Algorithms for Uncertainty Quantification

Transcription

1 Algorithms for Uncertainty Quantification Lecture 9: Sensitivity Analysis ST 2018 Tobias Neckel Scientific Computing in Computer Science TUM

2 Repetition of Previous Lecture Sparse grids in Uncertainty Quantification T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

3 Repetition of Previous Lecture Sparse grids in Uncertainty Quantification concept of sparse grids (SG) basic idea: truncate on diagonal SG save many grid points but often provide similar accuracy compared to full tensor grids rule of thumb: SG useful for 4 d 20 specific SG versions: depend on 1D grid point sequence (w/o nesting, point positions/stretching, boundary points) 1D discrete operator focus in this lecture: SG for quadrature in UQ adaptivity possible example: damped linear oscillator T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

4 Concept of Building Block: Time: 90 minutes Content: Global sensitivity analysis Sobol indices in context of gpce + pseudo-spectral approach T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

5 Concept of Building Block: Time: 90 minutes Content: Global sensitivity analysis Sobol indices in context of gpce + pseudo-spectral approach Expected Learning Outcomes The participants can explain the motivation of global sensitivity analysis (SA) in the context of UQ. They are able to distinguish and explain the concepts of global vs. local SA. They can indicate the ANOVA decomposition and relate it to the variance-based approach of global SA. The participants are able to formulate the total variance w.r.t. the ANOVA decomposition and to define the Sobol indices. In particular, they can explain the difference between local and total Sobol indices. They can formulate and explain the formulas for the computation of the Sobol indices via the coefficients of the gpce in the pseudo-spectral approach. T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

6 Agenda Topic Global sensitivity analysis (SA) in UQ Content motivation of SA categorisation: local vs. global SA variance-based global SA Sobol indices and gpc approximation example: damped linear oscillator T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

7 Multi-dimensional forward propagation of uncertainty stochastic model f (t, ω) stochastic inputs Ω Problem How sensitive is Y to changes in ω Ω? stochastic output(s) Y What is the relative contribution of ω i, i = 1,..., d to the output uncertainty? T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

8 Multi-dimensional forward propagation of uncertainty stochastic model f (t, ω) stochastic inputs Ω Problem How sensitive is Y to changes in ω Ω? stochastic output(s) Y What is the relative contribution of ω i, i = 1,..., d to the output uncertainty? What we want Compute sensitivities at a very low computational cost: Which uncertain parameters contribute most to the stochasic output Y? Reasonable measure of the output uncertainty T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

9 Motivation: Reasons for Sensitivity Analysis ascertain robustness of underlying model w.r.t. to various parameters example: safe flight mode for airplanes stochastic dimensionality reduction Can model be simplified by fixing insensitive parameters to deterministic value? specify regimes in parameters space that optimally impact responses or their uncertainties guide experimental design to determine measurement regimes that have greatest impact on parameter or response sensitivity T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

10 Motivation: Example of SA Usage/Results Fluid-structure interaction scenario scenario: bending beam in channel flow 5 stochastic parameters: fluid density and viscosity (ρ f and µ), structural density (ρ s ), elastic module (E), Poisson ratio (ν) Q.o.I.: displacement of beam tip (over time) total Sobol indices (at different times t) T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

11 Motivation: Example of SA Usage/Results Fluid-structure interaction scenario scenario: bending beam in channel flow 5 stochastic parameters: fluid density and viscosity (ρ f and µ), structural density (ρ s ), elastic module (E), Poisson ratio (ν) Q.o.I.: displacement of beam tip (over time) total Sobol indices (at different times t) t 0 = dim. reduction 5 2 > 50 h compute time saved T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

12 Motivation: Example of SA Usage/Results Fluid-structure interaction scenario scenario: bending beam in channel flow 5 stochastic parameters: fluid density and viscosity (ρ f and µ), structural density (ρ s ), elastic module (E), Poisson ratio (ν) Q.o.I.: displacement of beam tip (over time) total Sobol indices (at different times t) t 0 = t 0 = dim. reduction 5 2 > 50 h compute time saved dim. reduction 5 4 > 20 h compute time saved T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

