by using another basis specially adapted to the purpose, such as wavelets. Furthermore, by con-
struction, our emulator is only suitable for stochastic simulators whose response is correlated
throughout the input domain. If there is little to no correlation between the responses at dif-
ferent points in the input space, KLE (which is ultimately a dimension reduction technique),
would need infeasibly many modes to converge.

Our methodology can be extended in several ways. The computation of the sparse PCE ap-
proximation of the trajectories could be done jointly for all trajectories, instead of fitting each
trajectory separately and later discarding regressors. While in our study the dependence be-
tween KL-RV was not crucial for the accuracy of the emulator, an improved estimation of the
dependence structure would be desirable if for future models more modes turn out to be impor-
tant. Furthermore, an interesting question is under which circumstances one mode is enough to
represent the underlying stochasticity of stochastic simulators, and how the methodology can
be adapted to take advantage of this phenomenon. The general idea of representing trajectories
by their coefficients, and after dimension reduction modeling their joint distribution, can also
be applied outside spectral expansions, e.g. for Bayesian neural networks, at the cost of losing
some of the analytical properties following from orthogonality. Finally, in the spirit of common
random numbers (Pearce et al., 2022), the applicability of our stochastic emulator for purposes
such as optimization should be explored.

Acknowledgments

This paper is part of the project "Surrogate Modeling for Stochastic Simulators (SAMOS)"
funded by the Swiss National Science Foundation (Grant #200021_175524), whose support is
gratefully acknowledged.

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# Appendix

## 6.A Analytical derivations for extended KLE on PCE trajectories

The following is a detailed exposition of the analytical computations for extended KLE using the PCE basis in $L^2_{f_X}(\mathcal{D})$. A less detailed derivation for $L^2(\mathcal{D})$ can be found in Ramsay and Silverman (2005, Section 8.4.2).

We show in Section 6.A.1 that if the trajectories $x \mapsto \mathcal{M}(x, \omega)$ are represented by PCE, and extended KLE is applied, then the sample covariance function is a polynomial, the integral eigenvalue problem reduces to PCA in the expansion coefficients, and the eigenfunctions are polynomials. In Section 6.A.2, we show that the realizations of the random KLE coefficients can be determined analytically.

Let for $r = 1, \dots, R$

$$ \tilde{\mathcal{M}}_{r}^{\text{PCE}}(\boldsymbol{x}) = \sum_{\alpha \in \mathcal{A}} \tilde{a}_{\alpha} \psi_{\alpha}(\boldsymbol{x}) \qquad (6.30) $$

be the centered PCE trajectory computed from discrete evaluations of the trajectory $T_r$ (Eq. (6.13)). The sample covariance function is a polynomial given by

$$ \hat{c}(\boldsymbol{x}, \boldsymbol{x}') = \frac{1}{R-1} \sum_{r=1}^{R} \tilde{\mathcal{M}}_{r}^{\text{PCE}}(\boldsymbol{x}) \tilde{\mathcal{M}}_{r}^{\text{PCE}}(\boldsymbol{x}'). \qquad (6.31) $$

### 6.A.1 Analytical solution of the extended KLE eigenvalue problem

Computing an extended KLE in the function space $L^2_{f_X}(\mathcal{D})$ corresponds to solving the following eigenproblem:

$$ \langle \hat{c}(\boldsymbol{x}, \cdot), \phi_i(\cdot) \rangle_{L^2_{f_X}(\mathcal{D})} = \int_{\mathcal{D}} \hat{c}(\boldsymbol{x}, \boldsymbol{x}') \phi_i(\boldsymbol{x}') f_X(\boldsymbol{x}') \, d\boldsymbol{x}' = \lambda_i \phi_i(\boldsymbol{x}). \qquad (6.32) $$

Since $\hat{c}$ is polynomial, also the eigenfunctions will be polynomials and can be represented in terms of the PCE basis as follows:

$$ \phi_i(\boldsymbol{x}) = \sum_{\alpha \in \mathcal{A}} b_{\alpha}^{(i)} \psi_{\alpha}(\boldsymbol{x}). \qquad (6.33) $$
Solving Eq. (6.32) reduces to finding $(\lambda_i, (b_\alpha^{(i)})_{\alpha \in \mathcal{A}})$ for $i = 1, \dots, M$.

