – p.7/63 B E College, India, January 2007
Matrix variate distributions The probability density function of a random matrix can be defined in a manner similar to that of a random variable. If A is an n × m real random matrix, the matrix variate probability density function of A ∈ R n,m , denoted as p A ( A ) , is a mapping from the space of n × m real matrices to the real line, i.e., p A ( A ) : R n,m → R . Random Matrix Method – p.8/63 B E College, India, January 2007
Gaussian random matrix The random matrix X ∈ R n,p is said to have a matrix variate Gaussian distribution with mean matrix M ∈ R n,p and covariance matrix Σ ⊗ Ψ , where Σ ∈ R + n and Ψ ∈ R + p provided the pdf of X is given by p X ( X ) = (2 π ) − np/ 2 | Σ | − p/ 2 | Ψ | − n/ 2 � � − 1 2 Σ − 1 ( X − M ) Ψ − 1 ( X − M ) T etr (1) This distribution is usually denoted as X ∼ N n,p ( M , Σ ⊗ Ψ ) . Random Matrix Method – p.9/63 B E College, India, January 2007
Wishart matrix A n × n symmetric positive definite random matrix S is said to have a Wishart distribution with parameters p ≥ n and Σ ∈ R + n , if its pdf is given by � − 1 � � 1 � � � − 1 2 np Γ n 1 1 1 2 p 2 ( p − n − 1) etr 2 Σ − 1 S p S ( S ) = 2 2 p | Σ | | S | (2) This distribution is usually denoted as S ∼ W n ( p, Σ ) . Note: If p = n + 1 , then the matrix is non-negative definite. Random Matrix Method – p.10/63 B E College, India, January 2007
Matrix variate Gamma distribution A n × n symmetric positive definite matrix random W is said to have a matrix variate gamma distribution with parameters a and Ψ ∈ R + n , if its pdf is given by ℜ ( a ) > 1 Γ n ( a ) | Ψ | − a � − 1 | W | a − 1 2 ( n +1) etr {− ΨW } ; � p W ( W ) = 2( n − (3) This distribution is usually denoted as W ∼ G n ( a, Ψ ) . Here the multivariate gamma function: n � � a − 1 1 � 4 n ( n − 1) Γ n ( a ) = π Γ 2( k − 1) ; for ℜ ( a ) > ( n − 1) / 2 (4) k =1 Random Matrix Method – p.11/63 B E College, India, January 2007
Distribution of the system matrices The distribution of the random system matrices M , C and K should be such that they are symmetric positive-definite, and the moments (at least first two) of the inverse of the dynamic stiffness matrix D ( ω ) = − ω 2 M + iω C + K should exist ∀ ω Random Matrix Method – p.12/63 B E College, India, January 2007
Distribution of the system matrices The exact application of the last constraint requires the derivation of the joint probability density function of M , C and K , which is quite difficult to obtain. We consider a simpler problem where it is required that the inverse moments of each of the system matrices M , C and K must exist. Provided the system is damped, this will guarantee the existence of the moments of the frequency response function matrix. Random Matrix Method – p.13/63 B E College, India, January 2007
Maximum Entropy Distribution Soize (2000,2006) used this approach and obtained the matrix variate Gamma distribution. Since Gamma and Wishart distribution are similar we have: Theorem 1. If ν -th order inverse-moment of a system matrix G ≡ { M , C , K } exists and only the mean of G is available, say G , then the maximum-entropy pdf of G follows the Wishart distribution with parameters p = (2 ν + n + 1) and Σ = G / (2 ν + n + 1) , that is � � G ∼ W n 2 ν + n + 1 , G / (2 ν + n + 1) . Random Matrix Method – p.14/63 B E College, India, January 2007