13 Remember: Multivariate Polynomial Chaos Expansion (gpce) random vector Ω consisting of independent random variables Ω i, i = 1,..., d multi-index n = (n 1,..., n d ), k = (k 1,..., k d ) N d 0 multivariate polynomials: product of univariate polynomials φ n (ω) = φ n1 (ω 1 ) φ nd (ω d ), < φ n (ω), φ m (ω) > w = δ nm, δ nm = δ n1 m 1 δ nd m d T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

14 Remember: Multivariate Polynomial Chaos Expansion (gpce) random vector Ω consisting of independent random variables Ω i, i = 1,..., d multi-index n = (n 1,..., n d ), k = (k 1,..., k d ) N d 0 multivariate polynomials: product of univariate polynomials φ n (ω) = φ n1 (ω 1 ) φ nd (ω d ), < φ n (ω), φ m (ω) > w = δ nm, δ nm = δ n1 m 1 δ nd m d multivariate polynomial chaos expansion: f (t, ω) ˆfn (t)φ n (ω) n use multivariate pseudo-spectral approach to obtain ˆf n : K 1 ˆfn (t) f (t, x k ) φ n (x k ) w k k=0 T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

15 Multivariate gpce multivariate polynomial chaos expansion f (t, ω) N 1 n 1 =0 ˆfn (t)φ n (ω) n typically chosen such as n n d N for a given N with this setup: P = ( ) d+n d is the number of serialised summation terms T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

16 Multivariate gpce multivariate polynomial chaos expansion f (t, ω) N 1 n 1 =0 ˆfn (t)φ n (ω) n typically chosen such as n n d N for a given N with this setup: P = ( ) d+n d is the number of serialised summation terms multivariate pseudo-spectral approach ˆfn (t) = K 1 k =0 f (t, x k ) φ n (x k ) w k T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

17 Multivariate gpce multivariate polynomial chaos expansion f (t, ω) N 1 n 1 =0 ˆfn (t)φ n (ω) n typically chosen such as n n d N for a given N with this setup: P = ( ) d+n d is the number of serialised summation terms multivariate pseudo-spectral approach ˆfn (t) = K 1 k =0 f (t, x k ) φ n (x k ) w k standard approach: M = K d where K is number of quadrature points in one direction sparse grid quadrature: reduce number of quadrature nodes T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

18 Agenda Topic Global sensitivity analysis (SA) in UQ Content motivation of SA categorisation: local vs. global SA variance-based global SA Sobol indices and gpc approximation example: damped linear oscillator T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

19 Local and global sensitivity analysis Local sensitivity analysis aim: asses the sensitivity of output w.r.t. inputs perturbed about a nominal value generally: using gradients of output w.r.t. inputs can be used to screen out the insensitive uncertain inputs one input at a time tool for stochastic dimensionality reduction before doing the actual forward uncertainty propagation however: may be computationally expensive T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

20 Local and global sensitivity analysis Local sensitivity analysis aim: asses the sensitivity of output w.r.t. inputs perturbed about a nominal value generally: using gradients of output w.r.t. inputs can be used to screen out the insensitive uncertain inputs one input at a time tool for stochastic dimensionality reduction before doing the actual forward uncertainty propagation however: may be computationally expensive Global sensitivity analysis based on analyzing a suitable measure of uncertainty, e.g. the variance quantifies the contribution of each input to the output uncertainty relies solely on properties of the model and not on experimental data in this lecture variance-based global sensitivity analysis T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

21 Agenda Topic Global sensitivity analysis (SA) in UQ Content motivation of SA categorisation: local vs. global SA variance-based global SA Sobol indices and gpc approximation example: damped linear oscillator T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

