Structure exploiting methods for fast uncertainty quantification in multiphase flow through heterogeneous media

Abstract

We present a computational framework for dimension reduction and surrogate modeling to accelerate uncertainty quantification in computationally intensive models with high-dimensional inputs and function-valued outputs. Our driving application is multiphase flow in saturated-unsaturated porous media in the context of radioactive waste storage. For fast input dimension reduction, we utilize an approximate global sensitivity measure, for function-valued outputs, motivated by ideas from the active subspace methods. The proposed approach does not require expensive gradient computations. We generate an efficient surrogate model by combining a truncated Karhunen-Loéve (KL) expansion of the output with polynomial chaos expansions, for the output KL modes, constructed in the reduced parameter space. We demonstrate the effectiveness of the proposed surrogate modeling approach with a comprehensive set of numerical experiments, where we consider a number of function-valued (temporally or spatially distributed) QoIs.

Appendix

1.1 A.1 Proof of upper bound on total error in product space

Proof

Let f(s,ξ) be in L² of the product space ${{\varTheta }}\times {\mathscr{X}}$ and $\left \|\cdot \right \|$ be the L² error in the product space ${{\varTheta }}\times {\mathscr{X}}$. The truncated KLE of f is given by

$$ f_{N_{\text{qoi}}}^{\text{PC}}(s, \boldsymbol\xi) = \bar{f}(s) + \sum\limits_{i=1}^{{{N}_{\text{qoi}}}} \sqrt{\lambda_{i}} f_{i}^{\text{PC}}(\boldsymbol\xi) {{\varPhi}}_{i}(s). $$

The total error in the product space is given by

$$ \left\|f-f_{N_{\text{qoi}}}^{\text{PC}}\right\|^{2} \leq 2\left\|f-f_{{{{{N}_{\text{qoi}}}}\!}}\right\|^{2} + 2\left\|f_{{{{{N}_{\text{qoi}}}}\!}} - f_{N_{\text{qoi}}}^{\text{PC}}\right\|^{2} $$

We consider the first term

$$ \begin{array}{@{}rcl@{}} && 2\left\|f-f_{{{{{N}_{\text{qoi}}}}\!}}\right\|^{2}\\ &=& ~2\left\|\sum\limits_{i=1}^{\infty}\sqrt{\lambda_{i}}f_{i}(\boldsymbol\xi){{\varPhi}}_{i}(s) - \sum\limits_{i=1}^{{{N}_{\text{qoi}}}}\sqrt{\lambda_{i}}f_{i}(\boldsymbol\xi){{\varPhi}}_{i}(s)\right\|^{2}\\ &=& ~2{\int}_{{{\varTheta}}}{\int}_{\mathscr{X}} \left( \sum\limits_{i={{N}_{\text{qoi}}}+1}^{\infty}\sqrt{\lambda_{i}} f_{i}(\boldsymbol\xi){{\varPhi}}_{i}(s)\right)^{2} ds \!\upmu(d\boldsymbol\xi)\\ &=& ~2\sum\limits_{i,j ={{N}_{\text{qoi}}}+1}^{\infty} \sqrt{\lambda_{i}}\sqrt{\lambda_{j}} {\int}_{{{\varTheta}}} f_{i}(\boldsymbol\xi)f_{j}(\boldsymbol\xi){\int}_{\mathscr{X}}\\&& {{\varPhi}}_{i}(s){{\varPhi}}_{j}(s) ds \!\upmu (d\boldsymbol\xi)\\ &=& ~2\sum\limits_{i={{N}_{\text{qoi}}}+1}^{\infty}\lambda_{i} {\int}_{{{\varTheta}}} f_{i}(\boldsymbol\xi)^{2} \!\upmu (d\boldsymbol\xi) = ~2\sum\limits_{i={{N}_{\text{qoi}}}+1}^{\infty} \lambda_{i}. \end{array} $$

Changing the order of infinite sums and integral is a consequence of the Dominated Convergence Theorem and reordering of integrals is a justified by Fubini’s Theorem. The orthogonality of the eigenfunctions in $L^{2}({\mathscr{X}})$ justifies the simplification in the second to last line, and the last step is a consequence of the KL modes properties.

Next, we consider the second error term. Let

$$ f_{i}^{\text{PC}} = \sum\limits_{k=0}^{{N}_{\text{PC}}} \hat{c}_{i,k} {{\varPsi}}_{k}(\boldsymbol\xi), $$

where $\hat {c}_{i,k}$ represents the numerical approximation of the exact PCE coefficients c_i,k and recall, $f_{i} = {\sum }_{k=0}^{\infty } c_{i,k}{{\varPsi }}_{k}(\boldsymbol \xi )$ we have