Dropping the $i$-subscript of the eigenfunction for convenience, we compute

$$
\begin{align*}
\int_D \hat{c}(\mathbf{x}, \mathbf{x}') \phi(\mathbf{x}') f_X(\mathbf{x}') \, d\mathbf{x}' &= \int_D \frac{1}{R-1} \sum_{r=1}^R \tilde{\mathcal{M}}_r^{\text{PCE}}(\mathbf{x}) \tilde{\mathcal{M}}_r^{\text{PCE}}(\mathbf{x}') \phi(\mathbf{x}') f_X(\mathbf{x}') \, d\mathbf{x}' \\
&= \frac{1}{R-1} \sum_{r=1}^R \tilde{\mathcal{M}}_r^{\text{PCE}}(\mathbf{x}) \int_D \tilde{\mathcal{M}}_r^{\text{PCE}}(\mathbf{x}') \phi(\mathbf{x}') f_X(\mathbf{x}') \, d\mathbf{x}' \\
&= \frac{1}{R-1} \sum_{r=1}^R \tilde{\mathcal{M}}_r^{\text{PCE}}(\mathbf{x}) \int_D \left( \sum_{\alpha \in \mathcal{A}} \tilde{a}_\alpha^r \psi_\alpha(\mathbf{x}') \right) \left( \sum_{\beta \in \mathcal{A}} b_\beta \psi_\beta(\mathbf{x}') \right) f_X(\mathbf{x}') \, d\mathbf{x}' \\
&= \frac{1}{R-1} \sum_{r=1}^R \tilde{\mathcal{M}}_r^{\text{PCE}}(\mathbf{x}) \left( \sum_{\alpha \in \mathcal{A}} \tilde{a}_\alpha^r b_\alpha \right) \quad (\text{orthonormality of PCE basis}) \\
&= \frac{1}{R-1} \sum_{r=1}^R \left( \sum_{\beta \in \mathcal{A}} \tilde{a}_\beta^r \psi_\beta(\mathbf{x}) \right) \left( \sum_{\alpha \in \mathcal{A}} \tilde{a}_\alpha^r b_\alpha \right) \\
&= \sum_{\beta \in \mathcal{A}} \left( \frac{1}{R-1} \sum_{r=1}^R \tilde{a}_\beta^r \left( \sum_{\alpha \in \mathcal{A}} \tilde{a}_\alpha^r b_\alpha \right) \right) \psi_\beta(\mathbf{x}).
\end{align*}
$$

The eigenvalue problem reduces to

$$
\sum_{\beta \in \mathcal{A}} \left( \frac{1}{R-1} \sum_{r=1}^{R} \tilde{a}_{\beta}^{r} \underbrace{\left( \sum_{\alpha \in \mathcal{A}} \tilde{a}_{\alpha}^{r} b_{\alpha} \right)}_{=(\tilde{a}^{r})^{T} \mathbf{b} \in \mathbb{R}} \right) \psi_{\beta}(x) \stackrel{!}{=} \lambda \sum_{\beta \in \mathcal{A}} b_{\beta} \psi_{\beta}(x).
$$

Because the PCE basis functions $\psi_\beta$ are of different orders, we can rewrite this into matrix form:

$$
\frac{1}{R-1} \sum_{r=1}^{R} \begin{pmatrix} \tilde{a}_{\beta_1}^r (\tilde{a}^r)^T \\ \tilde{a}_{\beta_2}^r (\tilde{a}^r)^T \\ \vdots \\ \tilde{a}_{\beta_P}^r (\tilde{a}^r)^T \end{pmatrix} \mathbf{b} = \left( \frac{1}{R-1} \sum_{r=1}^{R} \tilde{a}^r (\tilde{a}^r)^T \right) \mathbf{b} = \left( \frac{1}{R-1} \tilde{a} \ \tilde{a}^T \right) \mathbf{b} \stackrel{!}{=} \lambda \mathbf{b}, \quad (6.34)
$$

where $\tilde{a}$ is the $P \times R$-matrix of active and centered PCE trajectory coefficients. Defining the matrix $\boldsymbol{\Sigma} = \frac{1}{R-1} \tilde{a} \tilde{a}^T$, which is the empirical covariance matrix of the centered PCE coefficients, we see that Eq. (6.34) is nothing else than principal component analysis (PCA) on the coefficients.

Note that it was necessary to apply KLE in $L^2_{f_X}(\mathcal{D})$ to arrive at this equation, since the PCE basis is orthonormal in $L^2_{f_X}(\mathcal{D})$ but in general not in $L^2(\mathcal{D})$.

