Uncertainty Quantification Accounting for Model Discrepancy Within a Random Effects Bayesian Framework

Abstract

With the advent of integrated computational materials engineering, there is a drive to exchange statistical confidence in a design obtained from repeated experimentation with one developed through modeling and simulation. Since these models are often missing physics or include incomplete knowledge or simplifying assumptions into their mathematical construct, they may not capture the physical system or process adequately over the entire domain. This can lead to a systematic discrepancy, otherwise known as model misfit, between the model output and the system it represents over at least part of the domain. However, by accounting for this discrepancy in uncertainty analyses, reliable inference, and prediction on the model parameters and material behavior can be made despite the missing physics. The previous statement is contingent on two conditions: (1) the discrepancy is systematic, and (2) the structure of the discrepancy is well understood, which is required to minimize issues with non-identifiability among unknown model components. We illustrate these insights via a case study of inference and prediction in a phenomenological meso-scale VPSC crystal plasticity model, which does not contain physics describing the elastic regime of deformation. Inference is performed via a Bayesian approach enabled by posterior simulation. Posterior uncertainty about unknown model parameters takes into account observation error, uncertainty stemming from aleatoric sample-to-sample variability, and model form error. Posterior uncertainty in the unknown Voce hardening parameters is propagated to generate a distribution of the stress–strain response in both the elastic and plastic regimes. Additionally, posterior predictive distributions are simulated to establish uncertainty bounds for future unobserved data.

Appendices

Appendix 1: MCMC Sampler Implementation

Appendix 2: Derivation of Full-Conditionals

Derivation of full conditional distributions over $\delta ^{2}, {\varPsi }, R$ and ${\varDelta }$.

Independent Error Precision,$\delta ^{s}$

$$\begin{aligned} \begin{aligned}&f\left( \delta ^{2} \mid \{y^{[s]}\}_{s=1}^{S}, \{\theta ^{[s]}\}_{s=1}^{S},\theta , {\varDelta } \right) \propto \prod _{s=1}^{S} \; MVN_{N} \left( y^{[s]} \right. \\&\quad \left. - m \left( \theta ^{[s]} \right) ; \; \; {\varDelta },\delta ^{-2} {\mathbb {I}}_N\right) \; \mathrm{Gamma}\left( \delta ^{2}; \, a,b\right) , \; \; a,\, b> 0 \\&\quad \propto \det \left( \delta ^{2} {\mathbb {I}}_N\right) ^{\frac{S}{2}}\exp \left\{ \right. \\&\qquad \left. -\frac{1}{2}\sum _{s=1}^{S} \left\Vert y^{[s]}-m\left( \theta ^{[s]}\right) - {\varDelta }\right\Vert _{2}^{2}(\delta ^2 {\mathbb {I}}_N) \right\} \; \left( \delta ^{2}\right) ^{a-1}\exp \left\{ -b\delta ^{2}\right\} , \; \; a, \, b> 0 \\&\quad =\left( \delta ^{2}\right) ^{\left( \frac{SN}{2} + a - 1\right) }\exp \left\{ -\delta ^{2}\left( b \right. \right. \\&\qquad \left. \left. + \frac{1}{2}\sum _{s=1}^{S}\left\Vert y^{[s]} - m\left( \theta ^{[s]}\right) - {\varDelta } \right\Vert _{2}^{2}\right) \right\} , \; \; a,\, b> 0 \\&\quad \propto \mathrm{Gamma}\left( \delta ^{2};\, \frac{SN}{2} + a, b\right. \\&\qquad \left. + \frac{1}{2}\sum _{s=1}^{S}\left\Vert y^{[s]} - m\left( \theta ^{[s]}\right) - {\varDelta }\right\Vert _{2}^{2}\right) , \; \; a,\, b > 0 \end{aligned} \end{aligned}$$

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Full Error Precision Matrix,${\varPsi }$

$$\begin{aligned} \begin{aligned}&f({\varPsi } \mid \{y^{[s]}\}_{s=1}^{S}, \{\theta ^{[s]}\}_{s=1}^{S},\theta , {\varDelta } ) \propto \prod _{s=1}^{S} \; MVN_{N}\left( y^{[s]} \right. \\&\qquad \left. - m\left( \theta ^{[s]}\right) ; \; \; {\varDelta }, \, {\varPsi }^{-1}\right) \; \mathrm{Wishart}_{N}\left( {\varPsi }; \, V_{0},v_{0}\right) , \; v_{0} \ge N + 1 \\&\quad \propto \det ({\varPsi })^{\frac{S}{2}}\exp \left\{ \right. \\&\qquad \left. -\frac{1}{2}\sum _{s=1}^{S}\left\Vert y^{[s]} - m\left( \theta ^{[s]}\right) - {\varDelta } \right\Vert _{2}^{2}\left( {\varPsi }\right) \right\} \, \det ({\varPsi })^{(v_{0}-N-1)/2}\exp \left\{ \right. \\&\qquad \left. -\hbox {Tr}\left( \frac{V_{0}^{-1}{\varPsi }}{2}\right) \right\} , \; v_{0} \ge N + 1 \\&\quad \propto \det ({\varPsi })^{(v_{0}-N-1+S)/2}\exp \left\{ -\frac{1}{2}\hbox {Tr}\left( {\varPsi }\left[ V_{0}^{-1}\right. \right. \right. \\&\qquad \left. \left. \left. + \sum _{s=1}^{S}\left\Vert y^{[s]} - m\left( \theta ^{[s]}\right) - {\varDelta } \right\Vert _{2}^{2}\right] \right) \right\} , \; v_{0} \ge N + 1\\&\quad \propto \mathrm{Wishart}_{N}\left( {\varPsi }; \left[ V_{0}^{-1} \right. \right. \\&\qquad \left. \left. + \sum _{s=1}^{S}\left\Vert y^{[s]} - m\left( \theta ^{[s]}\right) - {\varDelta } \right\Vert _{2}^{2}\right] ^{-1}, \, v_{0} + S\right) , \; v_{0} \ge N + 1 \end{aligned} \end{aligned}$$

