Quasi-Monte Carlo integration over the Euclidean space and applications

Transcription

1 Quasi-Monte Carlo integration over the Euclidean space and applications University of New South Wales, Sydney, Australia joint work with James Nichols (UNSW) Journal of Complexity 30 (204)

2 MC v.s. QMC in the unit cube [0,] s g(x)dx n n i= g(t i ) Monte Carlo method Quasi-Monte Carlo methods t i random uniform t i deterministic n /2 convergence close to n convergence or better 0 64 random points 0 First 64 points of a 2D Sobol sequence 0 A lattice rule with 64 points more effective for earlier variables and lower-order projections order of variables irrelevant order of variables very important use randomized QMC methods for error estimation

3 QMC Two main families of QMC methods: (t,m,s)-nets and (t,s)-sequences lattice rules 0 First 64 points of a 2D Sobol sequence 0 A lattice rule with 64 points (0,6,2)-net Having the right number of points in various sub-cubes A group under addition modulo Z and includes the integer points Niederreiter book (992) Sloan and Joe book (994) Important developments: component-by-component (CBC) construction higher order digital nets Dick and Pillichshammer book (200) Dick, Kuo, Sloan Acta Numerica (203)

4 Standard QMC analysis Worst case error bound g(x)dx [0,] s n n g(t i ) ewor γ (t,...,t n ) g γ i= Standard setting weighted Sobolev space g 2 γ = u g 2 (x u ;0) dx x u u u {:s} γ u [0,] u 2 s subsets anchor at 0 (also unanchored ) weights Mixed first derivatives are square integrable Small weight γ u means that g depends weakly on the variables x u Choose weights to minimize the error bound ( ) /(2λ) ( 2 γ λ u n a b u u u {:s} }{{} bound on worst case error (CBC) u {:s} γ u ) /2 } {{ } bound on norm γ u = Construct points (CBC) to minimize the worst case error ( bu a u ) /(+λ)

5 Practical problems do not fit... Practical integral over the Euclidean space q(z)ρ(z)dz. Transformation R s (translation, rotation, rescaling) = f(y) φ(y j )dy φ is any univariate pdf R s = [0,] s f(φ (x))dx 2. Mapping to unit cube φ - pdf Φ - cdf Φ - icdf n n f(φ (t i )) 3. Applying QMC i= Transformed integrand g = f Φ rarely falls in the standard setting!

6 Application : option pricing Black-Scholes model: stock price follows a geometric Brownian motion = = R s max ( s s ( max R s s [0,] s max ) exp( 2 S j (z) K,0 zt Σ z) (2π) s det(σ) dz s ( s S j (Ay) K,0 s RW/BB/PCA. Transformation Write Σ = AA T Sub. z = Ay ) s S j (AΦ nor(x)) K,0 φ nor (y j )dy ) dx 2. Mapping to the cube Sub. y = Φ nor (x) standard error S j (z) = exp( a j z j ) MC QMC + PCA naive QMC n Issues: unbounded near the boundary of the unit cube, and kink, i.e., no square-integrable mixed first derivatives

7 Application 2: maximum likelihood Generalized linear mixed model [K., Dunsmuir, Sloan, Wand, Womersley (2008)] R s ( s exp(τ j (β + z j ) e β+z j) τ j! ) exp( 2 zt Σ z) (2π) s det(σ) dz If we write Σ = AA T and substitute z = Ay followed by y = Φ nor (x), then we get very bad results... no re-scaling (bad) no centering and no re-scaling (worse)

8 Application 2: maximum likelihood Generalized linear mixed model [K., Dunsmuir, Sloan, Wand, Womersley (2008)] exp(t(z))dz R s = c exp(t(a y + z )) φ(y j )dy R s φ(y j ) }{{} f(y) = c exp(f(a Φ (x) + z )) (0,) s φ(φ du (x j )) }{{} g(x)=f(φ (x)) φ normal (good). Transformation [centering, re-scaling] Sub. z = A y + z 2. Mapping to the cube [φ free to choose] Sub. y = Φ (x) φ logistic (better) Issues: unbounded near the boundary of the unit cube, or huge derivatives near the boundary of the unit cube φ Student-t (best)