The solution vectors $\mathbf{b}^{(i)}$ to the eigenvalue problem $\boldsymbol{\Sigma}\mathbf{b}^{(i)} = \lambda_i\mathbf{b}^{(i)}$ are the PCE coefficients of the KLE eigenfunctions: $\phi_i(\mathbf{x}) = \sum_{\alpha \in \mathcal{A}} b_\alpha^{(i)} \psi_\alpha(\mathbf{x})$. Since the PCE basis is orthonormal, and assuming that the eigenvectors $\mathbf{b}^{(i)}$ are normalized to unit norm, it follows that the eigenfunctions $\{\phi_i\}$ are orthonormal.
6.A.2 Analytical computation of the realizations of the KL-RV

Let $\lambda_i$ be an eigenvalue and

$$ \phi_i(x) = \sum_{\alpha \in \mathcal{A}} b_{\alpha}^{(i)} \psi_{\alpha}(x) \qquad (6.35) $$

the associated eigenfunction expressed in the PCE basis. The projection of the PCE trajectories onto the KLE basis is given by

$$ \begin{aligned} \xi_i^r &= \frac{1}{\sqrt{\lambda_i}} \int_{\mathcal{D}} \tilde{M}_r^{\text{PCE}}(\mathbf{x}) \phi_i(x) f_{\mathbf{X}}(\mathbf{x}) \, d\mathbf{x} \\ &= \frac{1}{\sqrt{\lambda_i}} \sum_{\alpha \in \mathcal{A}} \tilde{a}_{\alpha}^r b_{\alpha}^{(i)} = \frac{1}{\sqrt{\lambda_i}} (\tilde{a}^r)^T \mathbf{b}^{(i)} \in \mathbb{R}. \end{aligned} $$

Let $\tilde{a} \in \mathbb{R}^{P \times R}$ the matrix of coefficients of centered PCE trajectories and $\mathbf{b} \in \mathbb{R}^{P \times M}$ the matrix of PCE coefficients of the KLE functions. Then we can compute the matrix $\Xi \in \mathbb{R}^{M \times R}$ of KLE coefficient realizations by

$$ \Xi = \left( \text{diag} \left( \frac{1}{\sqrt{\lambda_1}}, \dots, \frac{1}{\sqrt{\lambda_M}} \right) \mathbf{b} \right)^T \tilde{a}. \qquad (6.36) $$
# CHAPTER 7

## Discussion and conclusions

We live in exciting times. Computers can execute billions of calculation steps per second and thus dramatically extend our human thinking capabilities (with respect to anything that can be expressed by numbers, at least). When sitting at a laptop or desktop computer, or even holding a smartphone, this power is literally at our fingertips. The challenge is now to do something meaningful with this power: to contribute to the better understanding or even to the solution of real problems, while at the same time doing so efficiently with regard to time and energy. There is no doubt that carefully prepared computer simulations can help address this challenge, advance science and guide decisions – as long as the ever-present uncertainties of our real world are not neglected in the process.

Dealing with these uncertainties is precisely the task of the field of uncertainty quantification (UQ), which provides methods for investigating the influence of stochasticity on the model in question. In order to be useful, UQ methods must be efficient, well-tested, and accessible to practitioners, i.e., creators of computational models that might themselves not be experts in UQ.

This thesis made contributions towards these goals. Its overarching topic, surrogate modeling, is a general methodology for improving the efficiency of UQ techniques by approximating the original simulator with a less expensive replacement model, based on well-chosen model evaluations and regularity assumptions. In particular, we investigated the popular sparse polynomial chaos expansions (PCE) method. To make the large literature on sparse PCE more accessible, we performed an extensive literature survey and benchmark of various methods, which resulted in problem-dependent recommendations. We also investigated two new surrogate models. The first, Poincaré chaos expansions, is a spectral expansion technique in the spirit of sparse PCE, but using a non-polynomial basis that is especially well suited for sensitivity analysis when partial derivatives of the model are available. The second method is a surrogate for stochastic simulators, i.e., computational models with latent stochasticity. Taking the random function view, we use trajectory samples to approximate the spatial as well as the stochastic behavior of the model.
We now discuss the conclusions and outlooks associated to each of these topics. These discussions are not meant to merely repeat the conclusions of the respective papers, but to summarize the results and to point out connections with related work from a more global perspective. In particular, we will discuss the nature and role of sparse expansions, and share considerations regarding benchmark design.

# 7.1 Sparse polynomial chaos expansions

The literature on sparse regression-based PCE, a spectral surrogate modeling method making use of orthonormal polynomials and compressed sensing ideas, has grown tremendously in the recent years. To provide an overview of the various available approaches and to benchmark their performance with respect to the global mean-square error between surrogate and true model, we conducted an extensive literature review and a thorough benchmark, resulting in the two publications presented in Chapters 3 and 4.

