Efficient Aircraft Design Using Bayesian History Matching. Institute for Risk and Uncertainty

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1 Efficient Aircraft Design Using Bayesian History Matching F.A. DiazDelaO, A. Garbuno, Z. Gong Institute for Risk and Uncertainty DiazDelaO (U. of Liverpool) DiPaRT 2017 November 22, / 32

2 Outline 1 THE PROBLEM 2 HISTORY MATCHING 3 SUBSET SIMULATION 4 SUS-BASED SAMPLING DiazDelaO (U. of Liverpool) DiPaRT 2017 November 22, / 32

3 THE PROBLEM The problem was posed during the First Study Group with Industry at the Institute for Risk and Uncertainty at Liverpool. Modern aircraft must operate within strict performance and regulatory limits. Quickly assertaining options from the vast, uncertain design space is key to increasing the concept design process. Aim: To determine, visualise and act on uncertainties that propagate through the design process to comply with performance and regulatory limits. Objectives: Perform robust design to narrow the set of possible configurations. Discover the parameters that strongly contribute to variations in measures of aptness. Manage key parameters to drive reliably towards desired properties and behaviours. DiazDelaO (U. of Liverpool) DiPaRT 2017 November 22, / 32

4 THE PROBLEM Measures of Aptness Input Units L. Bounds U. Bounds Fan Pressure Ratio (FPR) N/A Overall Pressure Ratio (OPR) N/A Bypass Ratio (B) N/A 6 8 SLS Thrust (ST) lb 26,000 32,000 Wing Area (WA) ft 2 1,300 1,400 Wing Aspect Ratio (WAR) N/A DiazDelaO (U. of Liverpool) DiPaRT 2017 November 22, / 32

5 THE PROBLEM Figures of Merit Output Flyover Noise Sideline Noise Cruise Fuel Consumption Emissions (NOx) Units EPNLdb EPNLdb lb lb DiazDelaO (U. of Liverpool) DiPaRT 2017 November 22, / 32

6 THE PROBLEM DiazDelaO (U. of Liverpool) DiPaRT 2017 November 22, / 32

7 THE PROBLEM DiazDelaO (U. of Liverpool) DiPaRT 2017 November 22, / 32

8 THE PROBLEM DiazDelaO (U. of Liverpool) DiPaRT 2017 November 22, / 32

9 HISTORY MATCHING History Matching Aims at identifying subset of input space X X for which the evaluation of a simulator η : X R gives an acceptable match to observed data. Iterative process that starts by sampling from input space X and by applying some implausibility measure. Cutoffs are imposed in order to obtain successive non-implausible sets X... X 2 X 1. If the simulator is expensive, an emulator can be employed. Subtle difference. Calibration will result in a posterior distribution over the input space, whilst History Matching might conclude that the set of acceptable matches is empty. DiazDelaO (U. of Liverpool) DiPaRT 2017 November 22, / 32

10 HISTORY MATCHING Types of Uncertainty Observational uncertainty. Experimental error. Subject to instrument accuracy. Model Discrepancy. However carefully the model is built, there will always be a difference between the real system and the simulator. Models are always wrong. Code uncertainty. For any choice of inputs, the output is known when the model is run. However, the simulator can be computationally expensive. Ensemble Variability. The simulator can be stochastic. DiazDelaO (U. of Liverpool) DiPaRT 2017 November 22, / 32

11 HISTORY MATCHING HM Workflow (Andrianakis et al., 2015) DiazDelaO (U. of Liverpool) DiPaRT 2017 November 22, / 32

12 Non-Implausible Sampling HISTORY MATCHING Method 1. Acceptance-rejection. Draw samples uniformly from the input space and reject the implausible ones. Easy to implement. Becomes inefficient as non-implausible space shrinks in later waves. Method 2. Perturbation (Andrianakis et al., 2015). Draw samples from a multivariate normal centered at non-implausible seeds. Needs tuning of variance to produce good samples. Method 3. Implausibility Driven Evolutionary Monte Carlo (Williamson and Vernon, 2013). Samples uniformly from non-implausible space. Mixing and efficiency depend on choosing a suitable implausibility ladder. DiazDelaO (U. of Liverpool) DiPaRT 2017 November 22, / 32

