A Spline Dimensional Decomposition for High-Dimensional Uncertainty - PowerPoint PPT Presentation

INTRODUCTION SDD EXAMPLES CLOSURE A Spline Dimensional Decomposition for High-Dimensional Uncertainty Quantification Sharif Rahman and Ramin Jahanbin The University of Iowa, Iowa City, IA 52242 HDA 2019: 8th Workshop on HDA Zurich, Switzerland September 9-13, 2019 Acknowledgment: NSF (CMMI-1607398)

INTRODUCTION SDD EXAMPLES CLOSURE Outline 1 INTRODUCTION 2 SDD 3 EXAMPLES 4 CLOSURE

INTRODUCTION SDD EXAMPLES CLOSURE Uncertainty Quantification Complex System (jet engine) Input X = ( X 1 , . . . , X N ) Output Y = y ( X ) X : (Ω , F ) → ( A N , B N ) Y ∈ L 2 (Ω , F , P ) → → A N ⊆ R N , N ∈ N y ∈ L 2 ( A N , B N , f X d x ) Goals & Objectives � Y l � � Ω Y l d P = � A N y l ( x ) f X ( x ) d x Moments: E := � Probability distribution: P [ Y ≤ y 0 ] := { x : y ( x ) ≤ y 0 } f X ( x ) d x Stochastic design optimization (RDO/RBDO)

INTRODUCTION SDD EXAMPLES CLOSURE UQ Challenges & Methods Challenges (Works at Iowa) High-dimensional random input ( N > 10) Locally prominent (nonsmoothness, discontinuity) responses Statistical dependence among random input Data-driven problems Polynomial Expansion Methods (PCE & PDD) � y p ( X ) = C i Ψ i ( X ) (PCE) 0 ≤ i ≤ p � � C u i u Ψ u y S , p ( X ) = y ∅ + i u ( X u ) (PDD) 0 ≤ i u ≤ p u ∅� = u ⊆{ 1 ,..., N } 1 ≤| u |≤ S Explore spline basis equipped with local support

INTRODUCTION SDD EXAMPLES CLOSURE Assumptions The random vector X := ( X 1 , . . . , X N ) ⊺ : (Ω , F ) → ( A N , B N ) satisfies the following conditions: 1 All component random variables X k , k = 1 , . . . , N , are statistically independent, but not necessarily identical. 2 Each input random variable X k has absolute continuous marginal CDF and continuous marginal PDF. 3 Each input random variable X k is defined on a closed bounded interval [ a k , b k ] ⊂ R , b k > a k , so that all moments exist, i.e. , for l ∈ N 0 , � b k � � � X l X l x l := k ( ω ) d P ( ω ) = k f X k ( x k ) dx k < ∞ . E k Ω a k

INTRODUCTION SDD EXAMPLES CLOSURE Univariate B-Splines (Cox & de Boor, 1972) For a knot sequence ξ k = { a k = ξ k , 1 , . . . , ξ k , n k + p k +1 = b k } , where ξ k , 1 ≤ · · · ≤ ξ k , n k + p k +1 , n k > p k ≥ 0 , the B-splines are i k , p k , ξ k ( x k ) := ( x k − ξ k , i k ) B k + ( ξ k , i k + p k +1 − x k ) B k i k , p k − 1 , ξ k ( x k ) i k +1 , p k − 1 , ξ k ( x k ) B k , ξ k , i k + p k − ξ k , i k ξ k , i k + p k +1 − ξ k , i k +1 1 ≤ k ≤ N , 1 ≤ i k ≤ n k , 1 ≤ p k < ∞ . p 1 = 2, ξ 1 ={ 0,0,0,0.2,0.4,0.6,0.8,1,1,1 } p 1 = 2, λ 1 ={ 0,0,0,0.2,0.4,0.6,0.6,0.8,1,1,1 } 1.0 1.0 B 11,2, ξ 1 B 11,2, λ 1 B 18,2 , λ 1 B 17,2 , ξ 1 B 15,2 , λ 1 B 13,2 , ξ 1 B 14,2 , ξ 1 B 15,2 , ξ 1 B 13,2 , λ 1 0.8 0.8 B 12,2 , ξ 1 B 16,2 , ξ 1 B 12,2 , λ 1 B 14,2 , λ 1 B 16,2 , λ 1 B 17,2 , λ 1 B 1 i 1 ,2 , ξ 1 ( x 1 ) B 1 i 1 ,2 , λ 1 ( x 1 ) 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x 1 x 1

