Multi-level Monte Carlo methods in Uncertainty Quantification

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von Schwerin (KAUST) 3 rd GAMM AGUQ workshop on Uncertainty Quantification Dortmund, March 12-14, 2018 EU-FP7 project:

1 Multi-level Monte Carlo methods in Uncertainty Quantification Fabio Nobile CSQI - Institute of Mathematics, EPFL, Switzerland Acknowlegments: M. Pisaroni, S. Krumscheid, P. Leyland (EPFL), A.L. Haji-Ali (Oxford) R. Tempone, E. von Schwerin (KAUST) 3 rd GAMM AGUQ workshop on Uncertainty Quantification Dortmund, March 12-14, 2018 EU-FP7 project: Uncertainty Management for Robust Industrial Design in Aeronautics (UMRIDA) Center for Advanced Modeling Science F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

2 Outline 1 Motivating example 2 Multilevel Monte Carlo for expectations 3 MLMC for moments and distributions 4 Risk averse optimization with MLMC 5 Conclusions F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

3 Outline Motivating example 1 Motivating example 2 Multilevel Monte Carlo for expectations 3 MLMC for moments and distributions 4 Risk averse optimization with MLMC 5 Conclusions F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

4 Motivating example UQ in aerodynamic design Compute aerodynamic coeffs. (lift, drag, Cp ) and optimize airfoil shape in presence of operational uncertainties (Mach number, angle of attack,...) and geometrical uncertainties (manufacturing tolerances, icing, fatigue,...) F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

Probabilistic framework: Mach, Reynolds, Angle of Attack, etc.

5 Motivating example Operational uncertainties Atmospheric fluctuations with respect to location, time (T, p, ρ, u) over long flights Probabilistic framework: Mach, Reynolds, Angle of Attack, etc. treated as random variables F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

6 Motivating example Geometrical uncertainties Production: manufacturing, assembly Temporary factors: deflection, icing Permanent/degrading factors: impacts, erosion, fouling Probabilistic Framework: Leading edge radius, thickness, curvature, etc. treated as random variables F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

7 Motivating example Forward Uncertainty propagation Random input parameters: y (with given distribution) (Complex) Model: L y u = F (e.g. Euler, Navier-Stokes,...) hence u = u(y) is a random solution Quantity of interest: Q = Q(u) (random output, e.g. lift, drag, etc.) Goal: compute µ(q) = E[Q] or other statistical quantities In practice, u is not accessible. Computational model L h,y u h = F h = computational output Q h = Q(u h ) F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

8 Motivating example Forward Uncertainty propagation Random input parameters: y (with given distribution) (Complex) Model: L y u = F (e.g. Euler, Navier-Stokes,...) hence u = u(y) is a random solution Quantity of interest: Q = Q(u) (random output, e.g. lift, drag, etc.) Goal: compute µ(q) = E[Q] or other statistical quantities In practice, u is not accessible. Computational model L h,y u h = F h = computational output Q h = Q(u h ) F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

9 Motivating example Forward Uncertainty propagation Random input parameters: y (with given distribution) (Complex) Model: L y u = F (e.g. Euler, Navier-Stokes,...) hence u = u(y) is a random solution Quantity of interest: Q = Q(u) (random output, e.g. lift, drag, etc.) Goal: compute µ(q) = E[Q] or other statistical quantities In practice, u is not accessible. Computational model L h,y u h = F h = computational output Q h = Q(u h ) y computational model Q h F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

10 Monte Carlo method Motivating example Generate M iid copies y (1),..., y (M) y Compute the corresponding outputs Q (i) h, i = 1,..., M Approximate expectation by sample average µ MC h = 1 M Q (i) h (biased estimator E[µ MC h ] = E[Q h ] E[Q]) M Mean squared error i=1 MSE(µ MC h ) := E[(µ(Q) µ MC Complexity analysis (error versus cost) Assume: h ) 2 ] = (E[Q Q h ]) 2 }{{} discret. error E[Q Q h ] = O(h α ), Var[Q h ] = O(1), cost to compute each Q (i) h : C h = O(h γ ) + Var[Q h] M }{{} MC error Then MSE = O(tol 2 ) = h = O(tol 1 α ), M = O(tol 2 ) Total work: Work(µ MC h ) = C h M tol γ α tol 2 F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

11 Monte Carlo method Motivating example Generate M iid copies y (1),..., y (M) y Compute the corresponding outputs Q (i) h, i = 1,..., M Approximate expectation by sample average µ MC h = 1 M Q (i) h (biased estimator E[µ MC h ] = E[Q h ] E[Q]) M Mean squared error i=1 MSE(µ MC h ) := E[(µ(Q) µ MC Complexity analysis (error versus cost) Assume: h ) 2 ] = (E[Q Q h ]) 2 }{{} discret. error E[Q Q h ] = O(h α ), Var[Q h ] = O(1), cost to compute each Q (i) h : C h = O(h γ ) + Var[Q h] M }{{} MC error Then MSE = O(tol 2 ) = h = O(tol 1 α ), M = O(tol 2 ) Total work: Work(µ MC h ) = C h M tol γ α tol 2 F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

12 Monte Carlo method Motivating example Generate M iid copies y (1),..., y (M) y Compute the corresponding outputs Q (i) h, i = 1,..., M Approximate expectation by sample average µ MC h = 1 M Q (i) h (biased estimator E[µ MC h ] = E[Q h ] E[Q]) M Mean squared error i=1 MSE(µ MC h ) := E[(µ(Q) µ MC Complexity analysis (error versus cost) Assume: h ) 2 ] = (E[Q Q h ]) 2 }{{} discret. error E[Q Q h ] = O(h α ), Var[Q h ] = O(1), cost to compute each Q (i) h : C h = O(h γ ) + Var[Q h] M }{{} MC error Then MSE = O(tol 2 ) = h = O(tol 1 α ), M = O(tol 2 ) Total work: Work(µ MC h ) = C h M tol γ α tol 2 F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

13 Motivating example Can we improve on Monte Carlo? Control variate Let Z be random variable correlated with Q h, and with known mean. Idea: Apply MC on Q h,z = Q h α(z E[Z]) (notice that E[Q h,z ] = E[Q h ]) µ CV h = 1 M M i=1 (Q (i) h αz (i) ) + αe[z] Var[Q h,z ] = Var[Q h αz] = Var[Q h ] + α 2 Var[Z] 2α Cov(Q h, Z) ( ) For optimal α: Var[Q h,z ] = Var[Q h ] 1 Cov(Q h,z) Var[Z] Var[Q h ] (always gives variance reduction) Two ideas for choosing Z Use a surrogate model Z = Q surr with numerically optimized α multi-fidelity Monte Carlo [Peherstorfer, Willcox, Gunzburger, 2016] Use coarser discretization e.g. Z = Q 2h (usually with α = 1) two level Monte Carlo [Heinrich 1998, Giles 2008,...] F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