13 HISTORY MATCHING The Analogy Given an observation z, History Matching defines the following measure of implausibility: z E[η(y x)] I(x) = [V 0 + V c (x) + Vs + V m ] 1/2 At each iteration, the non-implausible set can be defined as: Π = {x X : I(x) 3} There is a natural analogy that connects History Matching and reliability analysis: The set Π can be regarded as a failure domain. DiazDelaO (U. of Liverpool) DiPaRT 2017 November 22, / 32

14 HISTORY MATCHING The Analogy Given an observation z, History Matching defines the following measure of implausibility: z E[η(y x)] I(x) = [V 0 + V c (x) + Vs + V m ] 1/2 At each iteration, the non-implausible set can be defined as: Π = {x X : I(x) 3} There is a natural analogy that connects History Matching and reliability analysis: The set Π can be regarded as a failure domain. Use Subset Simulation! DiazDelaO (U. of Liverpool) DiPaRT 2017 November 22, / 32

15 SUBSET SIMULATION The Engineering Reliability Problem Let G : R d R be a system performance function. Aim: To estimate the probability of failure, i.e. the probability of demand exceeding the capacity of the system. Let y be a critical value such that the system fails if y = G(x 1,..., x d ) > y. The failure domain F can thus be defined as: F = {x : G(x) > y } The engineering reliability problem can be formulated as computing the probability of failure: p F = P(x F) = π(x)dx F DiazDelaO (U. of Liverpool) DiPaRT 2017 November 22, / 32

16 SUBSET SIMULATION Subset Simulation Developed by Au and Beck (2001) to simulate rare events and estimate small probabilities of failure. The idea is to decompose a rare event F into a sequence of progressively less rare events as: F = F m F m 1... F 1 where F 1 is a relatively frequent event. Given the above sequence of events, the small probability P(F) of the rare event can be represented as a product of larger probabilities as: P(F) = P(F m ) = P(F 1 ) P(F 2 F 1 )... P(F m F m 1 ) DiazDelaO (U. of Liverpool) DiPaRT 2017 November 22, / 32

17 SUBSET SIMULATION Subset Simulation Subset simulation explores the input space X by generating a relatively small number of i.i.d. samples x (1) 0,..., x (n) 0 π(x) and computing the corresponding system responses y (1) 0,..., y (n) 0. DiazDelaO (U. of Liverpool) DiPaRT 2017 November 22, / 32

18 SUBSET SIMULATION Subset Simulation Let p (0, 1) such that np N. Define the first intermediate failure domain as: { F 1 = x : G(x) > y1 = y (np) 0 + y (np+1) } 0 2 DiazDelaO (U. of Liverpool) DiPaRT 2017 November 22, / 32

19 SUBSET SIMULATION Subset Simulation By construction, x (1) 0,..., x (np) 0 F 1, whilst x (np+1) 0,..., x (n) 0 / F 1. Thus, the Monte Carlo estimate for the probability of F 1 is given by P(F 1 ) 1 n n i=1 I F1 (x (i) 0 ) = p F 1 provides a rough estimate to the failure domain F. Since F F 1, the failure probability can be written as: p F = P(F 1 )P(F F 1 ) In the next stage, instead of sampling in the whole input space, SuS populates F 1. DiazDelaO (U. of Liverpool) DiPaRT 2017 November 22, / 32

20 SUBSET SIMULATION Subset Simulation We start with x (1) 0,..., x (np) 0 π(x F 1 ) and need to draw n np samples from π(x F 1 ). This is done with an MCMC scheme. DiazDelaO (U. of Liverpool) DiPaRT 2017 November 22, / 32

