Analyticity and sparsity in uncertainty quantification
for PDEs with Gaussian random field inputs

To indicate some of the mathematical issues which are considered in this book, consider in the interval ${D}=(0,1)$ and in a probability space $(\Omega,\mathcal{A},\mathbb{P})$, a GRF $g:\Omega\times{D}\to\mathbb{R}$ which takes values in $L^{\infty}({D})$. That is to say, that the map $\omega\mapsto g(\omega,\cdot)$ is an element of the Banach space $L^{\infty}({D})$. Formally, at this stage, we represent realizations of the random element $g\in L^{\infty}({D})$ with a representation system $(\psi_{j})_{j=1}^{J}\subset L^{\infty}({D})$ in affine-parametric form

$$g(\omega,x)=\sum_{j=1}^{J}y_{j}(\omega)\psi_{j}(x)\,;,$$

(1.1)

where the coefficients $(y_{j})_{j=1}^{J}$ are assumed to be i.i.d. standard normal random variables (RVs for short) and $J$ may be a finite number or infinity. Representations such as (1.1) are widely used both in the analysis and in the numerical simulation of random elements $g$ taking values in a function space. The coefficients $y_{j}(\omega)$ being standard normal RVs, the sum $\sum_{j=1}^{J}y_{j}\psi_{j}(x)$ may be considered as a parametric deterministic map $g:\mathbb{R}^{J}\to L^{\infty}({D})$. The random element $g(\omega,x)$ in (1.1) can then be obtained by evaluating this deterministic map in random coordinates, i.e., by sampling it in Gaussian random vectors $(y_{j}(\omega))_{j=1}^{J}\in\mathbb{R}^{J}$.

Gaussian random elements as inputs for PDEs appear in particular, in coefficients of diffusion equations. Consider, for illustration, in ${D}$, and for given $f\in L^{2}({D})$, the boundary value problem: find a random function $u:\Omega\to V$ with $V:=\{w\in H^{1}({D}):w(0)=0\}$ such that

$$f(x)+\frac{\,\mathrm{d}}{\,\mathrm{d}x}\left(a(x,\omega)\frac{\,\mathrm{d}}{\,\mathrm{d}x}u(x,\omega)\right)=0\quad\mbox{in}\quad{D}\;,\quad a(1,\omega)u^{\prime}(1,\omega)=\bar{f}\;.$$

(1.2)

Here, $a(x,\omega)=\exp(g(x,\omega))$ with GRF $g:\Omega\to L^{\infty}({D})$, and $\bar{f}:=F(1)$ with

$$F(x):=\int_{0}^{x}f(\xi)\,\mathrm{d}\xi\in V,\quad x\in{D}.$$

In order to dispense with summability and measurability issues, let us temporarily assume that the sum in (1.1) is finite, with $J\in\mathbb{N}$ terms. We find that a random solution $u$ of the problem must satisfy

$$u^{\prime}(x,\omega)=-\exp(-g(x,\omega))F(x),\quad x\in{D},\omega\in\Omega\;.$$

Inserting (1.1), this is equivalent to the parametric, deterministic family of solutions $u(x,{\boldsymbol{y}}):{D}\times\mathbb{R}^{J}\to\mathbb{R}$ given by

$$u^{\prime}(x,{\boldsymbol{y}})=-\exp(-g(x,{\boldsymbol{y}}))F(x),\quad x\in{D},{\boldsymbol{y}}\in\mathbb{R}^{J}\;.$$

(1.3)

Hence

$$\|u^{\prime}(\cdot,{\boldsymbol{y}})\|_{L^{2}({D})}=\|\exp(-g(\cdot,{\boldsymbol{y}}))F\|_{L^{2}({D})}\;,\quad{\boldsymbol{y}}\in\mathbb{R}^{J}\;,$$

which implies the (sharp) bounds

$$\|u^{\prime}(\cdot,{\boldsymbol{y}})\|_{L^{2}({D})}\left\{\begin{array}[]{l}\geq\exp(-\|g(\cdot,{\boldsymbol{y}})\|_{L^{\infty}({D})})\|F\|_{L^{2}({D})}\\ \leq\exp(\|g(\cdot,{\boldsymbol{y}})\|_{L^{\infty}({D})})\|F\|_{L^{2}({D})}.\end{array}\right.$$

Due to the homogeneous Dirichlet condition at $x=0$, up to an absolute constant the same bounds also hold for $\|u(\cdot,{\boldsymbol{y}})\|_{V}$.