14 Motivating example Can we improve on Monte Carlo? Control variate Let Z be random variable correlated with Q h, and with known mean. Idea: Apply MC on Q h,z = Q h α(z E[Z]) (notice that E[Q h,z ] = E[Q h ]) µ CV h = 1 M M i=1 (Q (i) h αz (i) ) + αe[z] Var[Q h,z ] = Var[Q h αz] = Var[Q h ] + α 2 Var[Z] 2α Cov(Q h, Z) ( ) For optimal α: Var[Q h,z ] = Var[Q h ] 1 Cov(Q h,z) Var[Z] Var[Q h ] (always gives variance reduction) Two ideas for choosing Z Use a surrogate model Z = Q surr with numerically optimized α multi-fidelity Monte Carlo [Peherstorfer, Willcox, Gunzburger, 2016] Use coarser discretization e.g. Z = Q 2h (usually with α = 1) two level Monte Carlo [Heinrich 1998, Giles 2008,...] F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

15 Motivating example Can we improve on Monte Carlo? Control variate Let Z be random variable correlated with Q h, and with known mean. Idea: Apply MC on Q h,z = Q h α(z E[Z]) (notice that E[Q h,z ] = E[Q h ]) µ CV h = 1 M M i=1 (Q (i) h αz (i) ) + αe[z] Var[Q h,z ] = Var[Q h αz] = Var[Q h ] + α 2 Var[Z] 2α Cov(Q h, Z) ( ) For optimal α: Var[Q h,z ] = Var[Q h ] 1 Cov(Q h,z) Var[Z] Var[Q h ] (always gives variance reduction) Two ideas for choosing Z Use a surrogate model Z = Q surr with numerically optimized α multi-fidelity Monte Carlo [Peherstorfer, Willcox, Gunzburger, 2016] Use coarser discretization e.g. Z = Q 2h (usually with α = 1) two level Monte Carlo [Heinrich 1998, Giles 2008,...] F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

16 Motivating example Can we improve on Monte Carlo? Control variate Problem: E[Z] not known, in general! compute it with independent MC with larger sample size (cheaper problem). From two-level to multilevel: µ CV h = 1 M 1 M 1 M M i=1 M i=1 M i=1 (Q (i) h (Q (i) h (Q (i) h Q (i) 2h ) + E[Q 2h] Q (i) 2h ) + 1 M 2 Q (i) 2h ) + 1 M 2 M 2 i=1 M 2 i=1 Q (i,2) 2h, M 2 > M (Q (i,2) 2h Q (i,2) 4h ) M n M n i=1 Q (i,n) 2 n h F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

17 Motivating example Can we improve on Monte Carlo? Control variate Problem: E[Z] not known, in general! compute it with independent MC with larger sample size (cheaper problem). From two-level to multilevel: µ CV h = 1 M 1 M 1 M M i=1 M i=1 M i=1 (Q (i) h (Q (i) h (Q (i) h Q (i) 2h ) + E[Q 2h] Q (i) 2h ) + 1 M 2 Q (i) 2h ) + 1 M 2 M 2 i=1 M 2 i=1 Q (i,2) 2h, M 2 > M (Q (i,2) 2h Q (i,2) 4h ) M n M n i=1 Q (i,n) 2 n h F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

18 Motivating example Can we improve on Monte Carlo? Control variate Problem: E[Z] not known, in general! compute it with independent MC with larger sample size (cheaper problem). From two-level to multilevel: µ CV h = 1 M 1 M 1 M M i=1 M i=1 M i=1 (Q (i) h (Q (i) h (Q (i) h Q (i) 2h ) + E[Q 2h] Q (i) 2h ) + 1 M 2 Q (i) 2h ) + 1 M 2 M 2 i=1 M 2 i=1 Q (i,2) 2h, M 2 > M (Q (i,2) 2h Q (i,2) 4h ) M n M n i=1 Q (i,n) 2 n h F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

19 Motivating example Can we improve on Monte Carlo? Control variate Problem: E[Z] not known, in general! compute it with independent MC with larger sample size (cheaper problem). From two-level to multilevel: µ CV h = 1 M 1 M 1 M M i=1 M i=1 M i=1 (Q (i) h (Q (i) h (Q (i) h Q (i) 2h ) + E[Q 2h] Q (i) 2h ) + 1 M 2 Q (i) 2h ) + 1 M 2 M 2 i=1 M 2 i=1 Q (i,2) 2h, M 2 > M (Q (i,2) 2h Q (i,2) 4h ) M n M n i=1 Q (i,n) 2 n h F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

20 Outline Multilevel Monte Carlo for expectations 1 Motivating example 2 Multilevel Monte Carlo for expectations 3 MLMC for moments and distributions 4 Risk averse optimization with MLMC 5 Conclusions F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

21 Multilevel Monte Carlo for expectations Multilevel Monte Carlo l = 0 l = L Denoting Q l = Q hl, the MLMC estimator is L µ MLMC 1 M l L = M l l=0 i=1 Mean squared error (Q (i,l) l Q (i,l) l 1 ), Q 1 = 0 MSE(µ MLMC L ) = (E[Q Q L ]) 2 }{{} discret. error level L Sequence of refined discretizations h 0 > h 1 >... > h L Sequence of sample sizes + M 0 > M 1 > > M L L Var[Q l Q l 1 ] l=0 } M l {{ } statistical error F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

22 Multilevel Monte Carlo for expectations Multilevel Monte Carlo l = 0 l = L Sequence of refined discretizations h 0 > h 1 >... > h L Sequence of sample sizes Denoting Q l = Q hl, the MLMC estimator is µ MLMC L = 1 M 0 Q (i,0) M 1 (Q (i,1) 1 Q (i,1) 0 ) M 0 M 1 M L i=1 Mean squared error i=1 MSE(µ MLMC L ) = (E[Q Q L ]) 2 }{{} discret. error level L + M 0 > M 1 > > M L M L i=1 L Var[Q l Q l 1 ] l=0 } M l {{ } statistical error (Q (i,l) L Q (i,l) L 1 ) F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

23 Multilevel Monte Carlo for expectations Multilevel Monte Carlo l = 0 l = L Denoting Q l = Q hl, the MLMC estimator is L µ MLMC 1 M l L = M l l=0 i=1 Mean squared error (Q (i,l) l Q (i,l) l 1 ), Q 1 = 0 MSE(µ MLMC L ) = (E[Q Q L ]) 2 }{{} discret. error level L Sequence of refined discretizations h 0 > h 1 >... > h L Sequence of sample sizes + M 0 > M 1 > > M L L Var[Q l Q l 1 ] l=0 } M l {{ } statistical error F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

