Multilevel stochastic collocations with dimensionality reduction

Transcription

1 Multilevel stochastic collocations with dimensionality reduction Ionut Farcas TUM, Chair of Scientific Computing in Computer Science (I5)

2 Outline 1 Motivation 2 Theoretical background Uncertainty modeling Sparse grids Generalized polynomial chaos and sparse grids Multilevel collocation methods Stochastic dimensionality reduction 3 Test scenario 4 Discussion

3 Motivation problem: quantification of uncertainty in complex phenomena multiphysics (e.g. fluid-structure interaction) plasma physics...

4 Motivation problem: quantification of uncertainty in complex phenomena multiphysics (e.g. fluid-structure interaction) plasma physics... main challenge: curse of dimensionality curse of resources

5 Motivation problem: quantification of uncertainty in complex phenomena multiphysics (e.g. fluid-structure interaction) plasma physics... main challenge: curse of dimensionality curse of resources solution 1.1: delay the curse of dimensionality sparse grids

6 Motivation problem: quantification of uncertainty in complex phenomena multiphysics (e.g. fluid-structure interaction) plasma physics... main challenge: curse of dimensionality curse of resources solution 1.1: delay the curse of dimensionality sparse grids solution 1.2: try reducing the dimensionality sensitivity analysis

7 Uncertainty modeling probabilistic modeling probability space (Ω, F, P) θ = (θ 1, θ 2,..., θ d ) vector of continuous i.i.d. random variables supp(θ i ) = Γ i, supp(θ) = Γ 1 Γ 2... Γ d = Γ

8 Generalized polynomial chaos approximation idea: represent an arbitrary random variable (of interest) as a function of another random variable with given distribution

9 Generalized polynomial chaos approximation idea: represent an arbitrary random variable (of interest) as a function of another random variable with given distribution how: use a series expansion of orthogonal polynomials

10 Generalized polynomial chaos approximation idea: represent an arbitrary random variable (of interest) as a function of another random variable with given distribution how: use a series expansion of orthogonal polynomials let p = (p 1,..., p d ) N d : d i=1 p i < P

11 Generalized polynomial chaos approximation idea: represent an arbitrary random variable (of interest) as a function of another random variable with given distribution how: use a series expansion of orthogonal polynomials let p = (p 1,..., p d ) N d : d i=1 p i < P consider d-variate orthogonal polynomials Φ p (θ) := Φ p1 (θ 1 )... Φ pd (θ d )

12 Generalized polynomial chaos approximation idea: represent an arbitrary random variable (of interest) as a function of another random variable with given distribution how: use a series expansion of orthogonal polynomials let p = (p 1,..., p d ) N d : d i=1 p i < P consider d-variate orthogonal polynomials Φ p (θ) := Φ p1 (θ 1 )... Φ pd (θ d ) for simplicity, drop the multi-index subscript p and use instead a scalar index n = 1,..., N, N = ( ) d+p d orthogonality means E[Φ n (θ)φ m (θ)] = Φ n (θ)φ m (θ)ρ(θ)dθ = γ n δ nm, γ n R Γ

13 Generalized polynomial chaos let x - deterministic inputs, θ - stochastic inputs, f - model the gpc approximation of order N reads f (x, θ) f N (x, θ) = N 1 n=0 c n (x)φ n (θ) gpc coefficients via projection c n (x) = f (x, θ)φ n (θ)ρ(θ)dθ = E[f (x, θ)φ n (θ)] Γ

14 Post-processing expectation variance total Sobol indices E[f (x, θ)] = c 0 (x), Var[f (x, θ)] = S T i (x) = Var p[f (x, θ)] Var[f (x, θ)] N 1 n=1 = c 2 n(x). k A p c 2 k (x) Var[f (x, θ)], A p = {p N d : p i p, p i 0} d Si T (x) = 1 i=1

15 Sparse grid idea problem: discretize efficiently a tensor product space

16 Sparse grid idea problem: discretize efficiently a tensor product space standard approach: full grid O(N d ) dof, if N dof in one direction curse of dimensionality