9 Application 3: PDE with random coeff. Elliptic PDE with lognormal random coefficient = R s G(u(,y)) (a( x,y) u( x,y)) = forcing( x), x D R d, d =,2,3 ( s ) a( x,y) = exp µj ξ j ( x)y j, y j i.i.d. normal φ nor (y j )dy [0,] s G(u(,Φ nor (x))dx G linear functional [Graham, K., Nuyens, Scheichl, Sloan (200)] circulant embedding [K., Schwab, Sloan (20,202,204)] 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 [Dick, K., Le Gia, Nuyens, Schwab (204)] higher order [Dick, K., Le Gia, Schwab (204)] higher order, multilevel etc. j u γ j

10 A non-standard setting Change of variables R s q(z)ρ(z)dz = f(y) R s φ(y j )dy = f(φ (x))dx [0,] s g = f Φ rarely belongs to weighted Sobolev space Non-standard norm [Wasilkowski & Woźniakowski (2000)] f 2 γ = u f 2 (y u ;0) ψ 2 (y y u j )dy u u {:s} γ u R u weight function Nichols & K. (204) cf. [K., Sloan, Wasilkowski, Waterhouse (200)] 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)

11 std.err. Application 3: PDE with random coeff. Elliptic PDE with lognormal random coefficient = R s G(u(,y)) (a( x,y) u( x,y)) = forcing( x), x D R d, d =,2,3 ( s ) a( x,y) = exp µj ξ j ( x)y j, y j i.i.d. normal φ nor (y j )dy [0,] s G(u(,Φ nor (x))dx G linear functional Graham, K., Nichols, Scheichl, Schwab, Sloan (204) Differentiate PDE to obtain bound on the norm 0 - ν =0.75, λ C =0. Choose φ φ nor Choose ψ j (y j ) = exp( α j y j ), α j > 0 Choose POD weights γ u = Γ u j u γ j

12 Application 2: maximum likelihood Generalized linear mixed model exp(t(z))dz R s = c exp(t(a y + z )) R s φ(y j ) }{{} f(y) φ(y j )dy = c exp(f(a Φ (x) + z )) (0,) s φ(φ du (x j )) }{{} Sinescu, K., Sloan (203) g(x)=f(φ (x)) Differentiate integrand to obtain bound on the norm φ normal (good) φ logistic (better) Choose φ φ nor or φ logit or φ stud Choose ψ Choose POD weights γ u = Γ u j u γ j φ Student-t (best)

13 Application : option pricing Black-Scholes model = = R s max ( s s ( max R s s [0,] s max ) exp( 2 S j (z) K,0 zt Σ z) (2π) s det(σ) dz s ( s S j (Ay) K,0 s ) s S j (AΦ nor(x)) K,0 φ nor (y j )dy ) dx standard error S j (z) = exp( a j z j ) 0 0 MC naive QMC QMC + PCA Griebel, K., Sloan (200, 203, 204) ANOVA decomposition: g = u {:s} g u in [0,] s n or f = u f u in R s RW/BB: all g u with u s+ belong to Sobolev space; PCA: similar 2 RW/BB: all f u with u { : s} are smooth; PCA: similar ANOVA decomposition for -variate function f: all terms are smooth Remains to see how to apply the theory of Nichols & K. (204)

14 Summary Nice QMC theory over the unit cube Transformed integrand rarely falls in the standard setting q(z)ρ(z)dz = f(y) φ(y j )dy = f(φ (x))dx R s R s [0,] s Non-standard setting f 2 γ = u {:s} γ u R u u f 2 (y u ;0) y u φ and ψ and γ u are up to us to choose (tune) ψ 2 (y j )dy u Application Transformation φ ψ γ u 3. PDE φ nor exp( α j y) POD 2. likelihood centering, re-scaling φ nor,φ logit,φ stud POD. option RW, BB, PCA φ nor???