24 Multilevel Monte Carlo for expectations Multilevel Monte Carlo l = 0 l = L Denoting Q l = Q hl, the MLMC estimator is L µ MLMC 1 M l L = M l l=0 i=1 Mean squared error (Q (i,l) l Q (i,l) l 1 ), Q 1 = 0 MSE(µ MLMC L ) = (E[Q Q L ]) 2 }{{} discret. error level L Sequence of refined discretizations h 0 > h 1 >... > h L Sequence of sample sizes + M 0 > M 1 > > M L L Var[Q l Q l 1 ] l=0 } M l {{ } statistical error F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

25 Multilevel Monte Carlo for expectations Multilevel Monte Carlo V l = Var[Q l Q l 1 ] (variance of differences) C l = cost of computing each Q (i,l) l = Q (i,l) l Q (i,l) l 1 Optimal sample sizes M l : [Giles 2008] minimize W = L l=0 C lm l s.t. MSE tol 2 ( M l = tol 2 V L l ) C l k=0 Ck V k Complexity analysis for h l = h 0 s l : [Giles 2008, Cliffe-Giles-Scheichl-Teckentrup 2011] Assume E[Q Q l ] = O(h α l ), V l = Var[Q l Q l 1 ] = O(h β l ), (β = 2α for smooth problems/noise) C l = O(h γ l ), 2α min{β, γ} Then, choosing L = O(tol 1 α ) and M l as above gives MSE(µ MLMC L ) tol 2 and L tol 2, β > γ Work(µ MLMC L ) = C l M l tol l=0 2 (log tol) 2, β = γ γ β 2 tol α, β < γ F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

26 Multilevel Monte Carlo for expectations Multilevel Monte Carlo V l = Var[Q l Q l 1 ] (variance of differences) C l = cost of computing each Q (i,l) l = Q (i,l) l Q (i,l) l 1 Optimal sample sizes M l : [Giles 2008] minimize W = L l=0 C lm l s.t. MSE tol 2 ( M l = tol 2 V L l ) C l k=0 Ck V k Complexity analysis for h l = h 0 s l : [Giles 2008, Cliffe-Giles-Scheichl-Teckentrup 2011] Assume E[Q Q l ] = O(h α l ), V l = Var[Q l Q l 1 ] = O(h β l ), (β = 2α for smooth problems/noise) C l = O(h γ l ), 2α min{β, γ} Then, choosing L = O(tol 1 α ) and M l as above gives MSE(µ MLMC L ) tol 2 and L tol 2, β > γ Work(µ MLMC L ) = C l M l tol l=0 2 (log tol) 2, β = γ γ β 2 tol α, β < γ F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

27 Multilevel Monte Carlo for expectations Multilevel Monte Carlo V l = Var[Q l Q l 1 ] (variance of differences) C l = cost of computing each Q (i,l) l = Q (i,l) l Q (i,l) l 1 Optimal sample sizes M l : [Giles 2008] minimize W = L l=0 C lm l s.t. MSE tol 2 ( M l = tol 2 V L l ) C l k=0 Ck V k Complexity analysis for h l = h 0 s l : [Giles 2008, Cliffe-Giles-Scheichl-Teckentrup 2011] Assume E[Q Q l ] = O(h α l ), V l = Var[Q l Q l 1 ] = O(h β l ), (β = 2α for smooth problems/noise) C l = O(h γ l ), 2α min{β, γ} Then, choosing L = O(tol 1 α ) and M l as above gives MSE(µ MLMC L ) tol 2 and L tol 2, β > γ Work(µ MLMC L ) = C l M l tol l=0 2 (log tol) 2, β = γ γ β 2 tol α, β < γ F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

28 Multilevel Monte Carlo for expectations Multilevel Monte Carlo practical aspects Remark: MC complexity always improved for optimal choice of M l. For β = 2α we get either O(tol 2 ) (up to log terms) or O(tol γ α ). To achieve improved complexity, one needs to estimate error decay E[Q Q l ] : needed to determine optimal L estimate variance decay V l : needed to determine optimal {M l } L l=0 E[Q Q l ] can be estimated as µ MC l µ MC l 1 based on a pilot run V l can be estimated by sample variance estimator based on pilot runs Problem: on the finest levels we should run only very few simulations. Cost for estimation of V L might dominate the overall cost of the MLMC algorithm. Idea: use adaptive algorithms: extrapolate information from previous levels and correct it when samples become available. F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

29 Multilevel Monte Carlo for expectations Multilevel Monte Carlo practical aspects Remark: MC complexity always improved for optimal choice of M l. For β = 2α we get either O(tol 2 ) (up to log terms) or O(tol γ α ). To achieve improved complexity, one needs to estimate error decay E[Q Q l ] : needed to determine optimal L estimate variance decay V l : needed to determine optimal {M l } L l=0 E[Q Q l ] can be estimated as µ MC l µ MC l 1 based on a pilot run V l can be estimated by sample variance estimator based on pilot runs Problem: on the finest levels we should run only very few simulations. Cost for estimation of V L might dominate the overall cost of the MLMC algorithm. Idea: use adaptive algorithms: extrapolate information from previous levels and correct it when samples become available. F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

30 Multilevel Monte Carlo for expectations Multilevel Monte Carlo practical aspects Remark: MC complexity always improved for optimal choice of M l. For β = 2α we get either O(tol 2 ) (up to log terms) or O(tol γ α ). To achieve improved complexity, one needs to estimate error decay E[Q Q l ] : needed to determine optimal L estimate variance decay V l : needed to determine optimal {M l } L l=0 E[Q Q l ] can be estimated as µ MC l µ MC l 1 based on a pilot run V l can be estimated by sample variance estimator based on pilot runs Problem: on the finest levels we should run only very few simulations. Cost for estimation of V L might dominate the overall cost of the MLMC algorithm. Idea: use adaptive algorithms: extrapolate information from previous levels and correct it when samples become available. F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

31 Multilevel Monte Carlo for expectations Multilevel Monte Carlo practical aspects Remark: MC complexity always improved for optimal choice of M l. For β = 2α we get either O(tol 2 ) (up to log terms) or O(tol γ α ). To achieve improved complexity, one needs to estimate error decay E[Q Q l ] : needed to determine optimal L estimate variance decay V l : needed to determine optimal {M l } L l=0 E[Q Q l ] can be estimated as µ MC l µ MC l 1 based on a pilot run V l can be estimated by sample variance estimator based on pilot runs Problem: on the finest levels we should run only very few simulations. Cost for estimation of V L might dominate the overall cost of the MLMC algorithm. Idea: use adaptive algorithms: extrapolate information from previous levels and correct it when samples become available. F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

