Estimating functional uncertainty using polynomial chaos and adjoint equations
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1 0. Estimating functional uncertainty using polynomial chaos and adjoint equations February 24, Florida State University, Tallahassee, Florida, Usa 2 Moscow Institute of Physics and Technology, Moscow, Russia 1 inavon@fsu.edu, 2 inavon@fsu.edu Estimating functional uncertainty using polynomial chaos and adjoint equations 1/33
2 Lecture Plan 1 The polynomial chaos expansion for uncertainty quantification 2 Intrusive polynomial chaos 3 Non-intrusive polynomial chaos 4 Adjoint acceleration of Monte-Carlo method 5 Adjoint with polynomial chaos 6 Test problem for the thermal conduction model 7 Numerical results 8 Future research directions
3 1. The polynomial chaos expansion for uncertainty quantification Polynomial Chaos Local sensitivity analysis and moment methods are suitable in the limit of small uncertainty (Cacuci,2009). Polynomial chaos, also referred to as Wiener chaos expansion is a non-sampling method used to determine evolution of uncertainty in a dynamical framework where there is probabilistic uncertaintanty in the system parameters. { } Let ξ i (w) be a set of independent standard Gaussian i=1 random variables on Ω, the sample space and P a probability measure. Estimating functional uncertainty using polynomial chaos and adjoint equations 3/33
4 1. The polynomial chaos expansion for uncertainty quantification Polynomial Chaos Than we can represent any random variable X : Ω R with finite variance, i.e. X L 2 (Ω) as i 1 X (w) = a 0 Γ 0 + a i1 Γ 1 (ξ i1 ) + a i1 i 2 Γ 2 (ξ i1, ξ i2 )+ i 1 i=1 i 2 i 1 =1 i 2 =1 i 3 =1 i 1 =1 i 2 =1 a i1 i 2 i 3 Γ 3 (ξ i1, ξ i2, ξ i3 ) +..., where Γ b is the Wiener polynomial chaos expansion (Wiener 1998) and a () R. In practice one truncates the PCE in both order p and dimension n. The orthogonality of the Γ p w.r.t. inner product < u(w), v(w) >= Ω u(w)v(w)dp requires the usage of the multivariate Hermite polynomials. Estimating functional uncertainty using polynomial chaos and adjoint equations 4/33
5 1. The polynomial chaos expansion for uncertainty quantification Polynomial Chaos Cameron and Martin (1947) proved convergence of this Wiener - Hermite PCE for a general square integrable process χ(x, t, w). Estimating functional uncertainty using polynomial chaos and adjoint equations 5/33
6 Monte-Carlo acceleration by adjoint We calculate tail probability of an objective (cost) function. J(ξ) objective function depends on a vector of uncertain variables ξ. PDF of ξ is known. J c is a known constante. The goal is the calculation of P(J > J c ). Brute force Monte-Carlo is most commonly use but is computationally expensive and inefficient. If probability is small only small fraction of samples would fall into tail region (i.e. insufficient sampling). Estimating functional uncertainty using polynomial chaos and adjoint equations 6/33
7 Monte-Carlo acceleration by adjoint P(J > J c ) P N = 1 N N i=1 I (J(ξ i) > J c ). The indicator function is defined as follow { 1 if J(ξ i ) > J c is true, I (J(ξ i ) > J c ) = 0 otherwise. Variance of the estimator P N is Var[P N ] = 1 N Var[I (J(ξ i) > J c )] = P(J>Jc) P2 (J>J c) N. Estimating functional uncertainty using polynomial chaos and adjoint equations 7/33
8 Monte-Carlo acceleration by adjoint Relative error of MC method can be characterized by ratio of standard deviation and the mean of estimator P N i.e. Var[P n ] P(J > J c ) = 1 P(J > J c ) N P(J > J c ) If tail probability is 0.1, MC samples should be used. Estimating functional uncertainty using polynomial chaos and adjoint equations 8/33
9 Adjoint sensitivity gradient for accelerating Monte-Carlo Variance reduction techniques for calculating tail probability P(J > J c ) based on a single adjoint calculation. It approximates the objective function as a linear function of the random variables describing the sources of uncertainties. Adjoint evaluated at ξ 0 and sensitivity gradient J(ξ 0 ) is obtained from the adjoint solution J(ξ) J L (ξ) = J(ξ 0 ) + J(ξ 0 ) (ξ ξ 0 ). Estimating functional uncertainty using polynomial chaos and adjoint equations 9/33