22 Variance-based Global Sensitivity Analysis ANOVA decomposition measure for uncertainty of output = variance of output assumption (for simplicity): ω Ω i.i.d. starting point: ANOVA decomposition d f (t, ω) = f 0 (t) + f i (t, ω i ) + 1 i 1 <...<i s d i=1 1 i<j d f ij (t, ω i, ω j ) + f i1,...,i s (t, ω i1,..., ω is ) f 12...d (t, ω), where f 0 (t) is the mean, f i (t, ω i ) are univariate functions, f ij (t, ω i, ω j ) are bivariate functions, etc. T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

23 Variance-based Global Sensitivity Analysis ANOVA decomposition measure for uncertainty of output = variance of output assumption (for simplicity): ω Ω i.i.d. starting point: ANOVA decomposition d f (t, ω) = f 0 (t) + f i (t, ω i ) + 1 i 1 <...<i s d i=1 1 i<j d f ij (t, ω i, ω j ) + f i1,...,i s (t, ω i1,..., ω is ) f 12...d (t, ω), where f 0 (t) is the mean, f i (t, ω i ) are univariate functions, f ij (t, ω i, ω j ) are bivariate functions, etc. make ANOVA unique by imposing orthogonality: let Γ d = supp(ω) {i 1,..., i n } {i 1,..., i m } f i1...i n (t, ω i1,..., ω in )f i1...i m (t, ω i1,..., ω im )dω = 0 Γ d T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

24 Variance-based Global Sensitivity Analysis (2) computation of ANOVA terms let Γ d = supp(ω). Then: f i (t, ω i ) = f (t, ω)dω i f 0 (t), Γ d 1 f ij (t, ω i, ω j ) = f (t, ω)dω ij f i (t, ω i ) f j (t, ω j ) f 0 (t), Γ d 2..., where ( )dω Γ d 1 i means that the integration is performed over all variables, except the i th (proof: literature; Smith Chap. 13, e.g.) standard approach for evaluation Monte Carlo sampling T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

25 Variance-based Global Sensitivity Analysis (2) computation of ANOVA terms let Γ d = supp(ω). Then: f i (t, ω i ) = f (t, ω)dω i f 0 (t), Γ d 1 f ij (t, ω i, ω j ) = f (t, ω)dω ij f i (t, ω i ) f j (t, ω j ) f 0 (t), Γ d 2..., where ( )dω Γ d 1 i means that the integration is performed over all variables, except the i th (proof: literature; Smith Chap. 13, e.g.) standard approach for evaluation Monte Carlo sampling Total Variance denoting D i1...i s (t) = f 2 Γ s i 1...i s (f, ω i1... ω is )dω i1... dω is, we have d Var[f (t, ω)] = D i (t) + D ij (t) D 12..d (t) i=1 1 i<j d T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

26 Variance-based Global SA Sobol Indices Sobol indices remember Var[f (t, ω)] = d D i (t) + D ij (t) D 12..d (t) i=1 1 i<j d use the above equation to asses the contribution of each term to Var[f (t, ω)] Sobol indices local/partial Sobol indices: measure individual contributions OR interactions between inputs D i1...i S i1...i s (t) = s Var[f (t, ω)] total Sobol indices: measure individual contributions AND interactions between inputs Si T (t) = S i1,...,i s (t) i {i 1,...,i s} T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

27 Agenda Topic Global sensitivity analysis (SA) in UQ Content motivation of SA categorisation: local vs. global SA variance-based global SA Sobol indices and gpc approximation example: damped linear oscillator T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

28 Computation of Sobol Indices Standard approach remember ANOVA decomposition partial variances Sobol indices terms in ANOVA decomposition integrals f i1,...i s (t, ω i,..., ω s ) = f (t, ω)dω i1...i s f i1 (t, ω i1 )... f is (t, ω is ) f 0 (t) Γ d s therefore, standard approach Monte Carlo sampling computationally expensive! T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