32 Multilevel Monte Carlo for expectations Multilevel Monte Carlo practical aspects Remark: MC complexity always improved for optimal choice of M l. For β = 2α we get either O(tol 2 ) (up to log terms) or O(tol γ α ). To achieve improved complexity, one needs to estimate error decay E[Q Q l ] : needed to determine optimal L estimate variance decay V l : needed to determine optimal {M l } L l=0 E[Q Q l ] can be estimated as µ MC l µ MC l 1 based on a pilot run V l can be estimated by sample variance estimator based on pilot runs Problem: on the finest levels we should run only very few simulations. Cost for estimation of V L might dominate the overall cost of the MLMC algorithm. Idea: use adaptive algorithms: extrapolate information from previous levels and correct it when samples become available. F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

33 Multilevel Monte Carlo for expectations Continuation Multilevel Monte Carlo [Collier-HajiAli-N.-vonSchwerin-Tempone 2015, Pisaroni-N.-Leyland 2017] Idea: Solve the problem with decreasing tolerances tol (0) > tol (1) >... tol. Use collected samples on all levels to improve the estimate of V l and E[Q Q l ]. Estimator ˆV l of V l = Var[ Q l ] at iteration j: MAP Bayesian estimator we make the ansatz Q l N(µ l, V l ) based on acquired samples at previous iteration, we fit models (least squares) = c αhl α Vl model = c β h β l We take a Normal-Gamma prior for (µ l, V l ), with mode in (µ model l µ model l, V model l ) Then ˆV l is the MAP Bayesian estimator based on the Normal-Gamma prior and the actual samples acquired at iteration j Effectively, we have M l = 0 M l ˆVl = V model l ˆVl V MC l (prior model) (sample variance) ˆV l is then used to determine the sample sizes M l for the next iteration. F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

34 Multilevel Monte Carlo for expectations Continuation Multilevel Monte Carlo [Collier-HajiAli-N.-vonSchwerin-Tempone 2015, Pisaroni-N.-Leyland 2017] Idea: Solve the problem with decreasing tolerances tol (0) > tol (1) >... tol. Use collected samples on all levels to improve the estimate of V l and E[Q Q l ]. Estimator ˆV l of V l = Var[ Q l ] at iteration j: MAP Bayesian estimator we make the ansatz Q l N(µ l, V l ) based on acquired samples at previous iteration, we fit models (least squares) = c αhl α Vl model = c β h β l We take a Normal-Gamma prior for (µ l, V l ), with mode in (µ model l µ model l, V model l ) Then ˆV l is the MAP Bayesian estimator based on the Normal-Gamma prior and the actual samples acquired at iteration j Effectively, we have M l = 0 M l ˆVl = V model l ˆVl V MC l (prior model) (sample variance) ˆV l is then used to determine the sample sizes M l for the next iteration. F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

35 Multilevel Monte Carlo for expectations Computation of pressure coefficient for NACA 0012 / NASA SC(2)-0012 airfoils Operational Geometrical y Name Nominal value Uncertainty T T n = [K] T N (T n, 2%, 110%, 90%) p p n = [N/m 2 ] T N (p n, 2%, 110%, 90%) α α n = 1.25 T N (α n, 1%, 110%, 90%) M M n = 0.8 T N (M n, 2%, 110%, 90%) R p T N (R Pn, 2.5%, 110%, 90%) R S T N (R Sn, 2.5%, 110%, 90%) X P T N (X Pn, 2.5%, 110%, 90%) X S T N (X Sn, 2.5%, 110%, 90%) Y P T N (Y Pn, 2.5%, 110%, 90%) Y S T N (Y Sn, 2.5%, 110%, 90%) C P T N (C Pn, 2.5%, 110%, 90%) C S T N (C Sn, 2.5%, 110%, 90%) θ P RAE-2822 T N (θ Pn, 2.5%, 110%, 90%) θ S T N (θ Sn, 2.5%, 110%, 90%) C s R s x p y s θ s x α R p y p x s θ p M C p F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

36 y y y Multilevel Monte Carlo for expectations Computation of pressure coefficient for NACA 0012 / NASA SC(2)-0012 airfoils x x x Inviscid model (Euler); SU2 solver (Stanford) [Pisaroni-Leyland-N., AIAA Aviation, 2016] F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

37 Multilevel Monte Carlo for expectations MLMC vs MC for aerodynamic inviscid problems C[h] ncp U Computational Complexity of MC and MLMC ε 2 (γ/α) ε 2 (γ β)/α MC (OPER) MLMC (OPER) MC (OPER+GEOM) MLMC (OPER+GEOM) Deterministic Nlevel Levels and Samples per Level for εr = ε level F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

38 Multilevel Monte Carlo for expectations Robustness of C-MLMC estimator Variability over 10 repetitions of the C-MLMC algorithm for different parameters in the Normal-Gamma prior. F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

39 Outline MLMC for moments and distributions 1 Motivating example 2 Multilevel Monte Carlo for expectations 3 MLMC for moments and distributions 4 Risk averse optimization with MLMC 5 Conclusions F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

40 MLMC for moments and distributions Beyond expectations: computation of central moments Goal: compute µ p (Q) = E[(Q E[Q]) p ] How to apply and tune MLMC in this case? [Bierig-Chernov ] use biased central moments estimators. Alternatively, use h-statistics [Pisaroni-Krumscheid-N. 2017]. Given iid sample Q M = {Q (1),..., Q (M) }, h p ( Q M ) : unbiased estimator of µ p (Q) with minimal variance Multilevel estimator: h MLMC p = L (h p ( Q l,ml ) h p ( Q l 1,Ml )) with ( Q l,ml, Q l 1,Ml ) generated with the same noise (highly correlated) L Mean squared error: MSE(hp MLMC ) = (µ p (Q) µ p (Q L )) 2 V l,p + M l l=0 where V l,p = M l Var[h p ( Q l,ml ) h p ( Q l 1,Ml )]. Same formal structure as for expectation, but now we need to estimate µ p (Q) µ p (Q l ) and V l,p to tune the MLMC algorithm F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop l=0

41 MLMC for moments and distributions Beyond expectations: computation of central moments Goal: compute µ p (Q) = E[(Q E[Q]) p ] How to apply and tune MLMC in this case? [Bierig-Chernov ] use biased central moments estimators. Alternatively, use h-statistics [Pisaroni-Krumscheid-N. 2017]. Given iid sample Q M = {Q (1),..., Q (M) }, h p ( Q M ) : unbiased estimator of µ p (Q) with minimal variance Multilevel estimator: h MLMC p = L (h p ( Q l,ml ) h p ( Q l 1,Ml )) with ( Q l,ml, Q l 1,Ml ) generated with the same noise (highly correlated) L Mean squared error: MSE(hp MLMC ) = (µ p (Q) µ p (Q L )) 2 V l,p + M l l=0 where V l,p = M l Var[h p ( Q l,ml ) h p ( Q l 1,Ml )]. Same formal structure as for expectation, but now we need to estimate µ p (Q) µ p (Q l ) and V l,p to tune the MLMC algorithm F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop l=0