10 Adjoint sensitivity gradient for accelerating Monte-Carlo Approximate the target tail probability P(J > J C ) with the estimator P(J > J C ) P CV N = P(J L > J C )+ 1 N N (I (J(ξ i ) > J C ) I (J L (ξ i ) J C )). i=1 Estimating functional uncertainty using polynomial chaos and adjoint equations 10/33
11 Adjoint sensitivity gradient for accelerating Monte-Carlo Monte-Carlo method can give entire probability density function of any system variable - but is too expensive since a large number of samples are required for reassemble accuracy. For small errors the variance of a cost functional J may be calculated from the input data f i variances using: σ 2 l = N i=1 ( J f i σ fi ) 2. The adjoint method enables time for J f i calculation to be approximately double the time of J calculation. The CPU burden does not depend on parameters containing the errors, thus providing high computational efficiency. Estimating functional uncertainty using polynomial chaos and adjoint equations 11/33
12 If input data f i error probability distribution is know, we may calculate the J error distribution and moment using adjoint - gradients serving as a rapid meta model. J(f ) = J( f ) + i J f i fi. If the adjoint problem is solved and the gradient is stored, we may deal with a very big ensemble for distribution and moments calculation without any computational burden. See Alekseev and Navon (2003) The second order adjoint - The adjoint MC may not be appropriate in vicinity of the optimal solution where the gradient is zero. Second order estimations may be used for functional shift J = J( f ) N i=1 2 J fi, 2 σf 2 i,. Estimating functional uncertainty using polynomial chaos and adjoint equations 12/33
13 Adjoint sensitivity gradient for accelerating Monte-Carlo The variance may be approximated with σ 2 y = 1 2 ( 2 J( f ) ) 2. σ fi, σ fj, f i, f j, i These estimations are governed by the Hessian j H i,j = 2 J f i, f j,, which may be calculated via the second order adjoint problem. Estimating functional uncertainty using polynomial chaos and adjoint equations 13/33
14 Uncertainty Quantification (UQ) Uncertainty in model parameters UQ has 2 intimately coupled components. First pertains to forward propagation of uncertainty from model parameters to model outputs. Second component involves estimation of parametric uncertainties themselves based on available data Estimating functional uncertainty using polynomial chaos and adjoint equations 14/33
15 UQ methods Local sensitivity analysis and moment methods (Cacuci 2003) suitable in the limit of small uncertainty. For large degrees of uncertainty - methods using probability theory including polynomial chaos (PC) amongst other methods. Uncertainty can be epistemic when it results from lack of knowledge about quantity whose true value exhibits no actual variability. Uncertainty resulting from variability is termed aleatoric. Both epistemic and aleatoric uncertainties can be handled using probability theory in Bayesian framework. Estimating functional uncertainty using polynomial chaos and adjoint equations 15/33
16 Polynomial Chaos 8. Future research directions The underlying concept of polynomial chaos is the uncertainty in a random parameter can be represented as a series of orthogonal polynomials (Wiener 1938) This representation of the stochastic process will converge to the true uncertainty of the parameter (Cameron and Martin 1947). The Polynomial Chais (PC) involves a probability space (Ω, Σ, P) where Ω is a sample space, Σ is a σ - algebra on Ω and P is a probability measure on (Ω, σ). Let {ξ i (w)} i=1 be a set of independent standard Gaussian random variables (RVs) on Ω. Estimating functional uncertainty using polynomial chaos and adjoint equations 16/33
17 Polynomial Chaos Then we can represent any RV X : Ω R with finite variance, i.e., x L 2 (Ω) as X (w) = a 0 Γ 0 + a i1 Γ 1 (ξ i1 ) + i 1 i=1 i 2 i 1 =1 i 2 =1 i 3 =1 i 1 i 1 =1 i 2 =1 a i1 i 2 i 3 Γ 3 (ξ i1, ξ i2, ξ i3 ) +..., where Γ p is the Wiener PC of order p. This is a polynomial chaos expansion (PCE) X (w) = a k Ψ k (ξ 1, ξ 2,...). i=1 a i1 i 2 Γ 2 (ξ i1, ξ i2 )+ Estimating functional uncertainty using polynomial chaos and adjoint equations 17/33