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Random Effects Precision (${\varLambda }$) Decomposed Correlation, R

$$\begin{aligned} \begin{aligned}&f(R \mid \{y^{[s]}\}_{s=1}^{S},\{\theta ^{[s]}\}_{s=1}^{S},\theta ) \\&\quad \propto \left\{ \prod _{s=1}^{S} \; \mathrm{MVN}_{D}\left( \theta ^{[s]}; \; \; \theta , \, {\varLambda }^{-1}\right) \right\} \; \mathrm{Wishart}_{D}\left( R; \, R_{o},r_{o}\right) , \; r_{o} \ge D + 1 \\&\quad \propto \det ({\varLambda })^{\frac{S}{2}}\exp \left\{ \right. \\&\qquad \left. -\frac{1}{2}\sum _{s=1}^{S}\left\Vert \theta ^{[s]} - \theta \right\Vert _{2}^{2}\left( {\varLambda }\right) \right\} \, \det (R)^{(r_{o}-D-1)/2}\exp \left\{ \right. \\&\qquad \left. -\hbox {Tr}\left( \frac{R_{o}^{-1}R}{2}\right) \right\} , \; r_{o} \ge D + 1 \\&\quad \propto \det (R)^{\frac{1}{2}(S + r_{o} - D - 1 ) } \exp \left\{ \right. \\&\qquad -\frac{1}{2}\hbox {Tr}\left( R\left[ R_{o}^{-1} \right. \right. \\&\qquad \left. \left. \left. + \text{ diag }(t)\left( \sum _{s=1}^{S}\left\Vert \theta ^{[s]} - \theta \right\Vert _{2}^{2}\right) \text{ diag }(t)\right] \right) \right\} , \; r_{o} \ge D + 1\\&\quad \propto \mathrm{Wishart}_{D}\left( R; \left[ R_{o}^{-1} \right. \right. \\&\qquad \left. \left. + \text{ diag }(t)\left( \sum _{s=1}^{S}\left\Vert \theta ^{[s]} - \theta \right\Vert _{2}^{2}\right) \text{ diag }(t)\right] ^{-1}, \, r_{o} + S\right) , \; r_{o} \ge D + 1 \end{aligned} \end{aligned}$$

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Discrepancy,${\varDelta }$

$$\begin{aligned} \begin{aligned}&f({\varDelta } \mid {\varPsi }, \{y^{[s]}\}_{s=1}^{S},\{\theta ^{[s]}\}_{s=1}^{S},\theta ) \\&\quad \propto \prod _{s=1}^{S} \; \mathrm{MVN}_{N}\left( y^{[s]} - m\left( \theta ^{[s]}\right) ; \, {\varDelta } ,{\varPsi }^{-1}\right) \; MVN_{N}\left( {\varDelta }; \, 0, {\varGamma } \right) \\&\quad \text {which is just a Normal--Normal hierarchy with full-conditional}, \\&\quad {\varDelta } \mid \{y^{[s]}\}_{s=1}^{S},\{\theta ^{[s]}\}_{s=1}^{S},{\varPsi } \sim \mathrm{MVN}_{N}({\varDelta }; \; \; m,V) \\&\quad f\left( {\varDelta } \mid {\varPsi }, \{y^{[s]}\}_{s=1}^{S},\{\theta ^{[s]}\}_{s=1}^{S},\theta \right) \propto \mathrm{MVN}_{N}({\varDelta }; \; \; m,V)\\&\quad \text {where,}\\&\quad m = ({\varGamma }^{-1} + N{\varPsi })^{-1}N{\varPsi }\left( \overline{y^{[s]} - m(\theta ^{[s]}})\right) \\&\quad V = ({\varGamma }^{-1} + N{\varPsi })^{-1}. \end{aligned} \end{aligned}$$

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Appendix 3: Parameter Correlation

See Tables 3 and 4.

Table 3 Element-wise posterior correlation for a representative random effect from case study I

Table 4 Element-wise posterior correlation for the overall effect from case study I

Appendix 4: Additional Figures

Marginal posterior distributions: Fig. 12 compares the distributions between the overall (dashed lines) and random effects (solid lines).

Fig. 12

Scaled marginal posterior densities from case study I for random effects (solid lines) and the overall effect (dashed lines) for each Voce hardening parameter