42 MLMC for moments and distributions Beyond expectations: computation of central moments Goal: compute µ p (Q) = E[(Q E[Q]) p ] How to apply and tune MLMC in this case? [Bierig-Chernov ] use biased central moments estimators. Alternatively, use h-statistics [Pisaroni-Krumscheid-N. 2017]. Given iid sample Q M = {Q (1),..., Q (M) }, h p ( Q M ) : unbiased estimator of µ p (Q) with minimal variance Multilevel estimator: h MLMC p = L (h p ( Q l,ml ) h p ( Q l 1,Ml )) with ( Q l,ml, Q l 1,Ml ) generated with the same noise (highly correlated) L Mean squared error: MSE(hp MLMC ) = (µ p (Q) µ p (Q L )) 2 V l,p + M l l=0 where V l,p = M l Var[h p ( Q l,ml ) h p ( Q l 1,Ml )]. Same formal structure as for expectation, but now we need to estimate µ p (Q) µ p (Q l ) and V l,p to tune the MLMC algorithm F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop l=0

43 MLMC for moments and distributions Beyond expectations: computation of central moments Goal: compute µ p (Q) = E[(Q E[Q]) p ] How to apply and tune MLMC in this case? [Bierig-Chernov ] use biased central moments estimators. Alternatively, use h-statistics [Pisaroni-Krumscheid-N. 2017]. Given iid sample Q M = {Q (1),..., Q (M) }, h p ( Q M ) : unbiased estimator of µ p (Q) with minimal variance Multilevel estimator: h MLMC p = L (h p ( Q l,ml ) h p ( Q l 1,Ml )) with ( Q l,ml, Q l 1,Ml ) generated with the same noise (highly correlated) L Mean squared error: MSE(hp MLMC ) = (µ p (Q) µ p (Q L )) 2 V l,p + M l l=0 where V l,p = M l Var[h p ( Q l,ml ) h p ( Q l 1,Ml )]. Same formal structure as for expectation, but now we need to estimate µ p (Q) µ p (Q l ) and V l,p to tune the MLMC algorithm F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop l=0

44 MLMC for moments and distributions Beyond expectations: computation of central moments Goal: compute µ p (Q) = E[(Q E[Q]) p ] How to apply and tune MLMC in this case? [Bierig-Chernov ] use biased central moments estimators. Alternatively, use h-statistics [Pisaroni-Krumscheid-N. 2017]. Given iid sample Q M = {Q (1),..., Q (M) }, h p ( Q M ) : unbiased estimator of µ p (Q) with minimal variance Multilevel estimator: h MLMC p = L (h p ( Q l,ml ) h p ( Q l 1,Ml )) with ( Q l,ml, Q l 1,Ml ) generated with the same noise (highly correlated) L Mean squared error: MSE(hp MLMC ) = (µ p (Q) µ p (Q L )) 2 V l,p + M l l=0 where V l,p = M l Var[h p ( Q l,ml ) h p ( Q l 1,Ml )]. Same formal structure as for expectation, but now we need to estimate µ p (Q) µ p (Q l ) and V l,p to tune the MLMC algorithm F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop l=0

45 MLMC for moments and distributions Beyond expectations: computation of central moments Complexity result for h l = h 0 s l Assume µ 2p (Q l ) < for all l and there exist α, β, γ > 0, 2α min{β, γ} s.t. µ p (Q) µ p (Q l ) = O(h α l ), V l,p = O(h β l ), C l = Cost(Q (i,l) l, Q (i,l) l 1 ) = O(h γ ), Then, taking L = O(tol 1 α ) and M l = tol 2 Vl,p C l ( L k=0 Ck V k,p ) leads to tol 2, β > γ MSE(hp MLMC ) tol 2 and W (hp MLMC ) tol 2 log(tol) 2, β = γ γ β 2 tol α, β < γ F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

46 MLMC for moments and distributions Beyond expectations: computation of central moments Technical difficulty: how to estimate the variances V l,p Define X + l,m l = Q l,ml + Q l 1,Ml, X l,m l = Q l,ml Q l 1,Ml l h p = h p ( Q l,ml ) h p ( Q l 1,Ml ) can be expressed as a power sum l h p = a+b p S a,b ( X + l,m l, X l,m l ), S a,b ( X, Y ) = i (X (i) ) a (Y (i) ) b Unbiased estimators ˆV l,p of V l,p can be computed in closed form starting from the power terms S a,b ( X + l,m l, X l,m l ) [Pisaroni-Krumscheid-N. 2017]. F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

47 MLMC for moments and distributions Beyond expectations: computation of central moments Technical difficulty: how to estimate the variances V l,p Define X + l,m l = Q l,ml + Q l 1,Ml, X l,m l = Q l,ml Q l 1,Ml l h p = h p ( Q l,ml ) h p ( Q l 1,Ml ) can be expressed as a power sum l h p = a+b p S a,b ( X + l,m l, X l,m l ), S a,b ( X, Y ) = i (X (i) ) a (Y (i) ) b Unbiased estimators ˆV l,p of V l,p can be computed in closed form starting from the power terms S a,b ( X + l,m l, X l,m l ) [Pisaroni-Krumscheid-N. 2017]. F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

48 MLMC for moments and distributions Beyond expectations: computation of central moments Technical difficulty: how to estimate the variances V l,p Define X + l,m l = Q l,ml + Q l 1,Ml, X l,m l = Q l,ml Q l 1,Ml l h p = h p ( Q l,ml ) h p ( Q l 1,Ml ) can be expressed as a power sum l h p = a+b p S a,b ( X + l,m l, X l,m l ), S a,b ( X, Y ) = i (X (i) ) a (Y (i) ) b Unbiased estimators ˆV l,p of V l,p can be computed in closed form starting from the power terms S a,b ( X + l,m l, X l,m l ) [Pisaroni-Krumscheid-N. 2017]. F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