It is evident from the explicit expression (1.3) and the upper and lower bounds, that for every parameter ${\boldsymbol{y}}\in\mathbb{R}^{J}$, the solution $u\in V$ exists. However, we can not, in general, expect uniform w.r.t. ${\boldsymbol{y}}\in\mathbb{R}^{J}$ a-priori estimates, also of the higher derivatives, for smoother functions $x\mapsto g(x,{\boldsymbol{y}})$ and $x\mapsto f(x)$. Therefore, the parametric problem (1.2) is nonuniformly elliptic, [28, 70]. In particular, also a-priori error bounds for various discretization schemes will contain this uniformity w.r.t. ${\boldsymbol{y}}$. The random solution will be recovered from (1.3) by inserting for the coordinates $y_{j}$ samples of i.i.d. standard normal RVs.

This book focuses on developing a regularity theory for countably-parametric solution families ${u(\cdot,{\boldsymbol{y}}):{\boldsymbol{y}}\in\mathbb{R}^{J}}$ with a particular emphasis on the case $J=\infty$. This allows for arbitrary Gaussian random fields $g(\cdot,\omega)$ in (1.2). Naturally, our results also cover the finite-parametric setting where the number $J$ of random parameters is finite, but may be very large. Then, all constants in our error estimates are either independent of the parameter dimension $J$ or their dependence of $J$ is explicitly indicated. Previous works [8, 9, 59] addressed the $\ell^{p}$-summability of the Wiener-Hermite PC expansion coefficients of solution families $\{u(\cdot,{\boldsymbol{y}}):{\boldsymbol{y}}\in\mathbb{R}^{\infty}\}\subset V$ for the forward problem, based on moment bounds of derivatives of parametric solutions w.r.t. GM. Estimates for these coefficients and, in particular, for the summability, were obtained in [72, 8, 9, 59, 71]. In these references, all arguments were based on real-variable, bootstrapping arguments with respect to ${\boldsymbol{y}}$.

We make the following contributions to the area computational UQ for PDEs with GRF inputs. First, we provide novel proofs of some of the sparsity results in [72, 9, 8] of the infinite-dimensional parametric forward solution map to PDEs with GRF inputs. The presently developed proof technique is based on holomorphic continuation and complex variable arguments in order to bound derivatives of parametric solutions, and their coefficients in Wiener-Hermite PC expansions. This is in line with similar arguments in the so-called “uniform case” in [39, 32]. There, the random parameters in the representation of the input random fields range in compact subsets of ${\mathbb{R}}$. Unlike in these references, in the present text due to the Gaussian setup the parameter domain ${\mathbb{R}}^{\infty}$ is not compact. This entails significant modifications of mathematical arguments as compared to those in [39, 32].

Contrary to the analysis in [8, 9, 59], where parametric regularity results were obtained by real-variable arguments combined with induction-based bootstrapping with respect to the derivative order, the present text develops derivative-free, complex variable arguments which allow directly to obtain bounds of the Wiener-Hermite PC expansion coefficients of the parametric solutions in scales of Sobolev and Besov spaces in the physical domain ${D}$ in which the parametric PDE is posed. They also allow to treat in a unified manner parametric regularity of the solution map in several scales of Sobolev and Kondrat’ev spaces in the physical domain ${D}$ which is the topic of Section 3.8, resulting in novel sparsity results for the solution operators to linear elliptic and parabolic PDEs with GRF inputs in scales of Sobolev and Besov spaces. We apply the quantified holomorphy of parametric solution families to PDEs with GRF inputs and preservation of holomorphy under composition, to problems of Bayesian PDE inversion conditional on noise observation data in Section 5, establishing in particular quantified parametric holomorphy of the corresponding Bayesian posterior.