INTRODUCTION SDD EXAMPLES CLOSURE Univariate ON B-Splines Auxiliary B-Spline Vector � � ⊺ 1 , B k 2 , p k , ξ k ( x k ) , . . . , B k P k ( x k ) := n k , p k , ξ k ( x k ) Spline Moment Matrix k ( X k )] ∈ R n k × n k G k := E [ P k ( X k ) P ⊺ G k → symmetric , positive − definite Whitening Transformation ψ k ( x k ) = Q − 1 k P k ( x k ) , where G k = Q k Q ⊺ k For k = 1 , . . . , N , let S k , p k , ξ k be a space real-valued splines in x k of degree p k and knot sequence ξ k . Then � � ψ k S k , p k , ξ k = span i k , p k , ξ k ( x k ) . i k =1 ,..., n k

INTRODUCTION SDD EXAMPLES CLOSURE Multivariate ON B-Splines Given N ∈ N , let ∅ � = u ⊆ { 1 , . . . , N } . For i u := ( i k 1 , . . . , i k | u | ), p u := ( p k 1 , . . . , p k | u | ), Ξ u := ( ξ k 1 , . . . , ξ k | u | ), the tensor-product ON B-splines in x u = ( x k 1 , . . . , x k | u | ) are � i k , p k , ξ k ( x k ) , i u ∈ ¯ Ψ u ψ k i u , p u , Ξ u ( x u ) = I u , n u . k ∈ u � � ¯ I u , n u := i u = ( i k 1 , . . . , i k | u | ) : 2 ≤ i k l ≤ n k l , l = 1 , . . . , | u | The second-moment properties are � � Ψ u E i u , p u , Ξ u ( X u ) = 0 , � 1 , u = v and i u = j v , � � Ψ u i u , p u , Ξ u ( X u )Ψ v j v , p v , Ξ v ( X v ) = E 0 , otherwise .

INTRODUCTION SDD EXAMPLES CLOSURE Dimensionwise Spline Space Splitting For p = ( p 1 , . . . , p N ) ∈ N N 0 & Ξ = { ξ 1 , . . . , ξ N } , let S p , Ξ be the space of all real-valued splines of degree p in x = ( x 1 , . . . , x N ). Then N � 1 ⊕ ¯ � � S p , Ξ = S k , p k , ξ k k =1 � ¯ S u = 1 ⊕ p u , Ξ u ∅� = u ⊆{ 1 ,..., N } � Ψ u � � = 1 ⊕ span i u , p u , Ξ u ( x u ) I u , n u . i u ∈ ¯ ∅� = u ⊆{ 1 ,..., N } ¯ � ¯ S u Ψ u � � p u , Ξ u = S k , p k , ξ k = span i u , p u , Ξ u ( x u ) I u , n u (zero mean) i u ∈ ¯ k ∈ u � � ¯ ψ k S k , p k , ξ k = span i k , p k , ξ k ( x k ) (zero mean) i k =2 ,..., n k

INTRODUCTION SDD EXAMPLES CLOSURE Spline Dimensional Decomposition Theorem Under Assumptions 1-3, a random variable y ( X ) ∈ L 2 (Ω , F , P ) admits a hierarchical orthogonal expansion in multivariate ON spline basis { Ψ u i u , p u , Ξ u ( X u ) } , referred to as the SDD of � � C u i u , p u , Ξ u Ψ u y p , Ξ ( X ) := y ∅ + i u , p u , Ξ u ( X u ) , i u ∈ ¯ ∅� = u ⊆{ 1 ,..., N } I u , n u � where y ∅ := A N y ( x ) f X ( x ) d x , � C u A N y ( x )Ψ u i u , p u , Ξ u := i u , p u , Ξ u ( x u ) f X ( x ) d x . Moreover, the SDD of y ( X ) is the best approximation, i.e., E [ y ( X ) − y p , Ξ ( X )] 2 = g ∈S p , Ξ E [ y ( X ) − g ( X )] 2 . inf

INTRODUCTION SDD EXAMPLES CLOSURE Error Bound & Convergence Modulus of smoothness ( α k ≥ 1 ) � � L 2 [ a k , b k − α k u k ] , h k ≥ 0 , ω α k ( y ; h k ) L 2 [ a k , b k ] := sup � ∆ α k u k y ( x k ) � 0 ≤ u k ≤ h k � ∆ α ω α ( y ; h ) L 2 [ A N ] := sup u y ( x ) � L 2 [ A N α , u ] , h ≥ 0 0 ≤ u ≤ h L 2 -error � | y ( X ) − y p , Ξ ( X ) | 2 � ≤ C ω p + 1 ( y ; h ) L 2 ( A N ) E � | y ( X ) − y p , Ξ ( X ) | 2 � h → 0 E lim = 0 SDD converges in m.s., in probability and in distribution.