49 MLMC for moments and distributions Beyond expectations: char. function, CDF, and more Some derived quantities can be written as parametric expectations Example 1: Characteristic function of Q Φ(θ) = E[φ(θ, Q)], φ(θ, Q) = e iθq we can compute Φ(θ j ) by MLMC on a set of points θ j. Example 2: CDF of Q F (θ) = E[φ(θ, Q)], φ(θ, Q) = 1 {Q θ} Problem: φ(θ, Q) is not smooth! When applying MLMC, the variance of the differences, V l = Var[φ(θ, Q l ) φ(θ, Q l 1 )] will decay slowly. No much gain in MLMC. Remedies: [Giles-Nagapetyan-Ritter 2015] smoothing: F ɛ (θ) = E[φ ɛ (θ, Q)]. Technical difficulty: ɛ should depend on the required tolerance difficult tuning of MLMC [Bierig-Chernov 2016] approximate F or pdf based on moments (see Alexey s talk) [Krumscheid-N. 2017] anti-derivative approach: F (θ) = Φ (θ) with Φ(θ) = E[φ(θ, Q)] and φ(θ, ) Lipschitz continuous. F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

50 MLMC for moments and distributions Beyond expectations: char. function, CDF, and more Some derived quantities can be written as parametric expectations Example 1: Characteristic function of Q Φ(θ) = E[φ(θ, Q)], φ(θ, Q) = e iθq we can compute Φ(θ j ) by MLMC on a set of points θ j. Example 2: CDF of Q F (θ) = E[φ(θ, Q)], φ(θ, Q) = 1 {Q θ} Problem: φ(θ, Q) is not smooth! When applying MLMC, the variance of the differences, V l = Var[φ(θ, Q l ) φ(θ, Q l 1 )] will decay slowly. No much gain in MLMC. Remedies: [Giles-Nagapetyan-Ritter 2015] smoothing: F ɛ (θ) = E[φ ɛ (θ, Q)]. Technical difficulty: ɛ should depend on the required tolerance difficult tuning of MLMC [Bierig-Chernov 2016] approximate F or pdf based on moments (see Alexey s talk) [Krumscheid-N. 2017] anti-derivative approach: F (θ) = Φ (θ) with Φ(θ) = E[φ(θ, Q)] and φ(θ, ) Lipschitz continuous. F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

51 MLMC for moments and distributions Beyond expectations: char. function, CDF, and more Some derived quantities can be written as parametric expectations Example 1: Characteristic function of Q Φ(θ) = E[φ(θ, Q)], φ(θ, Q) = e iθq we can compute Φ(θ j ) by MLMC on a set of points θ j. Example 2: CDF of Q F (θ) = E[φ(θ, Q)], φ(θ, Q) = 1 {Q θ} Problem: φ(θ, Q) is not smooth! When applying MLMC, the variance of the differences, V l = Var[φ(θ, Q l ) φ(θ, Q l 1 )] will decay slowly. No much gain in MLMC. Remedies: [Giles-Nagapetyan-Ritter 2015] smoothing: F ɛ (θ) = E[φ ɛ (θ, Q)]. Technical difficulty: ɛ should depend on the required tolerance difficult tuning of MLMC [Bierig-Chernov 2016] approximate F or pdf based on moments (see Alexey s talk) [Krumscheid-N. 2017] anti-derivative approach: F (θ) = Φ (θ) with Φ(θ) = E[φ(θ, Q)] and φ(θ, ) Lipschitz continuous. F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

52 MLMC for moments and distributions Beyond expectations: char. function, CDF, and more Some derived quantities can be written as parametric expectations Example 1: Characteristic function of Q Φ(θ) = E[φ(θ, Q)], φ(θ, Q) = e iθq we can compute Φ(θ j ) by MLMC on a set of points θ j. Example 2: CDF of Q F (θ) = E[φ(θ, Q)], φ(θ, Q) = 1 {Q θ} Problem: φ(θ, Q) is not smooth! When applying MLMC, the variance of the differences, V l = Var[φ(θ, Q l ) φ(θ, Q l 1 )] will decay slowly. No much gain in MLMC. Remedies: [Giles-Nagapetyan-Ritter 2015] smoothing: F ɛ (θ) = E[φ ɛ (θ, Q)]. Technical difficulty: ɛ should depend on the required tolerance difficult tuning of MLMC [Bierig-Chernov 2016] approximate F or pdf based on moments (see Alexey s talk) [Krumscheid-N. 2017] anti-derivative approach: F (θ) = Φ (θ) with Φ(θ) = E[φ(θ, Q)] and φ(θ, ) Lipschitz continuous. F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

53 MLMC for moments and distributions Anti-derivative approach to CDF computation For any τ (0, 1) define Φ τ (θ) = E[φ τ (θ, Q)], φ τ (θ, Q) = θ τ (Q θ) + Then F (θ) = (1 τ)φ τ (θ) + τ and MLMC can be effectively used to approximate Φ τ (θ) and its derivatives. Moreover, from the approximation of Φ τ and its derivatives we can get for free pdf: p(θ) = F (θ) = (1 τ)φ τ (θ) τ-quantile: q τ = inf{θ : F (θ) τ} = argmin θ R Φ τ (θ) Conditional Value at Risk CVaR τ = 1 1 τ q τ xdf (x) = min θ R Φ τ (θ) F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

54 MLMC for moments and distributions Anti-derivative approach to CDF computation For any τ (0, 1) define Φ τ (θ) = E[φ τ (θ, Q)], φ τ (θ, Q) = θ τ (Q θ) + Then F (θ) = (1 τ)φ τ (θ) + τ and MLMC can be effectively used to approximate Φ τ (θ) and its derivatives. Moreover, from the approximation of Φ τ and its derivatives we can get for free pdf: p(θ) = F (θ) = (1 τ)φ τ (θ) τ-quantile: q τ = inf{θ : F (θ) τ} = argmin θ R Φ τ (θ) Conditional Value at Risk CVaR τ = 1 1 τ q τ xdf (x) = min θ R Φ τ (θ) F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

55 MLMC for moments and distributions Computing parametric expectations by MLMC Goal: given φ(θ, Q), approximate Φ(θ) = E[φ(θ, Q)] and its derivatives uniformly in Θ. Interpolation approach: x ˆΦ Φ introduce a grid ξ = {ξ 1,..., ξ n } Θ compute Φ MLMC L (ξ j ), j = 1,..., n by MLMC (same sample of Q l for every ξ j ) Interpolate values Φ MLMC L Assumptions on I n (valid for spline interpolation) f I n (f ( ξ)) L (Θ) c 1 n k+1, I n x L (Θ) c 2 x l, Cost(I n ( x)) c 3 n x R n ( ξ) = {Φ MLMC L (ξ j )} n j=1 ˆΦ L = I n (Φ MLMC L ( ξ)) e.g. by spline or polynomial interpolation if f C k+1 ( Θ) F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

