Sparse Pseudo Spectral Projection Methods with Directional Adaptation for Uncertainty Quantification

Abstract

We investigate two methods to build a polynomial approximation of a model output depending on some parameters. The two approaches are based on pseudo-spectral projection (PSP) methods on adaptively constructed sparse grids, and aim at providing a finer control of the resolution along two distinct subsets of model parameters. The control of the error along different subsets of parameters may be needed for instance in the case of a model depending on uncertain parameters and deterministic design variables. We first consider a nested approach where an independent adaptive sparse grid PSP is performed along the first set of directions only, and at each point a sparse grid is constructed adaptively in the second set of directions. We then consider the application of aPSP in the space of all parameters, and introduce directional refinement criteria to provide a tighter control of the projection error along individual dimensions. Specifically, we use a Sobol decomposition of the projection surpluses to tune the sparse grid adaptation. The behavior and performance of the two approaches are compared for a simple two-dimensional test problem and for a shock-tube ignition model involving 22 uncertain parameters and 3 design parameters. The numerical experiments indicate that whereas both methods provide effective means for tuning the quality of the representation along distinct subsets of parameters, PSP in the global parameter space generally requires fewer model evaluations than the nested approach to achieve similar projection error. In addition, the global approach is better suited for generalization to more than two subsets of directions.

Appendix: Global Sensitivity Analysis

1.1 Sensitivity Indices

Following notation of Sect. 2, consider $F : {\varvec{\xi }}\in \varXi \mapsto F(\varXi ) \in L_2(\varXi ,\rho )$, where ${\varvec{\xi }}=(\xi _1 \cdots \xi _d)$ is a vector of d independent real-valued random variable with joined density $\rho$. Let $\mathscr {D}= \{1,\ldots ,d\}$ and for ${\varvec{\mathfrak u}}\in \mathscr {D}$ denote $|{\varvec{\mathfrak u}}| = \mathrm {Card}({\varvec{\mathfrak u}})$ and ${\varvec{\mathfrak u}}_{\sim } = \mathscr {D}\setminus {\varvec{\mathfrak u}}$, such that ${\varvec{\mathfrak u}}\cup {\varvec{\mathfrak u}}_\sim = \mathscr {D}$, ${\varvec{\mathfrak u}}\cap {\varvec{\mathfrak u}}_\sim =\emptyset$. Given ${\varvec{\mathfrak u}}\in \mathscr {D}$ we denote ${\varvec{\xi }}_{\varvec{\mathfrak u}}$ the sub-vector of ${\varvec{\xi }}$ with components $(\xi _{{{\mathfrak u}}_1} \cdots \xi _{{{\mathfrak u}}_{|{\varvec{\mathfrak u}}|}})$, so ${\varvec{\xi }}= ({\varvec{\xi }}_{\varvec{\mathfrak u}}{\varvec{\xi }}_{{\varvec{\mathfrak u}}_\sim })$. Under the stated assumptions, the function $F({\varvec{\xi }})$ has a unique orthogonal decomposition of the form [15]

$$\begin{aligned} F({\varvec{\xi }}) = \sum _{{\varvec{\mathfrak u}}\in \mathscr {D}} f_{\varvec{\mathfrak u}}({\varvec{\xi }}_{\varvec{\mathfrak u}}), \quad \left\langle {{f_{\varvec{\mathfrak u}}},{f_{\varvec{\mathfrak v}}}}\right\rangle = 0\ \text{ if } {\varvec{\mathfrak u}}\ne {\varvec{\mathfrak v}}. \end{aligned}$$

(6.1)