We construct deterministic sparse-grid interpolation and quadrature methods for the parametric solution with convergence rate bounds that are free from the curse of dimensionality, and that afford possibly high convergence rates, given a sufficient sparsity in the Wiener-Hermite PC expansion of the parametric solutions. For sampling strategies in deterministic numerical quadrature, our findings show improved convergence rates, as compared to previous results in this area. Additionally, our novel sparsity results provided in scales of function spaces of varying spatial regularity enable us to construct apriori multilevel versions of sparse-grid interpolation and quadrature, with corresponding approximation rate bounds which are free from the curse of dimensionality, and explicit in terms of the overall number of degrees of freedom. Lastly, and in contrast to previous works, leveraging the preservation of holomorphy under compositions with holomorphic maps, our holomorphy-based arguments enable us to establish that our algorithms and bounds are applicable to posterior distributions in Bayesian inference problems involving GRF or Besov priors, as developed in [48, 94] and the references there.

We prove quantified holomorphy of countably-parametric solution families of linear elliptic and parabolic PDEs. The parameter range equals ${\mathbb{R}}^{\infty}$, corresponding to countably-parametric representations of GRF input data, taking values in a separable locally convex space, in particular, Hilbert or Banach space of uncertain input data, endowed for example with a Gaussian product measure $\gamma$ on ${\mathbb{R}}^{\infty}$.

The results established in this text and the related bounds on partial derivatives w.r.t. the parameters in Karhunen-Loève or Lévy-Cieselsky expansions of uncertain GRF inputs imply convergence rate bounds for several families of computational methods to numerically access these parametric solution maps. Importantly, we prove that in terms of $n\geq 1$, an integer measure of work and memory, an approximation accuracy $O(n^{-a})$ for some parameter $a>0$ can be achieved, where the convergence rate $a$ depends on the approximation process and on the amount of sparsity in the Wiener-Hermite PC expansion coefficients of the GRF under consideration. In the terminology of computational complexity, a prescribed numerical tolerance $\varepsilon>0$ can be reached in work and memory of order $O(\varepsilon^{-1/a})$. In particular, the convergence rate $a$ and the constant hidden in the Landau $O(\cdot)$ symbol do not depend on the dimension of the space of active parameters involved in the approximations which we construct. The approximations developed in the present text are constructive and linear and can be realized computationally by deterministic algorithms of so-called “stochastic collocation” or “sparse-grid” type. Error bounds are proved in $L^{2}$-type Bochner spaces with respect to the GM $\gamma$ on the input data space of the PDE, in natural Hilbert or Banach spaces of solutions of the PDEs under consideration. Here, it is important to notice that the sparsity of the Wiener-Hermite PC expansion coefficients used in constructive linear approximation algorithms and in estimating convergence rates, takes the form of weighted $\ell^{2}$-summability, but not $\ell^{p}$-summability as in best $n$-term approximations [72, 9, 8]. Furthermore, $\ell^{p}$-summability results are implied from the corresponding weighted $\ell^{2}$-summability ones.

All approximation rates for deterministic sampling strategies in the present text are free from the so-called curse of dimensionality, a terminology coined apparently by R.E. Bellmann (see [17]). The rates are in fact only limited by the sparsity of the Wiener-Hermite PC expansion coefficients of the deterministic, countably-parametric solution families. In particular, dimension-independent convergence rates $>1/2$ are possible, provided a sufficient Wiener-Hermite PC expansion coefficient sparsity, that the random inputs feature sufficient pathwise regularity, and the affine representation system (being a tight frame on space of admissible input realizations) are stable in a suitable smoothness scale of inputs.

We briefly describe the structure and content of the present text.

In Section 2, we collect known facts from functional analysis and GM theory which are required throughout this text. In particular, we review constructions and results on GMs on separable Hilbert and Banach spaces. Special focus will be on constructions via countable products of univariate GMs on countable products of real lines. We also review assorted known results on convergence rates of Lagrangian finite elements for linear, second order, divergence-form elliptic PDEs in polytopal domains ${D}$ with Lipschitz boundary $\partial{D}$.