29 Computation of Sobol Indices Standard approach remember ANOVA decomposition partial variances Sobol indices terms in ANOVA decomposition integrals f i1,...i s (t, ω i,..., ω s ) = f (t, ω)dω i1...i s f i1 (t, ω i1 )... f is (t, ω is ) f 0 (t) Γ d s therefore, standard approach Monte Carlo sampling computationally expensive! Using polynomial chaos expansion ANOVA decomposition and the polynomial chaos expansion orthogonal series/sums orthogonality uniqueness ANOVA polynomial chaos expansion idea: derive Sobol indices directly from polynomial chaos expansion coefficients T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

30 Computation of Sobol Indices (2) Using polynomial chaos expansion remember: multivariate polynomial chaos expansion N 1 f (t, ω) ˆfn (t)φ n (ω), n 1 =0 where n 1 = n n d Sobol indices for global sensitivity analysis S i (t) = D i (t) Var[f (t, ω)], D i(t) = n A i ˆf 2 n (t) (1) T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

31 Computation of Sobol Indices (2) Using polynomial chaos expansion remember: multivariate polynomial chaos expansion N 1 f (t, ω) ˆfn (t)φ n (ω), n 1 =0 where n 1 = n n d Sobol indices for global sensitivity analysis for first order indices S i (t) = D i (t) Var[f (t, ω)], D i(t) = n A i ˆf 2 n (t) (1) A i = {n N d : j i, n j = 0} T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

32 Computation of Sobol Indices (2) Using polynomial chaos expansion remember: multivariate polynomial chaos expansion N 1 f (t, ω) ˆfn (t)φ n (ω), n 1 =0 where n 1 = n n d Sobol indices for global sensitivity analysis for first order indices S i (t) = D i (t) Var[f (t, ω)], D i(t) = n A i ˆf 2 n (t) (1) A i = {n N d : j i, n j = 0} for higher order contributions A i = {n N d : n i > 0} T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

33 Agenda Topic Global sensitivity analysis (SA) in UQ Content motivation of SA categorisation: local vs. global SA variance-based global SA Sobol indices and gpc approximation example: damped linear oscillator T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

34 Model Problem Damped Linear Oscillator d 2 y (t) + c dy dt 2 dt (t) + ky(t) = f cos(ω Ot) y(0) = y 0 dy dt (0) = y 1 T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

35 Model Problem Damped Linear Oscillator c damping coefficient k spring constant f forcing amplitude ω O frequency y 0 initial position y 1 initial velocity d 2 y (t) + c dy dt 2 dt (t) + ky(t) = f cos(ω Ot) y(0) = y 0 dy dt (0) = y 1 T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

36 Damped Linear Oscillator: Sensitivity Analysis Stochastic setup: t [0, 30] T = 15 assume c = 0.10 k U(0.03, 0.04) f U(0.08, 0.12) ω O U(0.8, 1.2) y 0 U(0.45, 0.55) y 1 U( 0.05, 0.05) T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

37 Damped Linear Oscillator: Sensitivity Analysis Stochastic setup: t [0, 30] T = 15 assume c = 0.10 k U(0.03, 0.04) f U(0.08, 0.12) ω O U(0.8, 1.2) y 0 U(0.45, 0.55) y 1 U( 0.05, 0.05) Uncertainty propagation setup: Polynomial chaos expansion + pseudo-spectral approach Full Gaussian grid (7776 nodes) vs. sparse Gaussian grid (2203 nodes) T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

38 Damped linear oscillator: Sensitivity Analysis (2) First order indices: full vs. sparse grid T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

39 Damped linear oscillator: Sensitivity Analysis (2) First order indices: full vs. sparse grid Total indices: full vs. sparse grid T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

40 Literature Local sensitivity analysis: Chapter 14 in R. C. Smith, Uncertainty Quantification Theory, Implementation, and Applications, SIAM, 2014 Global sensitivity analysis: Chapter 15 in R. C. Smith, Uncertainty Quantification Theory, Implementation, and Applications, SIAM, 2014 T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST

41 Summary Sensitivity analysis categorisation: local vs. global SA global SA can provide useful insight variance-based global SA & Sobol indices (for gpc) example: damped linear oscillator T. Neckel Algorithms for Uncertainty Quantification L9: Sensitivity Analysis ST