The functions $f_{\varvec{\mathfrak u}}$ can be recursively expressed as [39]

$$\begin{aligned} f_{\varvec{\mathfrak u}}({\varvec{\xi }}_{\varvec{\mathfrak u}}) = \mathbb {E}\left\{ {{\left. {F}\,\right| \,{{\varvec{\xi }}_{{\varvec{\mathfrak u}}}}}}\right\} - \sum _{\begin{array}{c} {\varvec{\mathfrak v}}\in \mathscr {D}\\ {\varvec{\mathfrak v}}\subsetneq {\varvec{\mathfrak u}} \end{array}} f_{\varvec{\mathfrak v}}({\varvec{\xi }}_{\varvec{\mathfrak v}}), \end{aligned}$$

where $\mathbb {E}\left\{ {{\left. {F}\,\right| \,{{\varvec{\xi }}_{{\varvec{\mathfrak u}}}}}}\right\}$ is the conditional expectation of F given ${\varvec{\xi }}_{\varvec{\mathfrak u}}$, namely

$$\begin{aligned} \mathbb {E}\left\{ {{\left. {F}\,\right| \,{{\varvec{\xi }}_{{\varvec{\mathfrak u}}}}}}\right\} = \int F({\varvec{\xi }}_{\varvec{\mathfrak u}}{\varvec{\xi }}_{{\varvec{\mathfrak u}}_\sim }) \rho ({\varvec{\xi }}_{{\varvec{\mathfrak u}}_\sim }) d{\varvec{\xi }}_\sim . \end{aligned}$$

The decomposition (6.1) being orthogonal, the variance of F, $\mathbb {V}\left\{ {F}\right\}$, is decomposed into

$$\begin{aligned} \mathbb {V}\left\{ {F}\right\} = \sum _{\begin{array}{c} {\varvec{\mathfrak u}}\in \mathscr {D}\\ {\varvec{\mathfrak u}}\ne \emptyset \end{array}} V_{\varvec{\mathfrak u}}, \quad V_{\varvec{\mathfrak u}}= \mathbb {V}\left\{ {f_{\varvec{\mathfrak u}}}\right\} . \end{aligned}$$

(6.2)

The partial variance $V_{\varvec{\mathfrak u}}$ measures the contribution to $\mathbb {V}\left\{ {F}\right\}$ of the interactions between the variables ${\varvec{\xi }}_{\varvec{\mathfrak u}}$. Since there are $2^d$ such partial variances, the sensitivity analysis is usually reduced to a simpler characterization, based on first and total-order sensitivity indices associated to individual variables $\xi _i$ or group of variables ${\varvec{\xi }}_{\varvec{\mathfrak u}}$. The first-order, $\mathtt S_{\varvec{\mathfrak u}}$, and total-order, $\mathtt T_{\varvec{\mathfrak u}}$, sensitivity indices associated to ${\varvec{\xi }}_{\varvec{\mathfrak u}}$ are given by [16]

$$\begin{aligned} \mathbb {V}\left\{ {F}\right\} \mathtt S_{\varvec{\mathfrak u}}= \sum _{\begin{array}{c} {\varvec{\mathfrak v}}\in \mathscr {D}\\ {\varvec{\mathfrak v}}\subseteq {\varvec{\mathfrak u}} \end{array}} V_{\varvec{\mathfrak v}}= \mathbb {V}\left\{ {\mathbb {E}\left\{ {{\left. {F}\,\right| \,{{\varvec{\xi }}_{\varvec{\mathfrak u}}}}}\right\} }\right\} , \quad \mathbb {V}\left\{ {F}\right\} \mathtt T_{\varvec{\mathfrak u}}= \sum _{\begin{array}{c} {\varvec{\mathfrak v}}\in \mathscr {D}\\ {\varvec{\mathfrak v}}\cap {\varvec{\mathfrak u}}\ne \emptyset \end{array}} V_{\varvec{\mathfrak v}}= \mathbb {V}\left\{ {F}\right\} - \mathbb {V}\left\{ {\mathbb {E}\left\{ {{\left. {F}\,\right| \,{{\varvec{\xi }}_{{\varvec{\mathfrak u}}_\sim }}}}\right\} }\right\} . \end{aligned}$$

(6.3)

The first-order index $\mathtt S_{\varvec{\mathfrak u}}$ is then the fraction of variance that arises due to the individual variables in ${\varvec{\xi }}_{\varvec{\mathfrak u}}$ and their mutual interactions, only; the total-order sensitivity index $\mathtt T_{\varvec{\mathfrak u}}$ is the fraction of the variance arising due to the variables in ${\varvec{\xi }}_{\varvec{\mathfrak u}}$, their mutual interactions and all their interactions with all other variables in ${\varvec{\xi }}_{{\varvec{\mathfrak u}}_\sim }$. Clearly, $\mathtt S_{\varvec{\mathfrak u}}\le \mathtt T_{\varvec{\mathfrak u}}$, and $\mathtt T_{\varvec{\mathfrak u}}= 1 - \mathtt S_{{\varvec{\mathfrak u}}_\sim }$ [16].