In Section 3, we address the analyticity and sparsity for elliptic divergence-form PDEs with log-Gaussian coefficients. In Section 3.1, we introduce a model linear, second order elliptic divergence-form PDE with log-Gaussian coefficients, with variational solutions in the “energy space” $H^{1}_{0}({D})$. This equation was investigated with parametric input data in a number of references in recent years [38, 39, 72, 32, 9, 8, 42, 59, 83, 112]. It is considered in this work mainly to develop the holomorphic approach to establish our mathematical approach to parametric holomorphy and sparsity of Wiener-Hermite PC expansions of parametric solutions in a simple setting, and to facilitate comparisons with the mentioned previous works and results. We review known results on its well-posedness in Section 3.1, and Lipschitz continuous dependence on the input data in Section 3.2. We discuss regularity results for parametric coefficients in Section 3.3. Sections 3.4 and 3.5 describe uncertainty modelling by placing GMs on sets of admissible, countably parametric input data, i.e., formalizing mathematically aleatoric uncertainty in input data. Here, the Gaussian series introduced in Section 2.5 will be seen to take a key role in converting operator equations with GRF inputs to infinitely-parametric, deterministic operator equations. The Lipschitz continuous dependence of the solutions on input data from function spaces will imply strong measurability of corresponding random solutions, and render well-defined the uncertainty propagation, i.e., the push-forward of the GM on the input data. In Section 3.6, we connect the quantified holomorphy of the parametric, deterministic solution manifold $\{u({\boldsymbol{y}}):{\boldsymbol{y}}\in{\mathbb{R}}^{\infty}\}$ in the space $H^{1}_{0}({D})$ with a sparsity (weighted $\ell^{2}$-summability and $\ell^{p}$-summability) of the coefficients $(u_{\boldsymbol{\nu}})_{{\boldsymbol{\nu}}\in{\mathcal{F}}}$ of the ($H^{1}_{0}({D})$-valued) Wiener-Hermite PC expansion. With this methodology in place, we show in Section 3.7 how to obtain holomorphic regularity of the parametric solution family $\{u({\boldsymbol{y}}):{\boldsymbol{y}}\in{\mathbb{R}}^{\infty}\}$ in Sobolev spaces $H^{s}({D})$ of possibly high smoothness order $s\in{\mathbb{N}}$ and how to derive from here the corresponding sparsity. The argument is self-contained and provides parametric holomorphy for any differentiation order $s\in{\mathbb{N}}$ in a unified way, in domains ${D}$ of sufficiently high regularity and for sufficiently high almost sure regularity of coefficient functions. In Section 3.8, we extend these results for linear second order elliptic differential operators in divergence form in a bounded polygonal domain ${D}\subset\mathbb{R}^{2}$. Here, corners are well-known to obstruct high almost sure pathwise regularity in the usual Sobolev and Besov spaces in ${D}$ for both, PC coefficients and parametric solutions. Therefore, we develop summability of the Wiener-Hermite PC expansion coefficients $(u_{\boldsymbol{\nu}})_{{\boldsymbol{\nu}}\in{\mathcal{F}}}$ of the random solutions in terms of corner-weighted Sobolev spaces, originating with V.A. Kondrat’ev (see, e.g., [61, 24, 90] and the references there). In Section 3.9, we briefly recall some known related results [32, 37, 38, 39, 72, 10, 11, 9, 8] on $\ell^{p}$-summability and weighted $\ell^{2}$-summability of the generalized PC expansion coefficients of solutions to parametric divergence-form elliptic PDEs, as well as applications to best $n$-term approximation.