18 Polynomial Chaos 8. Future research directions The orthogonality of the Γ p (or Ψ k ) w.r.t. inner product < u(w), v(w) >= Ω u(w)v(w)dp requires they be multivariate Hermite polynomials. Cameron and Martin (1947) proved the convergence of the Wiener - Hermite (WH) PCE for a general square integrable stochastic process X (X, t, w). In practical computational context one truncates the PCE in both order p and dimension n. item Next we introduce two different approaches for solving differential equations involving uncertain parameters using PCE. Estimating functional uncertainty using polynomial chaos and adjoint equations 18/33
19 Polynomial Chaos Intrusive method - we substitute PCE of the solution into the differential equation. Then by a Galerkin projection in the probability space we obtain a set of coupled equations governing the coefficients of the PCE of the solution. In a second approach, the non - intrusive method (based on approximating the PCE - we do not substitute the PCE into the differential equation - but use samples of the solutions of the differential equations to construct the coefficients of the PCE. Unlike the intrusive approach - this approach does not require solving a different (and usually more complicated) set of coefficient equations. This approach can use existing software and codes of solving deterministic differential equations. Estimating functional uncertainty using polynomial chaos and adjoint equations 19/33
20 Large-errors - Adjoint Polynomial Chaos First stage: Variables (providing major input in error of cost function) are determined using gradient - obtained via adjoint equations. This allows reducing dimension of random variables space Second stage: Coefficient of PCE or Hermite polynomials are determined using a least - squares nonintrusive variant of (PC). In a final stage - moments and PDF are obtained using the PC expansion (for large - errors) and adjoint based gradient for small errors. Input data parameters containing errors are expressed as f i =< f i > +ξ i σ i, where ξ i are normally distributed random variables with unit variance and σ i is the standard deviation of f i. Estimating functional uncertainty using polynomial chaos and adjoint equations 20/33
21 Large-errors - Adjoint Polynomial Chaos The linear estimate of the uncertainty of the functional J is represented as N J = i Jσ i ξ i, i=1 where the gradient i J is obtained using the adjoint method. New random values (RV) ξ i which satisfy the condition i Jσ i > σ, where σ is a certain critical magnitude of the error - are taken to be the leading variables providing main input to the total error of the functional. Estimating functional uncertainty using polynomial chaos and adjoint equations 21/33
22 Large-errors - Adjoint Polynomial Chaos To estimate nonlinear impact of large errors the cost functional as function of RVs is expand over Hermite polynomials as J = b 0 + N b i1 Hi 1 1 (ξ i1 ) + i 1 =1 N i 1 i 2 i 1 =1 i 2 =1 i 3 =1 N i 1 i 1 =1 i 2 =1 b i1 i 2 i 3 H 3 i 1 i 2 i 3 (ξ i1, ξ i2, ξ i3 ), b i1 i 2 H 2 i 1 i 2 (ξ i1, ξ i2 )+ where H p i 1...i n (ξ 1,..., ξ n ) = e ξ ξ/2 ( 1) p p ( ξ 1 ) i 1...( ξ n) in e ξ ξ/2, with ξ ξ/2 = (ξ ξ2 N )/2. Estimating functional uncertainty using polynomial chaos and adjoint equations 22/33
23 Large-errors - Adjoint Polynomial Chaos We used Hermite polynomials of the third and fourth order only. The Cameron - Martin theorem allows calculation of the mean and variance M E[y(ξ)] = b 0, σy 2 = bi 2. i=1 The random variables space is divided into 2 subspace ξ R Nm ξ R N Nm large errors small errors So the expansion is truncated assuming the following form N N i 1 J = b 0 + b i1 Hi 1 1 (ξ i1 ) + b i1 i 2 H 2 (ξ i 1 i i1, ξ i2 )+ 2 i 1 =1 N i 1 i 2 i 1 =1 i 2 =1 Estimating functional uncertainty using polynomial chaos 3 and adjoint equations 1 23/33
24 Large-errors - Adjoint Polynomial Chaos So the expansion is truncated assuming the following form J = b 0 + N i 1 N b i1 Hi 1 1 (ξ i1 ) + i 1 =1 i 2 i 1 =1 i 2 =1 i 3 =1 N i 1 i 1 =1 i 2 =1 b i1 i 2 i 3 H 3 i 1 i 2 i 3 (ξ i1, ξ i2, ξ i3 ) + b i1 i 2 H 2 i 1 i 2 (ξ i1, ξ i2 )+ N i 1 =N m b i1 H 1 i 1 (ξ i1 ). The first 3 terms account for large errors while the last term represents small errors and may be stated as N i=n m i Jσ i ξ i providing possibility of fast calculations using adjoint based gradient. Estimating functional uncertainty using polynomial chaos and adjoint equations 24/33