17 Sparse grid idea problem: discretize efficiently a tensor product space standard approach: full grid O(N d ) dof, if N dof in one direction curse of dimensionality idea: delay the curse of dimensionality use sparse grids: weaken the assumed coupling between the input dimensions O(N d ) O(N(logN) d 1 ) dof

18 Sparse grid idea

19 Hierarchical sparse grids ingredients grid level l = (l 1,..., l d ) N d spatial position i = (i 1,..., i d ) N d generic grid point u l,i = (u l1,i 1,..., u ld,i d ) equidistant grid with mesh size h li = 2 l i, i = 1,..., d basis functions ϕ l,i with support [u l,i h l, u l,i + h l ] ( u ihl ) ϕ l,i (u) = ϕ h l in d-dimensions, d ϕ l,i (u) = ϕ lj,i j (u j ) j=1

20 Hierarchical sparse grids preliminaries H l = span{ϕ l,i : 1 i 2 l 1} - nodal set W l = span{ϕ l,i : i I l } - hierarchical increment set I l = {i N d : 1 i k 2 l k 1, i k odd, k = 1... d} H l = k l W k

21 Hierarchical sparse grids preliminaries given the hierarchical increment spaces W l and given a level L, we can create further spaces V L V L = k J W k, for some multiindex set J

22 Hierarchical sparse grids preliminaries given the hierarchical increment spaces W l and given a level L, we can create further spaces V L V L = k J W k, for some multiindex set J if J = {l N d : l L} - full grid space

23 Hierarchical sparse grids preliminaries given the hierarchical increment spaces W l and given a level L, we can create further spaces V L V L = k J W k, for some multiindex set J if J = {l N d : l L} - full grid space if J = {l N d : l 1 L + d 1} - standard sparse grid space

24 Hierarchical sparse grids example L = 5

25 Interpolation on hierarchical sparse grids consider g : [0, 1] d R the sparse grid interpolant g I (u) of g(u) is g I (u) = l J,i I l α l,i ϕ l,i (u) (1) α l,i are the so-called hierarchical surpluses assume g H2 mix ([0, 1] d ) = {f : [0, 1] d R : D l f L 2 ([0, 1] d ), l 2}, D l f = l 1f / x l x l d d if full grid g(u) g I (u) L 2 O ( hl 2 ) if sparse grid g(u) g I (u) L 2 O ( hl 2 Ld 1)

26 Piecewise linear basis functions

27 Piecewise polynomial basis functions

28 Spatial refinement due to the hierarchical construction local refinement possible α l,i - good measure of the interpolation error

29 Spatial refinement due to the hierarchical construction local refinement possible α l,i - good measure of the interpolation error the absolute value of α l,i - good refinement metric select the grid points with the largest surpluses values add their hierarchical descendants to J if not all hierarchical parents exist add them

30 Spatial refinement due to the hierarchical construction local refinement possible α l,i - good measure of the interpolation error the absolute value of α l,i - good refinement metric select the grid points with the largest surpluses values add their hierarchical descendants to J if not all hierarchical parents exist add them multiple grid points can be refined in one step

31 Spatial refinement: Franke s function ( f (x 1, x 2 ) = 0.75 exp (9x 1 2) 2 4 (9x 2 2) 2 4 ( 0.75 exp (9x 1 + 1) 2 9x ) ( 0.5 exp (9x 1 7) 2 (9x 2 3) 2 ( 4 4 ) 0.2 exp (9x 1 4) 2 (9x 2 7) ) + )

32 Franke s function

33 Franke s function refinement part 1 L = 5 refine 20% of the grid points

34 Franke s function refinement part 2 L = 6 refine 20% of the grid points

35 gpc coefficients computation remember c n (x) = f (x, θ)φ n (θ)ρ(θ)dθ Γ

36 gpc coefficients computation remember c n (x) = Γ f (x, θ)φ n (θ)ρ(θ)dθ how can we use sparse grids?