In Section 4, we investigate sparsity of the Wiener-Hermite PC expansions coefficients of holomorphic functions. In Section 4.1, we introduce a concept of $({\boldsymbol{b}},\xi,\delta,X)$-holomorphy of parametric deterministic functions on the parameter domain $\mathbb{R}^{\infty}$ taking values in a separable Hilbert space $X$. This concept is fairly broad and covers a large range of parametric PDEs depending on log-Gaussian distributed data. In order to extend the results and the approach to bound Wiener-Hermite PC expansion coefficients via quantified holomorphy beyond the simple, second order diffusion equation introduced in Section 3, we address sparsity of the Wiener-Hermite PC expansions coefficients of $({\boldsymbol{b}},\xi,\delta,X)$-holomorphic functions. In Section 4.2, we show that composite functions of a certain type are $({\boldsymbol{b}},\xi,\delta,X)$-holomorphic under certain conditions. The significance of such functions is that they cover solution operators of a collection of linear, elliptic divergence-form PDEs in a unified way along with structurally similar PDEs with log-Gaussian random input data. This will allow to apply the ensuing results on convergence rates of deterministic collocation and quadrature algorithms to a wide range of PDEs with GRF inputs and functionals on their random solutions. In Section 4.3, we analyze some examples of holomorphic functions which are solutions to certain PDEs, including linear elliptic divergence-form PDEs with parametric diffusion coefficient, linear parabolic PDEs with parametric coefficient, linear elastostatics equations with log-Gaussian modulus of elasticity, Maxwell equations with log-Gaussian permittivity.

In Section 5, we apply the preceding abstract results on parametric holomorphy to establish quantified holomorphy of countably-parametric, posterior densities of corresponding BIPs where the uncertain input of the forward PDE is a countably-parametric GRF taking values in a separable Banach space of inputs. As an example, we analyze the BIP for the parametric diffusion coefficient of the diffusion equation with parametric log-Gaussian inputs.

In Section 6, we discuss deterministic interpolation and quadrature algorithms for approximation and numerical integration of $({\boldsymbol{b}},\xi,\delta,X)$-holomorphic functions. Such algorithms are necessary for the approximation of certain statistical quantities (expectations, statistical moments) of the parametric solutions with respect to a GM on the parameter space. The proposed algorithms are variants and generalizations of so-called “stochastic collocation” or “sparse-grid” type approximation, and proved to outperform sampling methods such as MC methods, under suitable sparsity conditions on coefficients of the Wiener-Hermite PC expansion of integrands. In the quadrature case, they are also known as “Smolyak quadrature” methods. Their common feature is a) the deterministic nature of the algorithms, and b) the possibility of achieving convergence rates $>1/2$ independent of the dimension of parameters and therefore the curse of dimensionality is broken. They offer, in particular, the perspective of deterministic numerical approximations for GRFs under nonlinear pushforwards (being realized via the deterministic data-to-solution map of the PDE of interest). The decisive analytic property to be established are dimension-explicit estimates of individual Wiener-Hermite PC expansion coefficients of parametric solutions, and based on these, sharp summability estimates of norms of the coefficients of Wiener-Hermite PC expansion of parametric, deterministic solution families are given. In Sections 6.1 and 6.2, we construct sparse-grid Smolyak-type interpolation and quadrature algorithms. In Sections 6.3 and 6.4, we prove the convergence rates of interpolation and quadrature algorithms for $({\boldsymbol{b}},\xi,\delta,X)$-holomorphic functions.