## 7.1.1 Conclusions

**Literature review and benchmark.** In Chapter 3, we identified a framework for the sparse PCE computation procedure consisting of the following components (see Figure 3.1 for an illustration): construction of the initial basis, construction of the initial experimental design, solution of the sparse regression problem (i.e., computation of the coefficients), evaluation of an error measure or model selection criterion, adaptation of the basis, and sequential enrichment of the experimental design. A large part of the literature on sparse PCE targets one or several of these components. Furthermore, there are a number of methods outside this framework that are meant to improve the sparse PCE computation procedure, e.g., by pre-conditioning the regression problem, or by identifying a more suitable coordinate system in the input domain.

Because the number of components and associated proposed methods is large, and the number of possible combinations even larger, we had to restrict our benchmark to a few components and to a selection of most promising methods. In Chapter 3, we investigated the impact of the initial experimental design and of the sparse regression solver on the global approximation quality of the final result. For the experimental design, we investigated the schemes MC, LHS, coherence-optimal sampling, D-optimal sampling, and near-optimal sampling. For the solvers, we considered least-angle regression (LARS), orthogonal matching pursuit (OMP), subspace pursuit (SP), Bayesian compressive sensing (BCS), and spectral projected gradient-$\ell^1$ (SPGL1). We tested 11 benchmark models of various dimensionality and complexity repeatedly for several sizes of the experimental design, while using a fixed basis.

We found that the performance of the methods varied considerably across the models and over the repetitions. To aggregate the results from the various models, experimental design sizes, and repetitions, we divided the experiments into four classes according to two criteria: small/large size of experimental design, and low/high dimensionality of the model. For each combination of sparse regression solver and sampling scheme, we then counted the fraction of experiments that resulted in an optimal or close-to-optimal error, and visualized this ranking.
Based on this ranking, we were able to identify solvers and sampling schemes that consistently showed better or worse performance. Our conclusions are explained in detail in Section 3.4. To highlight a few results: we found that the solvers OMP, LARS, and SPGL1 performed comparatively poorly, whereas BCS and SP based on leave-one-out cross-validation performed rather well. Regarding the sampling schemes, LHS was the best choice for high-dimensional models. For low-dimensional models, coherence-optimal sampling performed well for large design sizes, and the near-optimal scheme performed best for problems with a rather small basis.

**Basis adaptivity and automatic selection.** We continued the benchmark in Chapter 4, where we investigated the component of basis adaptivity. This introduces an iterative aspect into the PCE computation procedure: after computing an expansion, new regressors are chosen and added to the basis. Then, the expansion is recomputed and the procedure is repeated until convergence. We investigated three methods from the literature, namely degree- and q-norm, forward neighbor, and anisotropic degree basis adaptivity. The latter two choose the new regressors based on the previously computed solution. We used LHS for the experimental design, and paired the basis-adaptive schemes with the solvers LARS, OMP, SP, SPGL1, and BCS. For the evaluation of the results, we conducted the same aggregation as previously described, and focused on the relative performance, measured in the percentage of runs in which the combination attained an error within a certain factor of the best possible error on the same experimental design achieved by any other combination.

We found that the three basis-adaptive methods we tested generally produced a more accurate solution than a static (i.e., fixed) basis. This was especially pronounced for low-dimensional models, where in a large number of cases, basis adaptivity achieved an error that was more than one order of magnitude smaller than the one of a static basis. In contrast, in the case of high-dimensional models with a small experimental design (i.e., the case of very scarce data), the static basis produced more robust solutions than the adaptive basis schemes.

Among the tested solvers, SP and BCS performed well, whereas SPGL1 and OMP did not. This is consistent with the performance of these solvers in Chapter 3. Aggregated over all experiments, forward-neighbor basis adaptivity together with leave-one-out-based SP showed good performance most often. However, even this combination was far from the smallest possible error in a rather large percentage of cases.

Therefore, we implemented the automatic selection of one out of several sparse solutions generated by different methods based on cross-validation, which was able to achieve close-to-optimal errors, and outperformed any single recommendation of solver and basis adaptivity scheme. This result is another example for the good performance of cross-validation for model selection, which has already been utilized successfully e.g. for determining important regressors in sparse PCE (Liu et al., 2020b) and for ensemble modeling (Viana et al., 2009; Parisi et al., 2021).

