Quantifying Uncertainty in Engineering Analysis Mike Giles and - - PowerPoint PPT Presentation

Quantifying Uncertainty in Engineering Analysis

Mike Giles and Devendra Ghate

giles@comlab.ox.ac.uk

Oxford University Computing Laboratory

Quantifying Uncertainty – p. 1/18

Motivation

Engineering analysis assumes perfect knowledge of the geometry and mathematical model. Engineering design assumes we can manufacture exactly what is designed. In reality, there is much uncertainty: manufacturing tolerances uncertain modelling parameters Next big trend in engineering analysis is to quantify the consequences of these.

Quantifying Uncertainty – p. 2/18

Approaches

At one extreme, there are stochastic PDEs, able to cope with extremely large uncertainties (e.g. variation in rock porosity in oil reservoir modelling), but very complex and computationally demanding. We’re interested in the other extreme: limited to very small uncertainties relatively simple and computationally inexpensive usually only concerned with one or two output functionals (e.g. lift, drag)

• ften interested only in first order (variance) and second
• rder (mean perturbation) effects

sometimes, also interested in confidence limits

Quantifying Uncertainty – p. 3/18

Explicit Uncertainty Propagation

Monte Carlo simulations; Moment method with first, second and third order Taylor expansions; Moment method with adjoint error correction.

Quantifying Uncertainty – p. 4/18

Taylor Expansion

y = f(x) = f(µx) + f′(µx)(x − µx) + f′′(µx) 2! (x − µx)2 +f′′′(µx) 3! (x − µx)3 + O((x − µx)4)

where the primes denote derivative with respect to x, and µx is mean of x. First Order Taylor series approximation:

µy = f(µx) σ2

y

=

• f′(µx)

2 σ2

x

Quantifying Uncertainty – p. 5/18

Taylor Expansion

Second Order Taylor series approximation:

µy = f(µx) + f′′(µx) 2! σ2

x

σ2

y

= f′′(µx)2 σ2

x + f′(µx) f′′(µx) S(x) σ3 x

+ f′′(µx) 2!

2

(K(x)−1) σ4

x

Similarly third order Taylor series approximation can be derived involving skewness and Kurtosis terms.

Quantifying Uncertainty – p. 6/18

MC for Test Functions

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 2.5 x 10

4

Range Frequency

Histogram of MC simulations for sin(x)

−0.4 −0.2 0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 2.5 3 3.5 x 10

5

Range Frequency

Histogram of MC simulations for cos(x)

Frequency distribution for sin(x) and cos(x) for normally distributed x with µx = 0, σx = π

4 – sample size = 106.

Quantifying Uncertainty – p. 7/18

Mean: Moment Methods

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 −6 −4 −2 2 4 6 8 x 10

−4

Standard Deviation of x Mean of sin(x)

Mean of sin(x)

MC for sin(x) 1stOrder sin(x) 2ndOrder sin(x) 3rdOrder for sin(x) 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1

Standard Deviation of x Mean of cos(x)

Mean of cos(x)

MC for cos(x) 1stOrder cos(x) 2ndOrder cos(x) 3rdOrder for cos(x)

Figure 1: Prediction of µy with increasing σx

Quantifying Uncertainty – p. 8/18

Variance: Moment Methods

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Standard Deviation of x Variance of sin(x)

Variance of sin(x)

MC for sin(x) 1stOrder sin(x) 2ndOrder sin(x) 3rdOrder for sin(x) 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.002 0.004 0.006 0.008 0.01 0.012

Standard Deviation of x Variance of cos(x)

Variance of cos(x)

MC for cos(x) 1stOrder cos(x) 2ndOrder cos(x) 3rdOrder for cos(x)

Figure 2: Prediction of σ2

y with increasing σx

Quantifying Uncertainty – p. 9/18

Implicit Uncertainty Propagation

Let u be the solution of a set of non-linear algebraic equations

f(u(x), x) = 0,

and an output functional of interest, J(u(x), x). The adjoint equation corresponding to this functional is

∂f ∂u T f + ∂J ∂u T = 0.