Section 7 is devoted to multilevel interpolation and quadrature of parametric holomorphic functions. We construct deterministic interpolation and quadrature algorithms for generic $({\boldsymbol{b}},\xi,\delta,X)$-holomorphic functions. For linear second order elliptic divergence-form PDEs with log-Gaussian coefficients, the results on the weighted $\ell^{2}$-summability of the Wiener-Hermite PC expansion coefficients of parametric, deterministic solution families with respect to corner-weighted Sobolev spaces on spatial domain ${D}$ finally also allow to analyze methods for constructive, deterministic linear approximations of parametric solution families. Here, a truncation of Wiener-Hermite PC expansions is combined with approximating the Wiener-Hermite PC expansion coefficients in the norm of the “energy space” $H^{1}_{0}({D})$ of these solutions from finite-dimensional approximation spaces which are customary in the numerical approximation of solution instances. Importantly, required approximation accuracies of the Wiener-Hermite PC expansion coefficients $u_{\boldsymbol{\nu}}$ will depend on the relative importance of $u_{\boldsymbol{\nu}}$ within the Wiener-Hermite PC expansion. This observation gives rise to multilevel approximations where a prescribed overall accuracy in mean square w.r.t. the GM $\gamma$ with respect to $H^{1}_{0}({D})$ will be achieved by a ${\boldsymbol{\nu}}$-dependent discretization level in the physical domain. Multilevel approximation and integration and the corresponding error estimates will be developed in this section in an abstract setting: Besides $({\boldsymbol{b}},\xi,\delta,X)$-holomorphy, it is neccessary to require an assumption on the discretization error in the physical domain in the form of stronger holomorphy of the approximation error in this discretization. A combined assumption for guaranteeing constructive multilevel approximations is formulated in Section 7.1. In Section 7.2 we introduce multilevel algorithms for interpolation and quadrature of $({\boldsymbol{b}},\xi,\delta,X)$-holomorphic functions, and discuss work models and choices of discretization levels. A key for the sparse-grid integration and interpolation approaches is to efficiently numerically allocate discretization levels to Wiener-Hermite PC expansion coefficients. We develop such an approach in Section 7.3. It is based on greedy searches and suitable thresholding of (suitable norms of) Wiener-Hermite PC expansion coefficients and on a-priori bounds for these quantities which are obtained by complex variable arguments. In Sections 7.4 and 7.5, we establish convergence rate bounds of multilevel interpolation and quadrature algorithms for $({\boldsymbol{b}},\xi,\delta,X)$-holomorphic functions. In Section 7.6, we verify the abstract hypotheses of the sparse-grid multilevel approximations for the forward and inverse problems for concrete linear elliptic and parabolic PDEs on corner-weighted Sobolev spaces (Kondrat’ev spaces) with log-GRF inputs. In Section 7.7, we briefly recall some results from [43] (see also, [45] for some corrections) on linear multilevel (fully discrete) interpolation and quadrature in abstract Bochner spaces based on weighted $\ell^{2}$-summabilities. These results are subsequently applied to parametric divergence-form elliptic PDEs and to parametric holomorphic functions.

Additional to the real numbers $\mathbb{R}$, the complex numbers $\mathbb{C}$, and the positive integers $\mathbb{N}$, we set $\mathbb{R}_{+}:=\{x\in\mathbb{R}\,:\,x\geq 0\}$ and $\mathbb{N}_{0}:=\{0\}\cup\mathbb{N}$. We denote by ${\mathbb{R}}^{\infty}$ the set of all sequences ${\boldsymbol{y}}=(y_{j})_{j\in{\mathbb{N}}}$ with $y_{j}\in{\mathbb{R}}$, and similarly define $\mathbb{C}^{\infty}$, $\mathbb{R}_{+}^{\infty}$ and $\mathbb{N}_{0}^{\infty}$. Both, ${\mathbb{R}}^{\infty}$ and ${\mathbb{C}}^{\infty}$, will be understood with the product topology from ${\mathbb{R}}$ and ${\mathbb{C}}$, respectively. For ${\boldsymbol{\alpha}}$, ${\boldsymbol{\beta}}\in{\mathbb{N}}_{0}^{d}$, $d\in\mathbb{N}\cup\{\infty\}$, the inequality ${\boldsymbol{\beta}}\leq{\boldsymbol{\alpha}}$ is understood component-wise, i.e., ${\boldsymbol{\beta}}\leq{\boldsymbol{\alpha}}$ if and only if $\beta_{j}\leq\alpha_{j}$ for all $j$.

Denote by ${\mathcal{F}}$ the countable set of all sequences of nonnegative integers ${\boldsymbol{\nu}}=(\nu_{j})_{j\in{\mathbb{N}}}$ such that $\operatorname{supp}({\boldsymbol{\nu}})$ is finite, where $\operatorname{supp}({\boldsymbol{\nu}}):=\{j\in{\mathbb{N}}:\nu_{j}\neq 0\}$ denotes the “support” of the multi-index ${\boldsymbol{\nu}}$. Similarly, we define $\operatorname{supp}({\boldsymbol{\rho}})$ of a sequence ${\boldsymbol{\rho}}\in{\mathbb{R}}^{\infty}_{+}$. For ${\boldsymbol{\nu}}\in{\mathcal{F}}$, and for a sequence ${\boldsymbol{b}}=(b_{j})_{j\in{\mathbb{N}}}$ of positive real numbers, the quantities

$${\boldsymbol{\nu}}!:=\prod_{j\in{\mathbb{N}}}\nu_{j}!\,,\qquad|{\boldsymbol{\nu}}|:=\sum_{j\in{\mathbb{N}}}\nu_{j},\qquad\text{and}\qquad{\boldsymbol{b}}^{\boldsymbol{\nu}}:=\prod_{j\in{\mathbb{N}}}b_{j}^{\nu_{j}}$$

are finite and well-defined.