**The nature of sparse expansions.** The basis adaptivity scheme “forward neighbor” provides insight into the sparsity pattern of sparse PCE. This scheme views the set of regressors as a directed graph; see Section 4.2.3.2 for a detailed explanation. This scheme would not be successful if the target sparse expansion contained isolated regressors whose backward neighbors
all have negligible coefficients. This suggests that expansions of real-world models possess a
specific sparsity pattern: building on the exposition of forward neighbor basis adaptivity by
Jakeman et al. (2015) who state that “the magnitude of the ancestors of a PCE coefficient is a
reasonable indicator of the size of the child coefficient”, we hypothesize that sparse expansions
are actually dense with respect to the important input variables.<fnref n="1" /> The expansions are sparse
only within a larger basis of candidate regressors, such as a total-order basis: unimportant input
variables and low interaction between variables cause a substantial part of this candidate basis
to have negligible coefficients.

In the classification of models for global sensitivity analysis by Kucherenko et al. (2011), this
corresponds to function type A, i.e., functions of small effective dimension, of which they say that
“Type A functions are probably the most common type of functions encountered in practice”.

Following this reasoning, the type of basis is not necessarily essential for achieving a sparse
expansion: any tensor-product basis that includes the constant regressor in each univariate
basis would similarly allow sparse expansions. Of course, the type of basis has an influence on
compressibility, i.e., on the decay in coefficient magnitude. We will talk more about this topic
in Section 7.2 in the context of Poincaré chaos expansions.

Furthermore, this has implications for the design of benchmark cases to test sparse PCE algo-
rithms. It is not uncommon for methodological papers to use artificial test cases, such as the
“manufactured sparse PCE” model constructed as the weighted sum of a few randomly selected
polynomials, for a first test of the algorithm in question. However, for the result to be indica-
tive of the real-world performance of the tested method, such models should follow the sparsity
pattern described above.

**The role of sparsity.** We want to discuss the role of sparsity in sparse PCE using the example of BCS. In our benchmark, this solver performed especially well for high-dimensional models or for small experimental designs, in other words: whenever information is scarce. However, for low-dimensional models and a large design, it did not perform well. BCS uses a hierarchical Bayesian formulation of the regression problem, in which the parameters of the prior distribution of the coefficients are again random variables, resulting in a sparsity-encouraging effective prior for the coefficients. Empirically, we often found that BCS generates sparser solutions than other solvers, which fits the conclusions of Babacan et al. (2010) comparing BCS with another Bayesian method.

However, as opposed to compressed sensing, in UQ we are usually not interested in the sparsest
possible solution. Instead, sparsity is a device to achieve reasonably accurate approximations
under limited data. Sparsity avoids overfitting by only including the most relevant regressors
into the model, thus acting as a regularizer. Viewed from another angle, sparsity is a tool for
computational efficiency, allowing the solution of underdetermined systems of linear equations.
However, real computational models are almost never sparse in the PCE basis. Whenever there
is enough information to make it feasible (such as in the case of low-dimensional models with a

<footnote n="1">¹To give an example: if in a three-dimensional model the regressor with multi-index (3, 7, 0) has a non-negligible coefficient, then typically the same is true for all the smaller ones: {(*i*, *j*, 0) : *i* ≤ 3, *j* ≤ 7}. In the literature on least-squares approximation, such a set is also called *downward closed*.</footnote>
large experimental design), a denser model will in general give a better approximation than a
sparse one, which might explain the worse relative performance of BCS in this case.

**Benchmark design.** In an active methodological research field such as surrogate modeling, and in particular sparse PCE, it is important to perform comparative studies in order to identify the most promising methods for further research, to provide an incentive to study the reasons for better or worse performance of certain methods, and to create recommendations for practitioners looking for advice which method to use. Regarding the implementation of the benchmark, we found that it is important to clearly define the scope of the study, to include a sufficiently large number of test models, to use repetitions for the (stochastic) sampling of the experimental design, and to appropriately aggregate the results in order to see the big picture. For the latter, we developed a novel way to accumulate and visualize the relative performance of different methods using stacked bar plots. We concluded that the most relevant information to display is the fraction of experiments in which the respective method achieves a global validation error that is within a certain factor of the error of the best-performing method.

**Literature reviews.** Literature reviews can help make an established, specialized field of research more accessible to new researchers looking for the main directions and open questions. They can also help discover connections between topics and retrace the evolution of certain ideas. We hope that with our two review and benchmark papers, we were able to provide such guidance and overview for sparse regression-based PCE at least for the current point in time.