21 SUBSET SIMULATION Subset Simulation Define the second intermediate failure domain as: { F 2 = x : G(x) > y2 = y } (np) 1 + y (np+1) 1 2 DiazDelaO (U. of Liverpool) DiPaRT 2017 November 22, / 32

22 SUS-BASED SAMPLING 2-D Model (Williamson and Vernon, 2013) DiazDelaO (U. of Liverpool) DiPaRT 2017 November 22, / 32

SUS-BASED SAMPLING 10-D Model (Surjanovic and Bingham, 2016) ( W = 0.036Sw 0.758 Wfw 0.0035 A cos 2 (Λ) ) 0.6 q 0.

23 SUS-BASED SAMPLING 10-D Model (Surjanovic and Bingham, 2016) ( W = 0.036Sw Wfw A cos 2 (Λ) ) 0.6 q ( ) 0.3 λ tc (N z W dg) S w W p cos(λ) DiazDelaO (U. of Liverpool) DiPaRT 2017 November 22, / 32

24 SUS-BASED SAMPLING 10-D Model DiazDelaO (U. of Liverpool) DiPaRT 2017 November 22, / 32

25 SUS-BASED SAMPLING Airbus: Single Output, NOx = 240 ± 10 lb DiazDelaO (U. of Liverpool) DiPaRT 2017 November 22, / 32

26 Airbus: Multiple Output SUS-BASED SAMPLING DiazDelaO (U. of Liverpool) DiPaRT 2017 November 22, / 32

27 Further Improvements SUS-BASED SAMPLING We propose a full-probabilistic approach to History Matching: z η(x) I(x) = [V o + Vc(x) + Vs + Vm] 1/2 where η(x) GP(m(x), σ 2 (x)) is the Gaussian process emulator for the simulator output. The non-implausible space is described by statements of the form P{I(x) 3} Use measures such as entropy for active learning: H z (η(x)) = z+k σ(x) z k σ(x) ln f (y) f (y) dy DiazDelaO (U. of Liverpool) DiPaRT 2017 November 22, / 32

28 SUS-BASED SAMPLING Conclusions 1 There s a natural analogy between the Robust Design Problem and calibration of numerical models. 2 History Matching is a form of calibration that finds a subset (possibly empty) in the input space that provides a match between output and observed data. 3 There s also a natural analogy between the non-implausible space and a failure region, for which Subset Simulation is a suitable solution. DiazDelaO (U. of Liverpool) DiPaRT 2017 November 22, / 32

29 SUS-BASED SAMPLING References 1 Z. T. Gong, F. A. DiazDelaO, M. Beer (2016) Bayesian Model Calibration Using Subset Simulation. European Safety and Reliability Conference. Glasgow, UK. 2 Garbuno-Inigo, A. DiazDelaO, F.A., Zuev, K. (2017) Full Probabilistic History Matching, Under Review. 3 Andrianakis et al. (2015) Bayesian History Matching of Complex Infectious Disease Models Using Emulation, PLOS Computational Biology, 11 (1). 4 Au, S.K. and Beck, J. (2001) Estimation of small failure probabilities in high dimensions by subset simulation, Probabilistic Engineering Mechanics, 16 (4), DiazDelaO (U. of Liverpool) DiPaRT 2017 November 22, / 32

30 SUS-BASED SAMPLING Thank you! Acknowledgments to Sanjiv Sharma and Simon Coggon at Airbus, and Arturo Molina-Cristobal at Cranfield. DiazDelaO (U. of Liverpool) DiPaRT 2017 November 22, / 32

Risk Estimation and Uncertainty Quantification by Markov Chain Monte Carlo Methods

Risk Estimation and Uncertainty Quantification by Markov Chain Monte Carlo Methods Konstantin Zuev Institute for Risk and Uncertainty University of Liverpool http://www.liv.ac.uk/risk-and-uncertainty/staff/k-zuev/