Adaptive Algorithms for Stochastic Computation Fred J. Hickernell - PowerPoint PPT Presentation

Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples Discussion References Adaptive Algorithms for Stochastic Computation Fred J. Hickernell Department of Applied Mathematics, Illinois Institute of Technology hickernell@iit.edu mypages.iit.edu/~hickernell Joint work with Tony Jim´ enez Rugama Tony, Yuhan Ding, and Xuan Zhou will present posters on Tuesday afternoon Supported by NSF-DMS-1115392 Many thanks to the organizers September 15, 2014 hickernell@iit.edu Adaptive Algorithms for Stochastic Computation ICERM IBC & Stoch. Comp. 2014 1 / 18

Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples Discussion References Some Problems in Stochastic Computation financial risk, statistical physics, photon transport, . . . µ “ E p Y q “ ? ż µ “ E r f p X qs “ R d f p x q ̺ p x q d x “ ? µ “ P p a ď Y ď b q “ ? p “ P p Y ď µ q , µ “ ? hickernell@iit.edu Adaptive Algorithms for Stochastic Computation ICERM IBC & Stoch. Comp. 2014 2 / 18

Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples Discussion References Some Problems in Stochastic Computation financial risk, statistical physics, photon transport, . . . IID sampling, low discrepancy ÿ n µ n pt Y i uq : “ 1 sampling, error bounds, tractability, µ “ E p Y q « ˆ Y i n multi-level (Richtmyer, 1951; i “ 1 ż Niederreiter, 1992; Sloan and Joe, µ “ E r f p X qs “ R d f p x q ̺ p x q d x 1994; Hickernell, 1998; Dick and Pillichshammer, 2010; Novak and µ “ P p a ď Y ď b q “ ? Wo´ zniakowski, 2010; Dick et al., p “ P p Y ď µ q , µ “ ? 2014; Giles, 2014), . . . hickernell@iit.edu Adaptive Algorithms for Stochastic Computation ICERM IBC & Stoch. Comp. 2014 2 / 18

Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples Discussion References Some Problems in Stochastic Computation financial risk, statistical physics, photon transport, . . . IID sampling, low discrepancy ÿ n µ n pt Y i uq : “ 1 sampling, error bounds, tractability, µ “ E p Y q « ˆ Y i n multi-level (Richtmyer, 1951; i “ 1 ż Niederreiter, 1992; Sloan and Joe, µ “ E r f p X qs “ R d f p x q ̺ p x q d x 1994; Hickernell, 1998; Dick and Pillichshammer, 2010; Novak and µ “ P p a ď Y ď b q “ ? Wo´ zniakowski, 2010; Dick et al., p “ P p Y ď µ q , µ “ ? 2014; Giles, 2014), . . . guaranteed, adaptive Monte Carlo Given a tolerance ε how do we (Hickernell et al., 2014; Jiang and choose n adaptively to make Hickernell, 2014), trapezoidal rule (Clancy et al., 2014), quasi-Monte | µ ´ ˆ µ n | ď ε Carlo (Hickernell and Jim´ enez (with high probability)? Rugama, 2014; Jim´ enez Rugama and Hickernell, 2014), GAIL (Choi et al., 2013–2014) hickernell@iit.edu Adaptive Algorithms for Stochastic Computation ICERM IBC & Stoch. Comp. 2014 2 / 18

Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples Discussion References Recent Results § µ “ E p Y q “ ? (Hickernell et al., 2014), for somewhat different view see (Bayer et al., 2014) § Compute a highly probable upper bound on true variance, C 2 ˆ σ 2 n σ , using n σ IID samples. § Use a Berry-Esseen inequality (finite sample Central Limit Theorem) to find n such that P p | µ ´ ˆ µ n | ď ε q ě 99% . § Guaranteed for random variables in the cone of bounded kurtosis E rp Y ´ µ q 4 s{ σ 4 ď κ max p n σ , C q § Computational cost n — p σ { ε q 2 where σ is unknown. § µ “ E p Y q “ ? for Bernoulli Y (Jiang and Hickernell, 2014) § Can find n that guarantees that P p | µ ´ ˆ µ n | ď ε a q ě 99% or P p | µ ´ ˆ µ n | ď ε r | µ | q ě 99% . ş b § µ “ a f p x q d x “ ? (Clancy et al., 2014) § Can find n that guarantees that the trapezoidal rule with n trapezoids, ˆ µ n , gives | µ ´ ˆ µ n | ď ε . § Guaranteed for integrands in the cone Var p f 1 q ď τ � f 1 ´ r f p b q ´ f p a qs{p b ´ a q � 1 a § Computational cost n — Var p f 1 q{ ε where Var p f 1 q is unknown. hickernell@iit.edu Adaptive Algorithms for Stochastic Computation ICERM IBC & Stoch. Comp. 2014 3 / 18

Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples Discussion References Digital Net Cubature Error via Walsh Expansions ż 2 m ´ 1 � � 8 ÿ ÿ ω p ℓ q r r 0 , 1 q d f p x q d x ´ 1 ω p m q ˚ S m ´ ℓ,m p f q � ď p � � � ˆ � � f p z i q � ď f λ 2 m � � proof 2 m 1 ´ p ω p ℓ q ˚ ω p ℓ q � � � i “ 0 λ “ 1 digital net nodes Walsh coefficients in dual net 1 Walsh functions & coefficients 8 ÿ p´ 1 q x k p κ q , x y ˆ 0.75 f p x q “ f κ κ “ 0 2 m ´ 1 ÿ f m,κ : “ 1 0.5 p´ 1 qx ˜ k p κ q , z i y f p z i q ˜ 2 m i “ 0 ÿ 8 0.25 ˆ f κ ` λ 2 m aliasing “ λ “ 0 0 0 0.25 0.5 0.75 1 hickernell@iit.edu Adaptive Algorithms for Stochastic Computation ICERM IBC & Stoch. Comp. 2014 4 / 18

Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples Discussion References Digital Net Cubature Error via Walsh Expansions ż � 2 m ´ 1 � ÿ ÿ 8 ω p ℓ q r r 0 , 1 q d f p x q d x ´ 1 � ď p ω p m q ˚ S m ´ ℓ,m p f q � � � ˆ � � f p z i q � ď � � f λ 2 m proof 2 m 1 ´ p ω p ℓ q ˚ ω p ℓ q � � � i “ 0 λ “ 1 digital net nodes Walsh coefficients in dual net Walsh functions & coefficients ÿ 8 p´ 1 q x k p κ q , x y ˆ f p x q “ f κ κ “ 0 2 m ´ 1 ÿ f m,κ : “ 1 p´ 1 qx ˜ k p κ q , z i y f p z i q ˜ 2 m i “ 0 ÿ 8 ˆ “ f κ ` λ 2 m aliasing λ “ 0 hickernell@iit.edu Adaptive Algorithms for Stochastic Computation ICERM IBC & Stoch. Comp. 2014 4 / 18

Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples Discussion References Digital Net Cubature Error via Walsh Expansions ż 2 m ´ 1 � � 8 ÿ ÿ ω p ℓ q r r 0 , 1 q d f p x q d x ´ 1 ω p m q ˚ S m ´ ℓ,m p f q � ď p � � � ˆ � � f p z i q � ď f λ 2 m � � proof 2 m 1 ´ p ω p ℓ q ˚ ω p ℓ q � � � i “ 0 λ “ 1 digital net nodes Walsh coefficients in dual net 2 ℓ ´ 1 8 ÿ ÿ 0 10 p � ˆ � � S ℓ,m p f q : “ f κ ` λ 2 m � , κ “ t 2 ℓ ´ 1 u λ “ 1 ÿ 8 −5 10 q � ˆ � � S m p f q : “ f κ � f κ | | ˆ κ “ 2 m 2 ℓ ´ 1 ÿ −10 10 � ˆ error ≤ ˆ � � S 0 , 12 ( f ) S ℓ p f q : “ f κ � ˇ S 12 ( f ) κ “ t 2 ℓ ´ 1 u S 8 ( f ) −15 2 ℓ ´ 1 ÿ 10 0 1 2 3 4 r � ˜ � � S ℓ,m p f q : “ f m,κ 10 10 10 10 10 � κ κ “ t 2 ℓ ´ 1 u p ω p m ´ ℓ q q q Cone conditions: S ℓ,m p f q ď p S m p f q , S m p f q ď ˚ ω p ℓ q S m ´ ℓ p f q . hickernell@iit.edu Adaptive Algorithms for Stochastic Computation ICERM IBC & Stoch. Comp. 2014 4 / 18

Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples Discussion References Adaptively Attaining Absolute or Relative Error Tolerances ż 2 m ´ 1 � � ÿ ω p ℓ q r ´ 1 � ď p ω p m q ˚ S m ´ ℓ,m p f q � � Have r 0 , 1 q d f p x q d x f p z i q . � � 2 m 1 ´ p ω p ℓ q ˚ ω p ℓ q � � loooooooooooomoooooooooooon looooooomooooooon looooooomooooooon i “ 0 � err p m q µ µ m ˆ We want to find m and ˜ µ m that guarantees | µ ´ ˜ µ m | ď max p ε a , ε r | µ | q . Algorithm cubSobol g . Given tolerances ε a and ε r , fix ℓ and initalize m ą ℓ . Step 1. Compute the data-based error bound, err p m q , and ˆ µ m . Step 2. If err p m q is small enough such that err p m q ď 1 2 r max p ε a , ε r | ˆ µ m ´ err p m q | ` max p ε a , ε r | ˆ µ m ` err p m q | s , then return the shrinkage estimator µ m ` 1 µ m “ ˆ ˜ 2 r max p ε a , ε r | ˆ µ m ´ err p m q | ´ max p ε a , ε r | ˆ µ m ` err p m q | s . Step 3. Otherwise, increase m by one, and return to Step 1. Theorem. For integrands satifying the cone conditions cubSobol g proof , and the computational cost is O pr m ` $ p f qs 2 m q , for some succeeds m ď min t m 1 : r 1 ` p ω p m 1 q ˚ ω p ℓ q ˚ ω p ℓ qsp 1 ` ε r q p ω p ℓ q S m 1 ´ ℓ p f q ď max p ε a , ε r | µ | qr 1 ´ p ω p ℓ q ˚ ω p ℓ qsu proof more proof hickernell@iit.edu Adaptive Algorithms for Stochastic Computation ICERM IBC & Stoch. Comp. 2014 5 / 18