7.1.2 Outlook

The literature review and benchmarks could be extended in several ways. In our benchmarks, we investigated the effect of the initial experimental design, the sparse solver, and the basis adaptivity scheme. Further work could target the other components of the sparse PCE computation procedure, e.g., sequential enrichment of the experimental design, pre-conditioning, or the identification of potentially more suitable coordinates for the expansion.

Furthermore, for the components we tested, we were not able to consider all available methods. For example, in the solver benchmark we did not include the class of heuristic stepwise regression algorithms, which usually show superior performance in the numerical tests of the respective authors. Among the sampling schemes, techniques such as Leja sequences and Christoffel sparse approximation appear promising. At the same time, new methods are continuously being developed. Preferably, the benchmark should be continued on the same (or even on an extended) suite of models to allow for the fair comparison of methods.

Going one step further, the ideal solution would be a free public benchmarking service similar to the “RPrepo” platform, which Rozsas and Slobbe (2019) developed for black-box reliability algorithms. This service collects benchmark problems of various complexity and provides a server that allows the remote testing of new algorithms under exactly equal conditions.

Similar to a benchmark, a literature review becomes outdated as new methods continue to be published. A suitable format for keeping it up to date would be a live online document that is
regularly updated to collect the most relevant literature.

We have mentioned the difficulty of selecting a representative set of benchmark models, and the associated difficulty of inferring recommendations on which methods to use depending on the model characteristics and the available experimental design. For his benchmark of Sobol' sensitivity analysis methods, Becker (2020) has found an elegant solution called *metafunctions*: test cases are randomly constructed from a set of univariate basis functions, using specific distributions for the number of dimensions, the coefficient values, and the number of interaction terms. The resulting test functions are representative of real-world models with respect to the Pareto principle and the sparsity-of-effects principle (see also Section 2.5.3). While artificial, this principled way of constructing test cases allows detailed insight into the interplay between model properties and method performance.

Among the methods we tested, the solver BCS appeared to be especially promising. Apart from its good performance in the small-data regime, it is based on Bayesian inference, which has two beneficial consequences that are worth studying further: first, through the specification of the prior distributions it is possible to encode prior information about the magnitude of coefficients. This has already been utilized by Shao et al. (2017) in a stepwise regression algorithm. Second, the posterior distribution of the coefficients contains information about the confidence in the resulting estimates given the data, which can be used for active learning (Zhou et al., 2019; Pan et al., 2020).

Turning to model selection, with our automatic selection scheme choosing between several expansions generated by various sparse PCE algorithms, we have provided more evidence that cross-validation performs well for this purpose. While there is the danger of overfitting if used excessively (Cawley and Talbot, 2010), model selection by cross-validation, and more generally ensemble learning, are powerful concepts with great potential for improving surrogate modeling.

Among the sampling methods, coherence-optimal sampling showed good performance for low-dimensional models. However, so far this sampling scheme has only been analyzed and implemented for the uniform and the Gaussian distribution (Hampton and Doostan, 2015b). It is not yet known how to efficiently construct such a sample for other types of distributions or for dependent inputs.

More generally, many of the available sampling schemes are based either on optimizing the space-filling quality of the resulting sample (e.g., maximin LHS), or on optimizing a property of the regression matrix (e.g., D-optimal, coherence-optimal, or near-optimal sampling). While the authors proposing the latter schemes for PCE (Diaz et al., 2018; Hampton and Doostan, 2015b; Alemazkoor and Meidani, 2018a) demonstrate superior performance for these algorithms, Weise et al. (2022) find in their recent benchmark that for the considered test functions, LHS (with some adaptations improving the space-filling property) outperforms various optimal sampling schemes, including coherence- and D-optimal sampling. Therefore, it would be of special interest to further investigate in a principled way the impact of matrix properties on the approximation quality of the resulting PCE. Answering this question is particularly challenging because the result also crucially depends on the benchmark model and its input distribution.

Finally, we have listed several insights into the nature of sparse expansions. This information might be utilized to develop new algorithms.
## 7.2 Poincaré chaos expansions

In Chapter 5, we investigated Poincaré chaos expansions (PoinCE), a new type of spectral expansion especially suited for global sensitivity analysis. Similar to sparse PCE, we construct a tensor-product basis and compute the coefficients by sparse regression. The Poincaré basis is in general not polynomial (with the exception of the basis associated to the Gaussian distribution, which coincides with the Hermite polynomials). Poincaré functions are similar to polynomials in that they always include the constant function, and that the *n*-th order function has *n* roots. They arise as eigenfunctions to a differential eigenvalue problem that was originally derived for computing Poincaré constants for univariate measures with bounded support.

