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Stratified sampling Integration of indicator functions Numerical results Stratified Monte Carlo Integration and Applications R. El Haddad, R. Fakhereddine, C. L´ ecot, G. Venkiteswaran Universit´ e Saint-Joseph, Beirut, Lebanon & Universit´ e de Savoie, Le Bourget-du-Lac, France & Birla Institute of Technology and Science, Pilani, India MCQMC 2012 February 13 – 17, 2012, Sydney, Australia R. El Haddad, R. Fakhereddine, C. L´ ecot, G. Venkiteswaran Stratified Monte Carlo Integration and Applications

Stratified sampling Integration of indicator functions Numerical results Plan of the talk Stratified sampling 1 Integration of indicator functions 2 Numerical results 3 R. El Haddad, R. Fakhereddine, C. L´ ecot, G. Venkiteswaran Stratified Monte Carlo Integration and Applications

Stratified sampling Integration of indicator functions Numerical results Stratified sampling 1 Integration of indicator functions 2 Numerical results 3 R. El Haddad, R. Fakhereddine, C. L´ ecot, G. Venkiteswaran Stratified Monte Carlo Integration and Applications

Stratified sampling Integration of indicator functions Numerical results Numerical integration I := [0 , 1), for s ≥ 1, λ s is the s -dimensional Lebesgue measure, g : I s → R is square-integrable. We want to approximate � J := I s g ( x ) d λ s ( x ) . R. El Haddad, R. Fakhereddine, C. L´ ecot, G. Venkiteswaran Stratified Monte Carlo Integration and Applications

Stratified sampling Integration of indicator functions Numerical results Monte Carlo approximation { U 1 , . . . , U N } i.i.d. random variables uniformly distributed over I s , N � X := 1 g ◦ U ℓ . N ℓ =1 Var ( X ) = σ 2 ( g ) E [ X ] = J , and N where � � � � 2 � � 2 d λ s ( x ) − σ 2 ( g ) := g ( x ) I s g ( x ) d λ s ( x ) . I s R. El Haddad, R. Fakhereddine, C. L´ ecot, G. Venkiteswaran Stratified Monte Carlo Integration and Applications

Stratified sampling Integration of indicator functions Numerical results Stratified sampling { D 1 , . . . , D p } a partition of I s , N 1 , . . . , N p integers, { V k 1 , . . . , V k N k } i.i.d. random variables uniformly distributed over D k . N k p � � T k := 1 g ◦ V k T := λ s ( D k ) T k . ℓ N k ℓ =1 k =1 and ( proportional allocation 1 ) E [ T ] = J Var ( T ) ≤ Var ( X ) . 1 G.S. Fishman, Monte Carlo, Springer, New York (1996) R. El Haddad, R. Fakhereddine, C. L´ ecot, G. Venkiteswaran Stratified Monte Carlo Integration and Applications

Stratified sampling Integration of indicator functions Numerical results Simple stratified sampling { D 1 , . . . , D N } a partition of I s , λ s ( D 1 ) = · · · = λ s ( D N ) = 1 N , { V 1 , . . . , V N } independent random variables, V ℓ uniformly distributed over D ℓ . N � Y := 1 g ◦ V ℓ . N ℓ =1 and 2 3 E [ Y ] = J Var ( Y ) ≤ Var ( X ) 2 S. Haber, A modified Monte Carlo quadrature, Math. Comput. 20, 361–368 (1966) 3 R.C.H. Cheng, T. Davenport, The problem of dimensionality in stratified sampling, Manage. Sci. 35, 1278–1296 (1989) R. El Haddad, R. Fakhereddine, C. L´ ecot, G. Venkiteswaran Stratified Monte Carlo Integration and Applications

Stratified sampling Integration of indicator functions Numerical results Simple stratified sampling 1 0 1 R. El Haddad, R. Fakhereddine, C. L´ ecot, G. Venkiteswaran Stratified Monte Carlo Integration and Applications

Stratified sampling Integration of indicator functions Numerical results Latin hypercube sampling I ℓ := [( ℓ − 1) / N , ℓ/ N ), 1 ≤ ℓ ≤ N { V i 1 , . . . , V i N } independent random variables, V i ℓ uniformly distributed over I ℓ { π 1 , . . . , π s } independent random permutations of { 1 , . . . , N } . W ℓ := ( V 1 π 1 ( ℓ ) , . . . , V s π s ( ℓ ) ) N � Z := 1 g ◦ W ℓ . N ℓ =1 and 4 E [ Z ] = J Var ( Z ) ≤ Var ( X ) 4 M.D. McKay, R.J. Beckman, W.J. Conover, A comparison of three methods for selecting values of input variables in the analysis of output from a computer code, Technometrics, 21, 239–245 (1979) R. El Haddad, R. Fakhereddine, C. L´ ecot, G. Venkiteswaran Stratified Monte Carlo Integration and Applications

Stratified sampling Integration of indicator functions Numerical results Latin hypercube sampling R. El Haddad, R. Fakhereddine, C. L´ ecot, G. Venkiteswaran Stratified Monte Carlo Integration and Applications