INTRODUCTION SDD EXAMPLES CLOSURE Truncation S -variate, SDD Approximation (Poly. Complexity) � � C u i u , p u , Ξ u Ψ u y S , p , Ξ ( X ) := y ∅ + i u , p u , Ξ u ( X u ) i u ∈ ¯ ∅� = u ⊆{ 1 ,..., N } I u , n u 1 ≤| u |≤ S S s N � N � � � � No . of coeff ., L S , p , Ξ = 1 + ( n k − 1) ≤ n k s s =1 k =1 k =1 ( N = 15, n k = 5, S = 1 or 2: L S , p , Ξ = 61 or 1741 ≪ 5 15 ) Second-Moment Statistics E [ y S , p , Ξ ( X )] = y ∅ = E [ y ( X )] 2 ≤ var [ y ( X )] � � C u var [ y S , p , Ξ ( X )] = i u , p u , Ξ u i u ∈ ¯ ∅� = u ⊆{ 1 ,..., N } I u , n u 1 ≤| u |≤ S

INTRODUCTION SDD EXAMPLES CLOSURE Example 1: A Nonsmooth Function ( N = 2) Defined on the square A 2 = [ − 1 , 1] 2 , consider a nonsmooth function y ( X 1 , X 2 ) = g ( X 1 ) + g ( X 2 ) + 1 5 g ( X 1 ) g ( X 2 ) , X 1 , X 2 ∼ i . i . d . U [ − 1 , 1] , � 1 , − 1 ≤ x i ≤ 0 , g ( x i ) = exp( − 10 x i ) , 0 < x i ≤ 1 .

INTRODUCTION SDD EXAMPLES CLOSURE Example 1: Variance Errors 10 0 10 0 Univariate PDD Bivariate PDD / PCE ● ● ■ ● ● ◆ ■ 10 - 1 Univariate SDD ( p = 1 ) Bivariate SDD ( p = 1 ) ■ ■ ● ■ Univariate SDD ( p = 2 ) Bivariate SDD ( p = 2 ) ■ ● ◆ ● ◆ ● ■ ◆ ◆ ■ ● 10 - 1 10 - 2 ● ◆ ■ ● ● ● e 1, p , e 1, p , h ■ e 2, p , e 2, p , h ■ ● ◆ ■ ● ◆ ● 10 - 3 ■ ■ ■ ◆ ● ■ ● ◆ ◆ ■ 10 - 2 10 - 4 ● ● ◆ ■ ● ◆ ■ ● ◆ 0.00359877 ■ ■ ◆ ■ 10 - 5 ◆ ◆ ◆ ◆ ◆ ◆ ◆ 10 - 6 10 - 3 0 100 200 300 400 500 600 0 10 20 30 40 50 No. of coefficients No. of coefficients

INTRODUCTION SDD EXAMPLES CLOSURE Example 2: A Linear Elasticity Problem ( N = 15) A twisting horseshoe Stochastic PDE (Elliptical) ∇ · σ ( z ; X ) = 0 in D ⊂ R 3 , σ ( z ; X ) · n ( z ; X ) = ¯ t ( z ; X ) on ∂ D t , u ( z ; X ) = ¯ u ( z ; X ) on ∂ D u , ∂ D t ∪ ∂ D u = ∂ D , ∂ D t ∩ ∂ D u = ∅ . Random Input ( N = 15) E ( z ; · ) = C α exp[ α ( z ; · )] , � 1 + ν 2 C α = µ E / E , α ( z ; · ) → homogen . Gaussian RF , Γ α ( z , z ′ ) = σ 2 exp( −|| z − z ′ || / bL ) , 15 √ λ i φ i ( z ) X i . � α ( z ; · ) = i =1 SDD coeffs. est. by dim.-red. integ.

INTRODUCTION SDD EXAMPLES CLOSURE Example 2: St. Dev. of Displacement Field

INTRODUCTION SDD EXAMPLES CLOSURE Example 2: Probability Distribution of a Critical Stress

INTRODUCTION SDD EXAMPLES CLOSURE Conclusion A new ON spline expansion (SDD) is introduced. Comp. effort scales polynomially, not exponentially. SDD converges in m.s. and others weaker modes. A low-order SDD is more accurate than high-order PDD/PCE for nonsmooth functions. Future work Explore nonuniform knot sequences. Study unbounded domains without transformation.