Given approximate solutions u∗ and f

∗,

J(u(x), x) ≈ J(u∗, x) + (f

∗)T f(u∗, x).

Quantifying Uncertainty – p. 10/18

MC using Adjoint Error Correction

To avoid the cost of computing exact u, Monte-Carlo simulations are performed using adjoint error correction. The following options are available:

u∗ = uµx, f

∗ = fµx

u∗ = uµx + du dx(x−µx), f

∗ = fµx

u∗ = uµx + du dx(x−µx), f

∗ = fµx + df

dx(x−µx)

Quantifying Uncertainty – p. 11/18

Test Case

N(u(x), x) = u + u3 − x = 0, J(u(x), x) = u2.

nonlinear equation solved by Newton iteration; adjoints are calculated analytically;

du dx and df dx are calculated analytically.

Quantifying Uncertainty – p. 12/18

Results

µx = 5; σ2

x = 1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2.284 2.286 2.288 2.29 2.292 2.294 2.296 2.298 2.3

Standard Deviation of x Mean of function

Mean of u2

MC 1stOrder Taylor 2ndOrder Taylor Adjoint−1 Adjoint−2 Adjoint−3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Standard Deviation of x Variance of function

Variance of u2

MC 1stOrder Taylor 2ndOrder Taylor Adjoint−1 Adjoint−2 Adjoint−3

Quantifying Uncertainty – p. 13/18

Results

µx = 0; σ2

x = 1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −6 −5 −4 −3 −2 −1 1 2

Standard Deviation of x Mean of function

Mean of u2

MC 1stOrder Taylor 2ndOrder Taylor Adjoint−1 Adjoint−2 Adjoint−3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Standard Deviation of x Variance of function

Variance of u2

MC 1stOrder Taylor 2ndOrder Taylor Adjoint−1 Adjoint−2 Adjoint−3

Quantifying Uncertainty – p. 14/18

Computational Cost

One nonlinear solution One adjoint solution N linear solutions for du

dx (N = # of uncertain parameters)

N linear solutions for df

dx

M inexpensive approximate MC evaluations

Quantifying Uncertainty – p. 15/18

Cheap Hessian Evaluation

Functional and nonlinear equations:

j(x) = J(u, x) = ⇒ ∂j ∂xi = ∂J ∂u ∂u ∂xi + ∂J ∂xi . f(x, u) = 0 = ⇒ ∂f ∂u ∂u ∂xi + ∂f ∂xi = 0.

Adjoint equation and gradient:

∂f ∂u T f + ∂J ∂u T = 0, ∂j ∂xi = − ∂J ∂u ∂f ∂u −1 ∂f ∂xi + ∂J ∂xi = f

T ∂f

∂xi + ∂J ∂xi .

Quantifying Uncertainty – p. 16/18

Cheap Hessian Evaluation

Second derivative of functional and nonlinear equations:

∂2j ∂xi∂xj = ∂J ∂u ∂2u ∂xi∂xj + D2

i,jJ,

∂f ∂u ∂2u ∂xi∂xj + D2

i,jf = 0,

D2

i,jJ ≡ ∂2J

∂u2 ∂u ∂xi ∂u ∂xj + ∂2J ∂u∂xi ∂u ∂xj + ∂2J ∂u∂xj ∂u ∂xi + ∂2J ∂xi∂xj

and D2

i,jf is defined similarly.

= ⇒ ∂2j ∂xi∂xj = f

TD2 i,jf + D2 i,jJ.

Quantifying Uncertainty – p. 17/18

Cheap Hessian Evaluation

Computational cost: One nonlinear solution;

No adjoint solutions, one for each output; Ni linear solutions, one for each input;

inexpensive evaluation of D2

i,jf, D2 i,jJ using

forward-on-forward AD. Not a new idea (Taylor, Green, Newman and Putko, 2003) but worth pursuing?

Quantifying Uncertainty – p. 18/18