Stratified sampling Integration of indicator functions Numerical results Latin hypercube sampling R. El Haddad, R. Fakhereddine, C. L´ ecot, G. Venkiteswaran Stratified Monte Carlo Integration and Applications

Stratified sampling Integration of indicator functions Numerical results Latin hypercube sampling R. El Haddad, R. Fakhereddine, C. L´ ecot, G. Venkiteswaran Stratified Monte Carlo Integration and Applications

Stratified sampling Integration of indicator functions Numerical results Stratified sampling 1 Integration of indicator functions 2 Numerical results 3 R. El Haddad, R. Fakhereddine, C. L´ ecot, G. Venkiteswaran Stratified Monte Carlo Integration and Applications

Stratified sampling Integration of indicator functions Numerical results Monte Carlo approximation A ⊂ I s , J = λ s ( A ), g := 1 A , { U 1 , . . . , U N } i.i.d. random variables uniformly distributed over I s , N � X := 1 1 A ◦ U ℓ . N ℓ =1 � � Var ( X ) = 1 ≤ 1 1 − λ s ( A ) N λ s ( A ) 4 N . R. El Haddad, R. Fakhereddine, C. L´ ecot, G. Venkiteswaran Stratified Monte Carlo Integration and Applications

Stratified sampling Integration of indicator functions Numerical results ε -collars � x � ∞ := max 1 ≤ i ≤ s | x i | . For ε > 0, { x ∈ I s : ∀ y ∈ I s \ A � x − y � ∞ ≥ ε } , A − ε := { x ∈ I s : ∃ y ∈ A � x − y � ∞ < ε } . := A ε A − ε ⊂ A ⊂ A ε . Suppose there exists a nondecreasing γ : [0 , + ∞ ) → [0 , + ∞ ) such that � � ∀ ε > 0 max λ s ( A ε \ A ) , λ s ( A \ A − ε ) ≤ γ ( ε ) , R. El Haddad, R. Fakhereddine, C. L´ ecot, G. Venkiteswaran Stratified Monte Carlo Integration and Applications

Stratified sampling Integration of indicator functions Numerical results ε -collars 1 A 0 1 R. El Haddad, R. Fakhereddine, C. L´ ecot, G. Venkiteswaran Stratified Monte Carlo Integration and Applications

Stratified sampling Integration of indicator functions Numerical results ε -collars 1 A A - ε 0 1 R. El Haddad, R. Fakhereddine, C. L´ ecot, G. Venkiteswaran Stratified Monte Carlo Integration and Applications

Stratified sampling Integration of indicator functions Numerical results ε -collars 1 A ε A A - ε 0 1 R. El Haddad, R. Fakhereddine, C. L´ ecot, G. Venkiteswaran Stratified Monte Carlo Integration and Applications

Stratified sampling Integration of indicator functions Numerical results Simple stratified sampling N = n s ; for k = ( k 1 , . . . , k s ) with 1 ≤ k i ≤ n , � k i − 1 � s � , k i C k := , n n i =1 { V k : 1 ≤ k i ≤ n } independent random variables, V k uniformly distributed over C k . � Y := 1 1 A ◦ V k . N k R. El Haddad, R. Fakhereddine, C. L´ ecot, G. Venkiteswaran Stratified Monte Carlo Integration and Applications

Stratified sampling Integration of indicator functions Numerical results Simple stratified sampling R. El Haddad, R. Fakhereddine, C. L´ ecot, G. Venkiteswaran Stratified Monte Carlo Integration and Applications

Stratified sampling Integration of indicator functions Numerical results Simple stratified sampling R. El Haddad, R. Fakhereddine, C. L´ ecot, G. Venkiteswaran Stratified Monte Carlo Integration and Applications

Stratified sampling Integration of indicator functions Numerical results Simple stratified sampling : Jordan measurable set � � Proposition 1. If ∀ ε > 0 max λ s ( A ε \ A ) , λ s ( A \ A − ε ) ≤ γ ( ε ), then � � Var ( Y ) ≤ 1 1 2 N γ . N 1 / s Proof. We have 1 Var ( Y ) ≤ 4 N 2 # { k : C k ∩ A � = ∅ and C k �⊂ A } . Since � C k ⊂ A 1 / n \ A − 1 / n , C k ∩ A � = ∅ and C k �⊂ A we have � 1 � 1 N # { k : C k ∩ A � = ∅ and C k �⊂ A } ≤ 2 γ . n � � 1 for linear γ : Var ( Y ) ≤ O . N 1+1 / s R. El Haddad, R. Fakhereddine, C. L´ ecot, G. Venkiteswaran Stratified Monte Carlo Integration and Applications

Stratified sampling Integration of indicator functions Numerical results Under the hypersurface s − 1 → I and A f := { ( x ′ , x s ) ∈ I s : x s < f ( x ′ ) } . f : I 1 A f 0 1 We want to approximate � � I s − 1 f ( x ′ ) d λ s − 1 ( x ′ ) = I := I s 1 A f ( x ) d λ s ( x ) . R. El Haddad, R. Fakhereddine, C. L´ ecot, G. Venkiteswaran Stratified Monte Carlo Integration and Applications