Properties of the Distribution Covariance tensor of G : 1 � � cov ( G ij , G kl ) = G ik G jl + G il G jk 2 ν + n + 1 Normalized standard deviation matrix � G − E [ G ] � 2 � � � � } 2 G = E 1 + { Trace 1 G F δ 2 = � E [ G ] � 2 � 2 � 2 ν + n + 1 Trace G F 1 + n δ 2 2 ν + n + 1 and ν ↑ ⇒ δ 2 G ≤ G ↓ . Random Matrix Method – p.15/63 B E College, India, January 2007
Distribution of the inverse - 1 If G is W n ( p, Σ ) then V = G − 1 has the inverted Wishart distribution: P V ( V ) = 2 m − n − 1 n/ 2 | Ψ | m − n − 1 / 2 � � − 1 2 V − 1 Ψ Γ n [( m − n − 1) / 2] | V | m/ 2 etr where m = n + p + 1 and Ψ = Σ − 1 (recall that p = 2 ν + n + 1 and Σ = G /p ) Random Matrix Method – p.16/63 B E College, India, January 2007
Distribution of the inverse - 2 − 1 p G G − 1 � � Mean: E = p − n − 1 G − 1 ij , G − 1 � � cov = kl � � − 1 − 1 − 1 − 1 − 1 ilG − 1 2 ν + n + 1)( ν − 1 G ij G kl + G ik G jl + G kj 2 ν (2 ν + 1)(2 ν − 2) Random Matrix Method – p.17/63 B E College, India, January 2007
Application Suppose n = 101 & ν = 2 . So p = 2 ν + n + 1 = 106 and p − n − 1 = 4 . Therefore, E [ G ] = G and = 106 − 1 = 26 . 5 G − 1 !!!!!!!!!! G − 1 � � E 4 G From a practical point of view we do not expect them to be so far apart! One way to reduce the gap is to increase p . But this implies the reduction of variance. Random Matrix Method – p.18/63 B E College, India, January 2007
Matrix Factorization Approach (MFA) Because G is a symmetric and positive-definite random matrix, it can be always factorized as G = XX T (5) where X ∈ R n × p , p ≥ n is in general a rectangular matrix. The simplest case is when the mean of X is O ∈ R n × p , p ≥ n and the covariance tensor of X is given by Σ ⊗ I p ∈ R np × np where Σ ∈ R + n . X is a Gaussian random matrix with mean O ∈ R n × p , p ≥ n and covariance Σ ⊗ I p ∈ R np × np . Random Matrix Method – p.19/63 B E College, India, January 2007
Wishart Pdf After some algebra it can be shown that G is a W n ( p, Σ ) Wishart random matrix, whose pdf is given given by � − 1 � � 1 � � � − 1 2 np Γ n 1 1 1 2 p 2 ( p − n − 1) etr 2 Σ − 1 G p G ( G ) = 2 2 p | Σ | | G | (6) Random Matrix Method – p.20/63 B E College, India, January 2007
Parameter Estimation of Wishart Distribution The distribution of G must be such that E [ G ] and − 1 respectively. G − 1 � � E should be closest to G and G Since G ∼ W n ( p, Σ ) , there are two unknown parameters in this distribution, namely, p and Σ . This implies that there are in total 1 + n ( n + 1) / 2 number of unknowns. We define and subsequently minimize ‘normalized errors’: � � � � ε 1 = � G − E [ G ] F / � G � � F − 1 − E � G − 1 �� � − 1 � � ε 2 = F / � G � G � � � � � � F Random Matrix Method – p.21/63 B E College, India, January 2007
MFA Distribution Solving the optimization problem we have: Theorem 2. If ν -th order inverse-moment of a system matrix G ≡ { M , C , K } exists and only the mean of G is available, say G , then the distribution of G follows the Wishart distribution with parameters � p = (2 ν + n + 1) and Σ = G / 2 ν (2 ν + n + 1) , that is � � � G ∼ W n 2 ν + n + 1 , G / 2 ν (2 ν + n + 1) . Random Matrix Method – p.22/63 B E College, India, January 2007
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