### 7.2.1 Conclusions

While the Poincaré basis functions associated to the Gaussian distribution are the well-known Hermite polynomials, the basis associated to uniform input consists of cosine functions. Hence, PoinCE constitutes a connecting link between the two well-known spectral methods PCE and Fourier expansion with the potential of generalizing results for each of them.

Poincaré basis functions are orthonormal and have the special property that the partial derivatives of the basis form again an orthonormal basis with respect to the same measure. In Chapter 5, we were able to show that this is in fact a characterizing property, i.e., every basis with this property must be a Poincaré basis.

As a consequence, we derived an analytical formula for the computation of DGSM, which extends a formula by Sudret and Mai (2015) to the non-Gaussian case and provides insight into the inequality bound between total Sobol' indices and DGSM: namely, this bound is tight if the expansion does not include relevant terms of order larger than one in the respective variable. Otherwise, the gap can be significant, which we also observed in the numerical examples.

The characterizing property of orthogonal derivatives also provides a convenient way to utilize model derivative evaluations for the computation of expansion coefficients. We implemented an algorithm that combines the partial derivative evaluations with respect to the different input variables to compute accurate estimates of Sobol' indices. We found that this procedure generally resulted in more accurate and less biased estimates than the ones computed by PCE or by Poincaré chaos expansions (PoinCE) based on model evaluations alone.

When taking partial derivatives of a multivariate expansion with respect to a certain input variable, all regressors that do not involve this variable will vanish. This reduces the number of unknown coefficients particularly for high-dimensional problems and rather low-dimensional expansions, and might be one possible explanation for the good performance of derivative-based PoinCE.

We also compared the performance of PCE, PoinCE, and derivative-based PoinCE as a global surrogate model, based on the $L_{fX}^2(D)$-error. Surprisingly, we found that here PCE outperforms the other two methods, particularly for larger experimental design sizes. This difference in performance between sensitivity analysis and surrogate modeling requires more investigation.
Furthermore, we found that the two expansions PCE and PoinCE have almost the same num-
ber of nonzero coefficients for a given experimental design size. This supports the hypothesis
mentioned in Section 7.1 that expansions are sparse because of unimportant input variables and
low interaction order, not because of true sparsity in the basis. However, given that the same
level of sparsity in PCE and PoinCE leads to a different quality of global approximation, we
speculate that polynomial functions might be better suited than Poincaré chaos expansions to
approximate the model of interest with a low-degree expansion. In other words, it seems that
the model is less compressible in the Poincaré basis than in the PCE basis.

7.2.2 Outlook

While PoinCE is unlikely to replace PCE in the near future, it is an interesting example of a
non-polynomial orthonormal basis that is relatively easy to construct. The direct comparison
allows us to obtain insights also about PCE, e.g., regarding compressibility.

An open question about PoinCE is the performance gap between sensitivity analysis, where it performs well, and surrogate modeling, where it is outperformed by PCE. This should be investigated on a larger number of benchmark models. Answering this question might provide further insight into what is important for accurately estimating Sobol' sensitivity indices.

Furthermore, in our computations we have used either model evaluations or model partial deriva-
tives to compute the PoinCE coefficients. We are working on combining the information of both
sources in the spirit of derivative-enhanced PCE (Jakeman et al., 2015; Peng et al., 2016).

In general, the existence of the Poincaré basis with its favorable orthogonality property raises the question whether we could construct other non-polynomial orthonormal bases with beneficial properties for certain applications.

## 7.3 Spectral surrogate for stochastic simulators

In Chapter 6 we proposed a surrogate model for stochastic simulators based on the random function view of stochastic simulators. Fixing the seed, the response of a stochastic simulator becomes a deterministic function, which we surrogate using sparse PCE. Then, we perform Karhunen-Loève expansion (KLE) on the resulting trajectories and characterize the stochastic behavior by statistical inference of the random coefficients of the KLE (KL-RV).

7.3.1 Conclusions

The method approximates the full stochastic simulator including mean function, covariance func-
tion, marginal distributions, and trajectories. It does not assume that the stochastic simulator
is a Gaussian random field. Our numerical examples show that non-Gaussian distributions for
the KL-RV outperform Gaussian ones.

KLE requires the solution of an integral eigenvalue equation involving the covariance function, which is generally expensive to solve. In our case, several factors contribute to circumventing
this problem: we perform extended KLE in $L_{fX}^2(D)$ and utilize an orthonormal polynomial basis to approximate the trajectories. Here, sparsity is key for obtaining accurate approximations at low cost. The covariance function is computed from the sample trajectories. All of this results in the integral eigenvalue equation becoming a cheap-to-solve finite-dimensional matrix eigenvalue problem.