56 MLMC for moments and distributions Computing parametric expectations by MLMC Goal: given φ(θ, Q), approximate Φ(θ) = E[φ(θ, Q)] and its derivatives uniformly in Θ. Interpolation approach: x ˆΦ Φ introduce a grid ξ = {ξ 1,..., ξ n } Θ compute Φ MLMC L (ξ j ), j = 1,..., n by MLMC (same sample of Q l for every ξ j ) Interpolate values Φ MLMC L Assumptions on I n (valid for spline interpolation) f I n (f ( ξ)) L (Θ) c 1 n k+1, I n x L (Θ) c 2 x l, Cost(I n ( x)) c 3 n x R n ( ξ) = {Φ MLMC L (ξ j )} n j=1 ˆΦ L = I n (Φ MLMC L ( ξ)) e.g. by spline or polynomial interpolation if f C k+1 ( Θ) F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

57 MLMC for moments and distributions Computing parametric expectations by MLMC Goal: given φ(θ, Q), approximate Φ(θ) = E[φ(θ, Q)] and its derivatives uniformly in Θ. Interpolation approach: x ˆΦ Φ introduce a grid ξ = {ξ 1,..., ξ n } Θ compute Φ MLMC L (ξ j ), j = 1,..., n by MLMC (same sample of Q l for every ξ j ) Interpolate values Φ MLMC L Assumptions on I n (valid for spline interpolation) f I n (f ( ξ)) L (Θ) c 1 n k+1, I n x L (Θ) c 2 x l, Cost(I n ( x)) c 3 n x R n ( ξ) = {Φ MLMC L (ξ j )} n j=1 ˆΦ L = I n (Φ MLMC L ( ξ)) e.g. by spline or polynomial interpolation if f C k+1 ( Θ) F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

58 Error splitting MLMC for moments and distributions Define the mean squared error: MSE(ˆΦ L ) = E[ Φ ˆΦ L 2 L (Θ) ] Notation: for x R n define Var[ x] = E[ x E[ x] 2 l ] Useful result: for x (1),..., x (k) R n independent, k k Var[ x (i) ] c log(n) Var[ x (i) ] Error splitting i=1 i=1 MSE(ˆΦ L ) 3 Φ I n Φ I n Φ I n Φ L 2 + 3E[ I n Φ L I n Φ MLMC L 2 ] Φ I n Φ( ξ) 2 + Φ( ξ) }{{} Φ L ( L ξ) 2 V l + log(n) }{{} M l interp. error discret. error l=0 }{{} statistical error with V l = Var[φ( ξ, Q l ) φ( ξ, Q l 1 )]. All terms can be estimated in practice. Optimization of MLMC based on estimators ˆV l. [Pisaroni-Krumscheid-N. in preparation] F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

59 Error splitting MLMC for moments and distributions Define the mean squared error: MSE(ˆΦ L ) = E[ Φ ˆΦ L 2 L (Θ) ] Notation: for x R n define Var[ x] = E[ x E[ x] 2 l ] Useful result: for x (1),..., x (k) R n independent, k k Var[ x (i) ] c log(n) Var[ x (i) ] Error splitting i=1 i=1 MSE(ˆΦ L ) 3 Φ I n Φ I n Φ I n Φ L 2 + 3E[ I n Φ L I n Φ MLMC L 2 ] Φ I n Φ( ξ) 2 + Φ( ξ) }{{} Φ L ( L ξ) 2 V l + log(n) }{{} M l interp. error discret. error l=0 }{{} statistical error with V l = Var[φ( ξ, Q l ) φ( ξ, Q l 1 )]. All terms can be estimated in practice. Optimization of MLMC based on estimators ˆV l. [Pisaroni-Krumscheid-N. in preparation] F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

60 Complexity analysis MLMC for moments and distributions Complexity result for h l = h 0 s l [Krumscheid-N. 2017] Assume Φ Φ l L (Θ) c 1 h α l, ( ) E φ(, Q l ) φ(, Q l 1 ) 2 L (Θ) c 2 h β l, cost to simulate one realization of φ(θ, Q l ) c 3 h γ l. If Φ C k+1 (Θ), there exists an estimator ˆΦ L s.t. MSE (ˆΦ L ) = O(tol 2 ) and tol 2, if β > γ, W (ˆΦ L ) tol (2+ k+1) 1 log(tol) + log(tol) tol 2 log(tol) 2, if β = γ, γ β (2+ tol α ), if β < γ, The first term accounts for the cost of computing the spline interpolation. This is often negligible for heavy computational models. It can be removed by taking n = n l (different spline interpolant on each level). Neglecting the first term, the complexity is essentially the same as for simple expectations, up to an extra log factor. F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

61 Complexity analysis MLMC for moments and distributions Complexity result for h l = h 0 s l [Krumscheid-N. 2017] Assume Φ Φ l L (Θ) c 1 h α l, ( ) E φ(, Q l ) φ(, Q l 1 ) 2 L (Θ) c 2 h β l, cost to simulate one realization of φ(θ, Q l ) c 3 h γ l. If Φ C k+1 (Θ), there exists an estimator ˆΦ L s.t. MSE (ˆΦ L ) = O(tol 2 ) and tol 2, if β > γ, W (ˆΦ L ) tol (2+ k+1) 1 log(tol) + log(tol) tol 2 log(tol) 2, if β = γ, γ β (2+ tol α ), if β < γ, The first term accounts for the cost of computing the spline interpolation. This is often negligible for heavy computational models. It can be removed by taking n = n l (different spline interpolant on each level). Neglecting the first term, the complexity is essentially the same as for simple expectations, up to an extra log factor. F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

62 MLMC for moments and distributions Complexity result for derivatives [Krumscheid-N. 2017] If Φ C 2k+2 (Θ) and m 2k + 1, there exists an estimator ˆΦ L s.t. E[ d m dθ Φ d m ˆΦ m dθ m L 2 ] = O(tol 2 ) and 2k+2 2 tol 2k+2 m, if β > γ, 2k+2 2 W (ˆΦ L ) log(tol) tol 2k+2 m log(tol) 2, if β = γ, γ β (2+ tol α ) 2k+2 2k+2 m, if β < γ, (neglecting the cost of interpolation) This result applies to the approximation of CDF, quantiles and CVaR with m = 1 and PDF with m = 2. F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

63 MLMC for moments and distributions An example: the characteristic function An SDE model to describe a European call option, where the asset follows ds = rs dt + σs dw, S(0) = S 0, Quantity of interest is the discounted payoff : Q := e rt max ( S(T ) K, 0 ) Approximate characteristic function of Q: Φ(θ) = E ( cos(θq) ) + i E ( sin(θq) ) Φ 1 (θ) + i Φ 2 (θ), Milstein scheme with h l = 2 l T ; Θ = [ 1, 1], r = 1 20, σ = 1 5, T = 1, K = 10 = S 0. F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