37 gpc coefficients computation remember c n (x) = Γ f (x, θ)φ n (θ)ρ(θ)dθ how can we use sparse grids? let T : [0, 1] d Γ

38 gpc coefficients computation remember c n (x) = Γ f (x, θ)φ n (θ)ρ(θ)dθ how can we use sparse grids? let T : [0, 1] d Γ then, c n (x) = f (x, T (u))φ n (T (u)) detj T (u) ρ(t (u))du [0,1] d intuition Φ n (T (u)) - tensor structure if f (x, T (u)) would also have a tensor structure...

39 gpc coefficients computation if f (x, T (u)) ˆf (x, T (u)) = l J,i I l α l,i ϕ l,i (u) T (u) := (F 1 1 (u 1 ),..., F 1 d (u d )), F i cdf of θ i

40 gpc coefficients computation if f (x, T (u)) ˆf (x, T (u)) = l J,i I l α l,i ϕ l,i (u) T (u) := (F 1 1 (u 1 ),..., F 1 d (u d )), F i cdf of θ i then c n (x) = ˆf (x, T (u))φn (T (u))du [0,1] d ( ) = α l,i (x)ϕ l,i (u) Φ n (T (u))du [0,1] d l J,i I l = l J,i I l α l,i (x) d j=1 [0,1] Φ j (F 1 j (u j ))ϕ lj,i j (u j )du j

41 Multilevel approaches monolevel approach can we further reduce the computational cost? use multilevel approaches

42 Multilevel stochastic collocation: no refinement

43 Multilevel stochastic collocation: with refinement

44 Multilevel gpc coefficients let M h denote the level of the deterministic domain discretization let L l denote the sparse grid level

45 Multilevel gpc coefficients let M h denote the level of the deterministic domain discretization let L l denote the sparse grid level let c M h,l l n (x) denote the gpc coefficient computed using a deterministic grid of level M h sparse grid of level L l

46 Multilevel gpc coefficients let M h denote the level of the deterministic domain discretization let L l denote the sparse grid level let c M h,l l n (x) denote the gpc coefficient computed using a deterministic grid of level M h sparse grid of level L l then, for K + 1 levels c M K,L K n (x) =c M 0,L K n (x). + (c M 1,L K 1 n if nested sparse grids, c M K l,l l 1 n (x) c M 0,L K 1 n (x))+ + (c M K,L 0 n (x) c M K 1,L 0 n (x)) (x) c M K l,l l n (x)

47 Stochastic dimensionality reduction each uncertain input has a different contribution to the output uncertainty

48 Stochastic dimensionality reduction each uncertain input has a different contribution to the output uncertainty some inputs contribute very little they can be ignored (taken as deterministic)

49 Stochastic dimensionality reduction each uncertain input has a different contribution to the output uncertainty some inputs contribute very little they can be ignored (taken as deterministic) use sensitivity information to determine each input s contribution

50 Stochastic dimensionality reduction each uncertain input has a different contribution to the output uncertainty some inputs contribute very little they can be ignored (taken as deterministic) use sensitivity information to determine each input s contribution in the multilevel scheme, given K c < K and τ [0, 1] (e.g. τ = 5%) if S T i (x) τ, ignore input i determine the new stochastic dimensionality project computed result on the new (sparse) grid

51 Sparse grid projection if input k, 1 i d is ignored and ũ k is the corresponding deterministic value ˆf (x, T (u)) = α l,i ϕ l,i (u) l J,i I l = α l,i l J,i I l j=1 = d ϕ lj,i j (u j ) α l,i ϕ lk,i k (ũ k ) l J,i I l = α l,i ϕ l,i (u) l J,i I l d 1 j=1 ϕ lj,i j (u j )

