Quasi-Monte Carlo integration over the Euclidean space and applications Frances Kuo f.kuo@unsw.edu.au University of New South Wales, Sydney, Australia joint work with James Nichols (UNSW) Journal of Complexity 30 (2014) 444–468. Frances Kuo
MC v.s. QMC in the unit cube n � 1 � [0 , 1] s g ( x x x ) d x x x ≈ g ( t t t i ) n i =1 Monte Carlo method Quasi-Monte Carlo methods t t i deterministic t t t i random uniform t close to n − 1 convergence or better n − 1 / 2 convergence more effective for earlier variables and lower-order projections order of variables irrelevant order of variables very important 1 1 1 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 0 1 0 1 0 1 First 64 points of a 64 random points A lattice rule with 64 points 2D Sobol ′ sequence use randomized QMC methods for error estimation Frances Kuo
QMC Two main families of QMC methods: (t,m,s)-nets and (t,s)-sequences Niederreiter book (1992) Sloan and Joe book (1994) lattice rules Important developments: component-by-component ( CBC ) A group under addition modulo Z construction and includes the integer points higher order digital nets • • • • • • • • • • • • • • • 1 1 • • • • • • • Dick and Pillichshammer book (2010) • • • • • • • • • • • • • • • • • • • • • Dick, Kuo, Sloan Acta Numerica (2013) • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Having the right number of • 0 1 0 1 • • First 64 points of a • • A lattice rule with 64 points • points in various sub-cubes 2D Sobol ′ sequence • (0,6,2)-net Frances Kuo
Standard QMC analysis Worst case error bound � � � � n n � � � � x − 1 x − 1 � � � � � ≤ e wor ( t � ≤ e wor � � � � [0 , 1] s g ( x [0 , 1] s g ( x x x x ) d x x ) d x x x g ( t g ( t t t t i ) t i ) ( t t t t t t n ) � g � γ t n ) � g � t 1 , . . . ,t t 1 , . . . ,t γ γ � � � � γ γ γ n n � � i =1 i =1 “Standard” setting – weighted Sobolev space � � 2 � ∂ | u | g 1 � � � � g � 2 � � γ = ( x x x u ; 0) d x x x u � � γ γ x γ u ∂x x u [0 , 1] | u | � � u ⊆{ 1: s } “anchor” at 0 (also “unanchored”) 2 s subsets “weights” Mixed first derivatives are square integrable x Small weight γ u means that g depends weakly on the variables x x u Choose weights to minimize the error bound � b u � � 1 / (2 λ ) � � 1 / 2 � 1 / (1+ λ ) 2 b u � � γ λ u a u ⇒ γ u = n γ u a u u ⊆{ 1: s } u ⊆{ 1: s } � �� � � �� � bound on worst case error (CBC) bound on norm Construct points (CBC) to minimize the worst case error Frances Kuo
Practical problems do not fit... Practical integral over the Euclidean space � R s q ( z z z ) ρ ( z z z ) d z z z 1. Transformation (translation, rotation, rescaling) s � � φ is any univariate pdf = R s f ( y y y ) φ ( y j ) d y y y j =1 2. Mapping to unit cube � Φ − 1 - icdf φ - pdf Φ - cdf [0 , 1] s f (Φ − 1 ( x = x x )) d x x x n ≈ 1 � 3. Applying QMC f (Φ − 1 ( t t t i )) n i =1 Transformed integrand g = f ◦ Φ − 1 rarely falls in the “standard” setting! Frances Kuo