$$ \begin{array}{@{}rcl@{}} && 2\left\|f_{{{{{N}_{\text{qoi}}}}\!}} - f_{N_{\text{qoi}}}^{\text{PC}}\right\|^{2} \\&=& 2\left\|\sum\limits_{i = 1}^{{{N}_{\text{qoi}}}} \sqrt{\lambda_{i}}f_{i}(\boldsymbol\xi){{\varPhi}}_{i}(s) -{\sum}_{i=1}^{{{N}_{\text{qoi}}}}\sqrt{\lambda_{i}}f_{i}^{\text{PC}}(\boldsymbol\xi) {{\varPhi}}(s)\right\|^{2}\\ &=&~ 2{\int}_{{{\varTheta}}} {\int}_{\mathscr{X}} \left( \sum\limits_{i=1}^{{{N}_{\text{qoi}}}} \sqrt{\lambda_{i}}{{\varPhi}}_{i}(s)\left[f_{i}(\boldsymbol\xi)-f_{i}^{\text{PC}}(\boldsymbol\xi)\right]\right)^{2} ds \!\upmu (d\boldsymbol\xi) \end{array} $$

$$ \begin{array}{@{}rcl@{}} &=&~ 2\sum\limits_{i,j = 1}^{{{N}_{\text{qoi}}}} \sqrt{\lambda_{i}}\sqrt{\lambda_{j}} {\int}_{{{\varTheta}}} (f_{i}-f_{i}^{\text{PC}})(f_{j}-f_{j}^{\text{PC}}) {\int}_{\mathscr{X}}\\&& {{\varPhi}}_{i}(s){{\varPhi}}_{j}(s) ds \!\upmu (d\boldsymbol\xi)\\ &=&~ 2\sum\limits_{i=1}^{{{N}_{\text{qoi}}}} \lambda_{i}{\int}_{{{\varTheta}}} (f_{i}(\boldsymbol\xi)-f_{i}^{\text{PC}}(\boldsymbol\xi))^{2} \!\upmu(d\boldsymbol\xi)\\ &=&~ 2\sum\limits_{i=1}^{{{N}_{\text{qoi}}}}\lambda_{i} {\int}_{{{\varTheta}}} \left( \sum\limits_{k =0}^{\infty} c_{i,k}{{\varPsi}}_{k}(\boldsymbol\xi) - \sum\limits_{k=0}^{{N}_{\text{PC}}}\hat{c}_{i,k}{{\varPsi}}_{k}(\boldsymbol\xi)\right)^{2} \!\upmu (d\boldsymbol\xi)\\ &=& 2\sum\limits_{i = 1}^{{{N}_{\text{qoi}}}} \lambda_{i}{\int}_{{{\varTheta}}} \left( \sum\limits_{k=0}^{{N}_{\text{PC}}} (c_{i,k} - \hat{c}_{i,k}){{\varPsi}}_{k}(\boldsymbol\xi) + \sum\limits_{k=1+{N}_{\text{PC}}}^{\infty} c_{i,k}{{\varPsi}}_{k}(\boldsymbol\xi)\!\right)^{2} \!\upmu (d\boldsymbol\xi)\\ &=&~ 2\sum\limits_{i=1}^{{{N}_{\text{qoi}}}} \lambda_{i} \sum\limits_{k=1}^{{N}_{\text{PC}}}(c_{i,k}-\hat{c}_{i,k})^{2} \left\|{{\varPsi}}_{k}\right\|_{L^2({{\varTheta}})}^{2}\\ &&+~ 2\sum\limits_{i=1}^{{{N}_{\text{qoi}}}} \lambda_{i} \sum\limits_{j=1+{N}_{\text{PC}}}^{\infty} c_{i,j}^{2}\left\|{{\varPsi}}_{j}\right\|_{L^2({{\varTheta}})}^{2}. \end{array} $$

The simplification in the third line a consequence of the orthogonality of the PCE basis functions.

Thus, we have a bound on the total error

$$ \begin{array}{@{}rcl@{}} && \left\|f - {f_{{{{{N}_{\text{qoi}}}}\!}}}^{{\text{PC}}}\right\|^{2} \!\leq\! 2\left\|f - f_{{{{{N}_{\text{qoi}}}}\!}}\right\|^{2} + 2\left\|f_{{{{{N}_{\text{qoi}}}}\!}} - {f_{{{{{N}_{\text{qoi}}}}\!}}}^{{\text{PC}}}\right\|^{2}\\ &=&~ 2\sum\limits_{i={{N}_{\text{qoi}}}+1}^{\infty}\lambda_{i} + \sum\limits_{i=1}^{{{N}_{\text{qoi}}}} \lambda_{i} \sum\limits_{k=1}^{{N}_{\text{PC}}}(c_{i,k}-\hat{c}_{i,k})^{2} \left\|{{\varPsi}}_{k}\right\|_{L^2({{\varTheta}})}^{2}\\ &&+ 2\sum\limits_{i=1}^{{{N}_{\text{qoi}}}} \lambda_{i} \sum\limits_{j=1+{N}_{\text{PC}}}^{\infty} c_{i,j}^{2}\left\|{{\varPsi}}_{j}\right\|_{L^2({{\varTheta}})}^{2}. \end{array} $$

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