In our numerical examples, we observe that $M = 1$ term in the KL expansion is often sufficient to emulate a large part of the variance. We explain this phenomenon using the Hoeffding-Sobol' decomposition of the model considering both its explicit and latent parameters – assuming it exists – and find that one mode is sufficient if there are no interaction terms between the explicit and the latent group.

In general, by identifying the number of modes through the explained variance, our method can discover the effective dimensionality of the latent stochastic space.

### 7.3.2 Outlook

The random function view of stochastic simulators shares similarities with the “common random numbers” approach used in optimization (Pearce et al., 2022). This connection should be explored further; possibly, our stochastic emulator could be applied for optimization. Furthermore, the precise connection of our approach to kernel PCA (Sarma et al., 2008) should be investigated.

Navarro Jimenez et al. (2017) and Zhu and Sudret (2021c) have discussed Sobol' sensitivity analysis for stochastic simulators. This methodology can easily be applied to our simulator, and can even be computed analytically from the PCE trajectories.

While KLE works especially well for Gaussian random fields, our approach is not restricted to the latter, because our methodology characterizes the dependent distribution of the random coefficients of KLE. We use the copula framework to infer the dependence, which could benefit from the introduction of additional copula families that better describe the type of dependence found in KL-RV of stochastic simulators.

Finally, the observation that only one mode covers a large amount of the explained variance is worth exploring further. This phenomenon should be tested and analyzed for more complex and realistic engineering stochastic simulators.

## 7.4 Final conclusion

We have investigated several types of sparse spectral surrogate models in this thesis, with the goals of testing their performance in practice, giving recommendations, and understanding their behavior in more detail. Spectral methods, which utilize an orthonormal basis of $L_{fX}^2(D)$ for approximation, are global methods adapted to the distribution of the input random vector. The orthogonality of the basis is an important property for computational reasons (see e.g. Eq. (2.12) and the concept of isotropy in Section 2.5.2).
We focused on sparse expansions computed by sparse regression. Here, sparsity is a tool for computational efficiency, allowing the accurate calculation of coefficients even for rather small experimental design sizes, which is the data regime typically encountered in UQ. With our literature review and benchmarks of sparse polynomial chaos expansions, and the two methodological contributions of sparse Poincaré chaos expansions for global sensitivity analysis and our new trajectory-based surrogate for stochastic simulators, we have addressed the above-mentioned goals of performance evaluation, recommendation, and improved understanding, as discussed in the preceding sections.

Furthermore, to disseminate the results of our research, we have already made the most promising sparse regression solvers from our benchmark available in the general-purpose UQ software UQLab (Marelli et al., 2021a). We are planning to add further functionality related to basis adaptivity and automatic selection for PCE, as well as a module for the treatment of stochastic simulators.

Finally, we want to comment on the interdisciplinary nature of the field of uncertainty quantification, specifically on its location at the intersection of engineering and applied mathematics. During our research, we have seen various examples for exciting, rigorous results stemming from thorough mathematical investigation. To name a few examples, there is coherence-optimal sampling (Candès and Plan, 2011; Hampton and Doostan, 2015b), the inequality relation between Sobol' indices and DGSM (Lamboni et al., 2013), and the surprising fact that there is no polynomial basis associated to the lognormal distribution (Ernst et al., 2012). Rigorous mathematical analysis has been essential in obtaining these results.

On the other hand, the theoretical performance and proven validity of methods is not always indicative of their performance in practice. For example, the bound on the number *N* of samples in compressive sensing (Theorem 1 in Section 2.5.2) involves a constant whose value is not known, making the bound useful mainly for large problem sizes *P*. For the case of rather small degree, this theorem does not necessarily guarantee that *N* < *P* is sufficient for sparse recovery.

Furthermore, engineering research can be ahead of the associated mathematical results by several decades. For example, Xiu and Karniadakis (2002) proposed to compute PCE associated to distributions from the Wiener-Askey scheme ten years before Ernst et al. (2012) provided the formal proof of convergence. Also, polynomial approximation by least-squares regression (Isukapalli, 1999; Berveiller et al., 2006) has been used long before in the recent years Cohen and Migliorati (2017) and colleagues were able to prove rigorous bounds for its performance.

These small examples illustrate that applied mathematics and engineering research should go hand in hand to further advance the field of uncertainty quantification, finally resulting in methods that are able to efficiently obtain reliable and meaningful solutions to real-world problems from computer simulations.
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