52 Test scenario t [0, 20], w = 1.05 d 2 y (t) + c dy dt 2 dt (t) + ky(t) = f cos(wt) y(0) = y 0 dy dt (0) = y 1

53 Test scenario d 2 y (t) + c dy dt 2 dt (t) + ky(t) = f cos(wt) y(0) = y 0 dy dt (0) = y 1 t [0, 20], w = 1.05 five uncertain inputs damping coefficient c U(0.08, 0.12) spring constant k U(0.03, 0.04) forcing amplitude f U(0.08, 0.12) initial position y0 U(0.45, 0.55) initial velocity y1 U( 0.05, 0.05) underdamped regime

54 Tests setup sparse grid functionality: SG

55 Tests setup sparse grid functionality: SG ++1 finite difference discretization 1

56 Tests setup sparse grid functionality: SG ++1 finite difference discretization uniform inputs Legendre polynomials modified polynomial basis functions of deg 2 1

57 Tests setup sparse grid functionality: SG ++1 finite difference discretization uniform inputs Legendre polynomials modified polynomial basis functions of deg 2 t interest =

58 Tests setup sparse grid functionality: SG ++1 finite difference discretization uniform inputs Legendre polynomials modified polynomial basis functions of deg 2 t interest = 10 reference results with Gauss-Legendre nodes 1

59 Tests setup sparse grid functionality: SG ++1 finite difference discretization uniform inputs Legendre polynomials modified polynomial basis functions of deg 2 t interest = 10 reference results with Gauss-Legendre nodes multilevel approach with K = 2 c M 2,L 2 n (x) = c M 0,L 2 n (x) + (c M 1,L 1 n (x) c M 0,L 1 n (x)) + (c M 2,L 0 n (x) c M 1,L 0 n (x)) 1

60 Tests setup sparse grid functionality: SG ++1 finite difference discretization uniform inputs Legendre polynomials modified polynomial basis functions of deg 2 t interest = 10 reference results with Gauss-Legendre nodes multilevel approach with K = 2 c M 2,L 2 n (x) = c M 0,L 2 n (x) + (c M 1,L 1 n (x) c M 0,L 1 n (x)) + (c M 2,L 0 n (x) c M 1,L 0 n (x)) when using refinement, L i, i = 1, 2 means L 0 with i refinement steps 1

61 Tests setup M 0 = 500, M 1 = 2000, M 2 = 8000 reference results E ref [y(10)] = Var ref [y(10)] =

62 Tests setup M 0 = 500, M 1 = 2000, M 2 = 8000 reference results E ref [y(10)] = Var ref [y(10)] = error measurement err = qoi ref qoi qoi ref

63 Test case 1: no dimensionality reduction L 0 L 1 L 2 ref % err exp err var e e e e % e e % e e % e e-05

64 Test case 2: dimensionality reduction compute Sobol indices for c M 1,L 1 n (x), i.e. start with K = 1 τ = 5% err expectation O(10 3 ) L 0 L 1 L 2 ref % S1 T S2 T S3 T S4 T S5 T % 1.2% 56.5% 4.8% 34.0% % 1.2% 56.5% 4.8% 34.0% % 4.1% 0.65% 56.7% 4.8% 33.9% % 4.1% 1.2% 56.5% 4.8% 34.0% % 4.0% 1.2% 56.6% 4.8% 34.0%

65 Test case 3: dimensionality reduction compute Sobol indices for c M 0,L 2 n (x) + (c M 1,L 1 n (x) c M 0,L 1 n (x)) τ = 5% err expectation O(10 4 ) L 0 L 1 L 2 ref % S1 T S2 T S3 T S4 T S5 T % 1.2% 56.5% 4.8% 34.0% % 1.2% 56.5% 4.8% 34.0% % 4.1% 1.2% 56.5% 4.8% 34.0% % 4.0% 1.2% 56.6% 4.8% 34.0% % 4.0% 1.2% 56.6% 4.8% 34.0%

66 Discussion from polynomial chaos coefficients, we can compute mean, variance, Sobol (sensitivity) indices etc. spatially adaptive sparse grids suitable to delay the curse of dimensionality multilevel ideas can further reduce the computational cost uncertain inputs that contribute little to output uncertainty can be ignored all of these should be used with care

67 Thank you for your attention!