25 Large-errors - Adjoint Polynomial Chaos The adjoint equations enable us to calculate the gradient (N variables) of the cost functional by running a code whose computational cost is close to that of the forward code. This can accelerate computation of PC coefficients for large N. The adjoint PC approach provides approximately an N fold reduction of the CPU time when compared with the pure PC. Estimating functional uncertainty using polynomial chaos and adjoint equations 25/33
26 Test problems 8. Future research directions Let us consider the following conduction model c T t ( λ T ) = 0, x x with T (0, t) = T w, T (X, t) = T R, boundary conditions and initial condition T (x, 0) = T 0 (x). T(x,t) denotes temperature, λ is the thermal conductivity and X states for the specimen thickness. The cost functional has the following form J = = T δ(x x est )δ(t t test )dxdt, with S = (0, X ) (0, T ). S Estimating functional uncertainty using polynomial chaos and adjoint equations 26/33
27 Test problems 8. Future research directions x est and t est are coordinates of pointwise temperature estimation, i.e. temperatures at important moments in a sensitive location. We look for moments and pdf of functional caused by a normally distributed error in the thermal conductivity or in the initial state. To calculate the gradient we use the adjoint function Ψ(x, t) obtained by solving adjoint problem c Psi t + λ 2 Ψ x 2 δ(t t est)δ(x x est ) = 0, with the following boundary and final conditions: Ψ Ψ x = 0, x=0 x = 0, Ψ(x, t f ) = 0. x=x Estimating functional uncertainty using polynomial chaos and adjoint equations 27/33
28 Test problems 8. Future research directions The gradient of the cost functional w.r.t. initial temperature is J = Ψ(x, 0), and regarding conductivity coefficient is ( 2 ) T Ψ x 2 dxdt. S This gradient is connected with derivatives of functional over random variables (RVs) in T 0 : and in λ: J(ξ (m) ) ξ i = J T 0,i T 0 ξ i = Ψ(x i, 0)σ i, J = J λ = ξ i λ ξ i S Ψ 2 T x 2 σ idxdt. Estimating functional uncertainty using polynomial chaos and adjoint equations 28/33
29 Test problems We consider 3 test problems 1) We consider 4 random variables in the thermal conductivity λ = λ + σ i ξ i. Here ξ i are RVs with normal distribution of unit variance. The variation of cost function is δj = J < J >= T 0 S i Ψ 2 T x 2 σ iξ i dxdt. 2) The initial data contain normally distributed error T 0 (x i ) + δt 0 (x i ) = T 0 (x i ) + σ i ξ i. In this test when λ(t ) = λ 1 = Const and J = Ψ(x, 0) does not depend on T 0 - i.e. gradient info cannot be used in PC and adjoint PC cannot be applied. Estimating functional uncertainty using polynomial chaos and adjoint equations 29/33
30 Test problems 3) Here the thermal conductivity is a nonlinear function assuming form λ(t ) = λ 0 + λ 1 (T /T s ) 3 fort < 0, λ(t ) = λ 0. and initial data contain a normally distributed error T 0 (x i ) + δt 0 (x i ) = T 0 (x i ) + σ i ξ i Estimating functional uncertainty using polynomial chaos and adjoint equations 30/33
31 Numerical results 8. Future research directions Heat transfer equations and adjoint equations were solved by conservative finite differences scheme of 2 nd order over space and time. Numerical tests: Monte - Carlo (MC) Monte - Carlo using adjoint acceleration (AMC) Nonintrusive Polynomial Chaos (PC) Polynomial Chaos using adjoint equations (APC) Estimating functional uncertainty using polynomial chaos and adjoint equations 31/33
32 Error in thermal conductivity Fig. 1. Probability density. 1 -MC, 2 -AMC, 3-PC 1.2E-02 P(T) 1.0E E E E E E T Estimating functional uncertainty using polynomial chaos and adjoint equations 32/33
33 Error in thermal conductivity Thermal conductivity contained 4 normally distributed random variables with standard deviation equal 0.1 λ. PC implemented using third order Hermite polynomials. All tests used samples. Comparison of mean values ε and standard deviation σ is shown in next table. Table 1 MC AMC PC PC MC averaged PC coeff. (4) ε std. dev. σ Also shown in this table is the computational cost (CPU normalized) by time of single problem run. Estimating functional uncertainty using polynomial chaos and adjoint equations 33/33
The estimation of functional uncertainty using polynomial chaos and adjoint equations
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