Application 1: option pricing Black-Scholes model : stock price follows a geometric Brownian motion z ) = exp( · · · � a j z j ) � � � s exp( − 1 S j ( z z z T Σ − 1 z z z ) z 2 z 1 � z z R s max S j ( z z ) − K, 0 d z z � (2 π ) s det(Σ) s 0 j =1 10 MC � � s s � 1 � � naive QMC −1 10 = R s max S j ( Ay y ) − K, 0 y φ nor ( y j ) d y y y standard error s j =1 j =1 −2 10 � � s � 1 � S j ( A Φ − 1 x x = [0 , 1] s max nor ( x x )) − K, 0 d x x s −3 10 j =1 QMC + PCA −4 10 RW/BB/PCA 2 3 4 10 10 10 1. Transformation 2. Mapping to the cube n Write Σ = AA T y = Φ − 1 Sub. z z z = Ay y Sub. y y nor ( x x x ) y Issues: unbounded near the boundary of the unit cube, and kink, i.e., no square-integrable mixed first derivatives Frances Kuo
Application 2: maximum likelihood Generalized linear mixed model [K., Dunsmuir, Sloan, Wand, Womersley (2008)] � � � s exp( − 1 exp( τ j ( β + z j ) − e β + z j ) z T Σ − 1 z z z ) z 2 z � z d z z � (2 π ) s det(Σ) τ j ! R s j =1 If we write Σ = AA T and substitute z y = Φ − 1 z z = Ay y y followed by y y nor ( x x x ) , then we get very bad results... no re-scaling (bad) no centering and no re-scaling (worse) Frances Kuo
Application 2: maximum likelihood Generalized linear mixed model � [K., Dunsmuir, Sloan, Wand, Womersley (2008)] R s exp( T ( z z z )) d z z z s s � 1 � � = c R s exp( T ( A ∗ y y + z y z z ∗ )) φ ( y j ) d y y y φ ( y j ) j =1 j =1 � �� � f ( y y y ) φ normal (good) s � 1 � (0 , 1) s exp( F ( A ∗ Φ − 1 ( x = c x x ) + z z z ∗ )) d u u u φ (Φ − 1 ( x j )) j =1 � �� � x ) = f (Φ − 1 ( x g ( x x x x )) 1. Transformation 2. Mapping to the cube [centering, re-scaling] [ φ – free to choose] φ logistic (better) z = A ∗ y z ∗ y = Φ − 1 ( x Sub. z z y y + z z Sub. y y x x ) Issues: unbounded near the boundary of the unit cube, or huge derivatives near the boundary of the unit cube φ Student-t (best) Frances Kuo
Application 3: PDE with random coeff. Elliptic PDE with lognormal random coefficient x ∈ D ⊆ R d , d = 1 , 2 , 3 −∇ · ( a ( � x, y y y ) ∇ u ( � x, y y y )) = forcing ( � x ) , � � � s √ µ j ξ j ( � � a ( � x, y y y ) = exp x ) y j , y j ∼ i.i.d. normal j =1 s � � y y R s G ( u ( · , y y )) φ nor ( y j ) d y y G – linear functional j =1 � [0 , 1] s G ( u ( · , Φ − 1 = nor ( x x x )) d x x x [Graham, K., Nuyens, Scheichl, Sloan (2010)] – circulant embedding [K., Schwab, Sloan (2011,2012,2014)] – “uniform” case, POD weights, fast CBC, multilevel Differentiate the PDE to estimate the norm of integrand � Minimize the error bound ⇒ POD weights γ u = Γ | u | γ j j ∈ u [Dick, K., Le Gia, Nuyens, Schwab (2014)] – higher order [Dick, K., Le Gia, Schwab (2014)] – higher order, multilevel etc. Frances Kuo
A non-standard setting Change of variables s � � � � [0 , 1] s f (Φ − 1 ( x R s q ( z z ) ρ ( z z z z ) d z z z = R s f ( y y y ) φ ( y j ) d y y y = x x )) d x x x j =1 g = f ◦ Φ − 1 rarely belongs to weighted Sobolev space Non-standard norm [Wasilkowski & Wo´ zniakowski (2000)] � � 2 s � ∂ | u | f 1 � � � � � f � 2 ψ 2 ( y j ) d y � � γ = ( y y y u ; 0) y y u � � γ γ γ u ∂y y y u R | u | � � j =1 u ⊆{ 1: s } weight function Nichols & K. (2014) cf. [K., Sloan, Wasilkowski, Waterhouse (2010)] Also unanchored variant, coordinate dependent ψ j Randomly shifted lattice rules CBC error bound for general weights γ u Convergence rate depends on the relationship between φ and ψ � Fast CBC for POD weights γ u = Γ | u | j ∈ u γ j Important for applications : φ and ψ and γ u are up to us to choose (tune) Frances Kuo
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