1.2 Sensitivity Indices of PC Expansion

The partial variances and sensitivity indices of F can be easily computed from the PC expansion of F in the form (2.5)

$$\begin{aligned} F({\varvec{\xi }}) \approx \sum _{{\varvec{k}}\in \mathscr {K}} f_{\varvec{k}}\varPsi _{\varvec{k}}({\varvec{\xi }}), \end{aligned}$$

where $\mathscr {K}$ is the multi-index set of polynomial tensorizations as discussed in Sect. 2.1 (see (2.4)). Owing to the linear structure and polynomial form of the PC expansion, one can easily partition $\mathscr {K}$ into distinct subsets $\mathscr {K}_{\varvec{\mathfrak u}}$ contributing to the PC expansion of $f_{\varvec{\mathfrak u}}({\varvec{\xi }}_{\varvec{\mathfrak u}})$ [6, 42]. Specifically, the PC approximation of $f_{\varvec{\mathfrak u}}$ is

$$\begin{aligned} f_{\varvec{\mathfrak u}}({\varvec{\xi }}_{\varvec{\mathfrak u}}) \approx \sum _{{\varvec{k}}\in \mathscr {K}_{\varvec{\mathfrak u}}} f_{\varvec{k}}\varPsi _{\varvec{k}}({\varvec{\xi }}), \quad \mathscr {K}_{\varvec{\mathfrak u}}\doteq \{ {\varvec{k}}\in \mathscr {K}; k_{1\le i\le d} >0 \text{ if } i\in {\varvec{\mathfrak u}}, k_{1\le i\le d}=0 \text{ if } i\notin {\varvec{\mathfrak u}}\}. \end{aligned}$$

Observe that $\mathscr {K}_\emptyset = \{ (0 \cdots 0\})$. Then, $\forall {\varvec{\mathfrak u}}\in \mathscr {D}, {\varvec{\mathfrak u}}\ne \emptyset$, we have $V_{\varvec{\mathfrak u}}\approx \sum _{{\varvec{k}}\in \mathscr {K}_{\varvec{\mathfrak u}}} f_{\varvec{k}}^2$, while the variance of F is approximated by

$$\begin{aligned} \mathbb {V}\left\{ {F}\right\} \approx \sum _{{\varvec{k}}\in \mathscr {K}\setminus \mathscr {K}_\emptyset } f_{\varvec{k}}^2. \end{aligned}$$

The approximations for the first and total-order sensitivity indices $\mathtt S_{\varvec{\mathfrak u}}$ and $\mathtt T_{\varvec{\mathfrak u}}$ can be easily derived through (6.3), by taking the corresponding unions of subsets $\mathscr {K}_{\varvec{\mathfrak v}}$. For instance, in the case of singleton ${\varvec{\mathfrak u}}= \{i\}$, we have

$$\begin{aligned}&\mathtt S_{\{i\}} \approx \frac{ 1 }{\mathbb {V}\left\{ {F}\right\} } \sum _{{\varvec{k}}\in \mathscr {K}^\mathtt S_{{\{i\}}}} f_{\varvec{k}}^2, \quad \mathscr {K}^{\mathtt S}_{\{i\}} \doteq \{ {\varvec{k}}\in \mathscr {K}; k_i >0 , k_{j\ne i}=0 \}, \end{aligned}$$

(6.4)

$$\begin{aligned}&\mathtt T_{\{i\}} \approx \frac{ 1 }{\mathbb {V}\left\{ {F}\right\} } \sum _{{\varvec{k}}\in \mathscr {K}^\mathtt T_{{\{i\}}}} f_{\varvec{k}}^2, \quad \mathscr {K}^{\mathtt T}_{\{i\}} \doteq \{ {\varvec{k}}\in \mathscr {K}; k_i >0 \}. \end{aligned}$$

(6.5)