64 MLMC for moments and distributions An example: the characteristic function An SDE model to describe a European call option, where the asset follows ds = rs dt + σs dw, S(0) = S 0, Quantity of interest is the discounted payoff : Q := e rt max ( S(T ) K, 0 ) Approximate characteristic function of Q: Φ(θ) = E ( cos(θq) ) + i E ( sin(θq) ) Φ 1 (θ) + i Φ 2 (θ), Milstein scheme with h l = 2 l T ; Θ = [ 1, 1], r = 1 20, σ = 1 5, T = 1, K = 10 = S ( ) k = 0 k = 1 k = 2 k = 3 k = 5 k = 7 O (ε 3 ln ( ε 1)) O (ε ˆΦ1 2 ln ( ε 1)) ε = 0.1, k = ˆΦ ϑ F. Nobile (EPFL) MLMC for UQ ε GAMM AGUQ workshop

MLMC for moments and distributions NASA Common Research Model NASA CRM: aircraft configuration equipped with a contemporary supercritical transonic wing and a fuselage that is representative of a

65 MLMC for moments and distributions NASA Common Research Model NASA CRM: aircraft configuration equipped with a contemporary supercritical transonic wing and a fuselage that is representative of a wide-body commercial transport aircraft. Q.ty Reference Uncertainty M 0.85 B(2, 2, 0.05, M 0.025) Re c T ref [K] B(2, 2, 30, T ref 15) C L 0.3, 0.4, 0.5, 0.55 LEVEL Cells y+ CTime on 280 CPUs L [s] (0.11 [h]) L [s] (0.23 [h]) L [s] (0.35 [h]) L [s] (0.89 [h]) Spalart-Allmaras turbulence model, hybrid unstructured grids. F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

66 MLMC for moments and distributions NASA Common Research Model F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

67 MLMC for moments and distributions NASA Common Research Model F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

68 MLMC for moments and distributions NASA Common Research Model C L = 0.55 C L = 0.5 C L = 0.4 C L = CD T C M CD V CD I F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

69 Outline Risk averse optimization with MLMC 1 Motivating example 2 Multilevel Monte Carlo for expectations 3 MLMC for moments and distributions 4 Risk averse optimization with MLMC 5 Conclusions F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

70 Risk averse optimization with MLMC Risk averse optimization min R(Q(x)), x X X : feasible design space R: risk measure Examples R(Q) = E[Q] (mean-based risk) R(Q) = E[Q] ± α std[q] R(Q) = q α [Q] (α-quantile) R(Q) = CVaR α [Q] F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

71 Risk averse optimization with MLMC Risk averse optimization min R(Q(x)), x X X : feasible design space R: risk measure Examples R(Q) = E[Q] (mean-based risk) R(Q) = E[Q] ± α std[q] R(Q) = q α [Q] (α-quantile) R(Q) = CVaR α [Q] F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

72 Risk averse optimization with MLMC Combining MLMC with CMA-ES Optimization done by Covariance Matrix Adaptation Evolutionary Algorithm (CMA-ES) For each individual at each generation, risk measure computed by MLMC. F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

73 Risk averse optimization with MLMC Airfoil optimization under operating uncertainties { min R [C D(x)] x X s.t C L (x) = CL, thickness constraint R µ,σ [C D (x)] µ CD (x) + σ CD (x) R µ,σ,γ [C D (x)] µ CD (x) + σ CD (x) + µ CD (x) γ CD (x) R VaR 90 [C D (x)] VaRC 90 D (x) R CVaR 90 [C D (x)] CVaRC 90 D (x) Quantity Reference (r) Uncertainty C L 0.5 Operating M 0.75 B(2, 2, 0.1, M 0.05) parameters Re c p [Pa] T [K] y RAE-2822 Cs Rs xp ys θs x α Rp yp xs θp M Cp F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

74 Risk averse optimization with MLMC Qualitative comparison Model: steady state Euler + boundary layer equation (MSES software) F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

75 Risk averse optimization with MLMC Deterministic versus Robust optimization F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

76 Risk averse optimization with MLMC Multi-objective optimization under operating uncertainties P-min x X {µ CD (x) + σ CD (x), µ CL (x) + σ CL (x)} (Pareto front) Uncertainties in Mach number and Angle of Attack. F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

77 Outline Conclusions 1 Motivating example 2 Multilevel Monte Carlo for expectations 3 MLMC for moments and distributions 4 Risk averse optimization with MLMC 5 Conclusions F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

78 Conclusions Conclusions and outlook Multilevel Monte Carlo is a very powerful technique that can dramatically reduce the computational cost of a UQ analysis compared to plain MC. The tuning of MLMC requires adaptive algorithms and reliable error and variances estimators. We have presented a way to compute higher order moments as well as cdf, quantiles, CVaR with MLMC and properly tune the method. The methodology has been successfully applied to forward UQ propagation and robust optimization under uncertainty in compressible aerodynamics. F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

79 Conclusions Thank you for your attention! F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

80 Conclusions References M. Pisaroni. Multi Level Monte Carlo Methods for Uncertainty Quantification and Robust Design Optimization in Aerodynamics, PhD Thesis n. 8082, EPFL, 2017 M. Pisaroni, S. Krumscheid, F. Nobile. Quantifying uncertain system outputs via the multilevel Monte Carlo method - Part I: Central moment estimation, MATHICSE Technical report no M. Pisaroni, S. Krumscheid, F. Nobile. Quantifying uncertain system outputs via the multilevel Monte Carlo method Part 2: distribution and robustness measures, in preparation. S. Krumscheid, F. Nobile. Multilevel Monte Carlo approximation of functions, MATHICSE Technical report no M. Pisaroni, F. Nobile, P. Leyland. A Multilevel Monte Carlo Evolutionary Algorithm for Robust Aerodynamic Shape Design. 18th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Denver, Colorado, USA, M. Pisaroni, F. Nobile, P. Leyland. Continuation Multi-Level Monte-Carlo method for Uncertainty Quantifiction in Turbulent Compressible Aerodynamics Problems modeled by RANS, MATHICSE Technical report no M. Pisaroni, F. Nobile, P. Leyland. A Continuation Multi Level Monte Carlo (C-MLMC) method for uncertainty quantification in compressible inviscid aerodynamics, CMAME, vol. 326, p , A.-L. Haji-Ali, F. Nobile, R. Tempone. Multi-index Monte Carlo: when sparsity meets sampling, in Numer. Math., vol. 132(4), p , N. Collier, A.-L. Haji-Ali, F. Nobile, E. von Schwerin, R. Tempone. A continuation multilevel Monte Carlo algorithm, BIT Numerical Mathematics, vol. 55(2), p , F. Nobile (EPFL) MLMC for UQ GAMM AGUQ workshop

Estimation of probability density functions by the Maximum Entropy Method

Estimation of probability density functions by the Maximum Entropy Method Claudio Bierig, Alexey Chernov Institute for Mathematics Carl von Ossietzky University Oldenburg alexey.chernov@uni-oldenburg.de