Method of Distributions for Uncertainty Quantification

Abstract

Parametric uncertainty, considered broadly to include uncertainty in system parameters and driving forces (source terms and initial and boundary conditions), is ubiquitous in mathematical modeling. The method of distributions, which comprises PDF and CDF methods, quantifies parametric uncertainty by deriving deterministic equations for either probability density function (PDF) or cumulative distribution function (CDF) of model outputs. Since it does not rely on finite-term approximations (e.g., a truncated Karhunen-Loève transformation) of random parameter fields, the method of distributions does not suffer from the “curse of dimensionality.” On the contrary, it is exact for a class of nonlinear hyperbolic equations whose coefficients lack spatiotemporal correlation, i.e., exhibit an infinite number of random dimensions.

Appendix

It follows from (22.11) and the fact that u solves (22.10) that

$$\displaystyle\begin{array}{rcl} I& \equiv & \int \limits _{0}^{\infty }\int \limits _{ -\infty }^{\infty }f_{\mathbf{ u},\epsilon }(\mathbf{U};t) \frac{\partial \phi } {\partial t}(\mathbf{U},t)\,\mathrm{d}\mathbf{U}\,\mathrm{d}t = \mathbb{E}\left [\int \limits _{0}^{\infty }\int \limits _{ -\infty }^{\infty }\eta _{ \epsilon }(\mathbf{U} -\mathbf{u}) \frac{\partial \phi } {\partial t}(\mathbf{U},t)\,\mathrm{d}\mathbf{U}\,\mathrm{d}t\right ] {}\\ & =& \mathbb{E}\left [\int \limits _{0}^{\infty }\int \limits _{ -\infty }^{\infty }\eta _{ \epsilon }'(\mathbf{U} -\mathbf{u})\mathbf{G}(\mathbf{u})\phi (\mathbf{U},t)\,\mathrm{d}\mathbf{U}\,\mathrm{d}t\right ] -\int \limits _{-\infty }^{\infty }\mathbb{E}[\eta _{\epsilon }(\mathbf{U} -\mathbf{u}_{ 0})]\phi (\mathbf{U},0)\,\mathrm{d}\mathbf{U}, {}\\ \end{array}$$

for any $\phi \in \mathcal{ C\,}_{c}^{1}(\mathbb{R}^{N} \times [0,\infty ))$. Therefore

$$\displaystyle{I =\int \limits _{ 0}^{\infty }\int \limits _{ -\infty }^{\infty }\int \limits _{ -\infty }^{\infty }\eta _{ \epsilon }'(\mathbf{U}-\tilde{\mathbf{u}})\mathbf{G}(\tilde{\mathbf{u}})\phi (\mathbf{U},t)f_{\mathbf{u}}(\tilde{\mathbf{u}};t)\,\mathrm{d}\mathbf{U}\,\mathrm{d}\tilde{\mathbf{u}}\,\mathrm{d}t-\int \limits _{-\infty }^{\infty }f_{\mathbf{ u},\epsilon }(\mathbf{U};0)\phi (\mathbf{U},0)\,\mathrm{d}\mathbf{U}.}$$

Integration by parts in U yields

$$\displaystyle\begin{array}{rcl} I = -\int \limits _{0}^{\infty }\int \limits _{ -\infty }^{\infty }(\eta _{\epsilon } \star \mathbf{G}f_{\mathbf{ u}})(\mathbf{U},t)\nabla _{\mathbf{U}}\phi (\mathbf{U},t)\,\mathrm{d}\mathbf{U}\,\mathrm{d}t -\int \limits _{-\infty }^{\infty }f_{\mathbf{ u},\epsilon }(\mathbf{U};0)\phi (\mathbf{U},0)\,\mathrm{d}\mathbf{U}.& & {}\\ \end{array}$$

By the above definition of I, we have established that, for any $\phi \in \mathcal{ C\,}_{c}^{1}(\mathbb{R}^{N} \times [0,\infty ))$,

$$\displaystyle\begin{array}{rcl} \int \limits _{0}^{\infty }\int \limits _{ -\infty }^{\infty }f_{\mathbf{ u},\epsilon } \frac{\partial \phi } {\partial t}\mathrm{d}\mathbf{U}\,\mathrm{d}t +\int \limits _{ 0}^{\infty }\int \limits _{ -\infty }^{\infty }(\eta _{\epsilon } \star \mathbf{G}f_{\mathbf{ u}})\nabla _{\mathbf{U}}\phi \,\mathrm{d}\mathbf{U}\,\mathrm{d}t +\int \limits _{ -\infty }^{\infty }f_{\mathbf{ u},\epsilon }(\mathbf{U};0)\phi (\mathbf{U},0)\,\mathrm{d}\mathbf{U} = 0.& & {}\\ \end{array}$$

Using standard arguments [1], taking the limit 𝜖 → 0 leads to (22.12).