For a multi-index ${\boldsymbol{\alpha}}\in{\mathbb{N}}_{0}^{d}$ and a function $u({\boldsymbol{x}},{\boldsymbol{y}})$ of ${\boldsymbol{x}}\in{\mathbb{R}}^{d}$ and parameter sequence ${\boldsymbol{y}}\in{\mathbb{R}}^{\infty}$ we use the notation $D^{\boldsymbol{\alpha}}u({\boldsymbol{x}},{\boldsymbol{y}})$ to indicate the partial derivatives taken with respect to ${\boldsymbol{x}}$. The partial derivative of order ${\boldsymbol{\alpha}}\in{\mathbb{N}}_{0}^{\infty}$ with respect to ${\boldsymbol{y}}$ of finite total order $|{\boldsymbol{\alpha}}|=\sum_{j\in{\mathbb{N}}}\alpha_{j}$ is denoted by $\partial^{\boldsymbol{\alpha}}u({\boldsymbol{x}},{\boldsymbol{y}})$. In order to simplify notation, we will systematically suppress the variable ${\boldsymbol{x}}\in{D}\subset\mathbb{R}^{d}$ in mathematical expressions, except when necessary. For example, instead $\int_{D}v({\boldsymbol{x}})\,\mathrm{d}{\boldsymbol{x}}$ we will write $\int_{D}v\,\mathrm{d}{\boldsymbol{x}}$, etc. For a Banach space $X$, we denote by $X_{{\mathbb{C}}}:=X+{\rm i}X$ the complexification of $X$. The space $X_{\mathbb{C}}$ is also a Banach space endowed with the (minimal, among several possible equivalent ones, see [91]) norm $\|x_{1}+{\rm i}x_{2}\|_{X_{\mathbb{C}}}:=\sup_{0\leq t\leq 2\pi}\|x_{1}\cos t-x_{2}\sin t\|_{X}$. The space $X^{\infty}$ is defined in a similar way as ${\mathbb{R}}^{\infty}$.

By $\mathcal{L}(X,Y)$ we denote the vector space of bounded, linear operators between to Banach spaces $X$ and $Y$. With $\mathcal{L}_{{\rm is}}(X,Y)$ we denote the subspace of boundedly invertible, linear operators from $X$ to $Y$.

For a function space $X({{D}})$ defined on the domain ${{D}}$, if there is no ambiguity, when writing the norm of $x\in X({{D}})$ we will omit ${{D}}$, i.e., we write $\|x\|_{X}$ instead of $\|x\|_{X({{D}})}$.

For $0<p\leq\infty$ and a finite or countable index set $J$, we denote by $\ell^{p}(J)$ the quasi-normed space of all ${\boldsymbol{y}}=(y_{j})_{j\in J}$ with $y_{j}\in\mathbb{R}$, equipped with the quasi-norm $\|{\boldsymbol{y}}\|_{{\ell^{p}(J)}}:=\big{(}\sum_{j\in J}|y_{j}|^{p}\big{)}^{1/p}$ for $p<\infty$, and $\|{\boldsymbol{y}}\|_{{\ell^{\infty}(J)}}:=\sup_{j\in J}|y_{j}|$. Sometimes, we make use of the abbreviation $\ell^{p}=\ell^{p}(J)$ in a particular context if there is no misunderstanding of the meaning. We denote by $({\boldsymbol{e}}_{j})_{j\in J}$ the standard basis of $\ell^{2}(J)$, i.e., ${\boldsymbol{e}}_{j}=(e_{j,i})_{i\in J}$ with $e_{j,i}=1$ for $i=j$ and $e_{j,i}=0$ for $i\not=j$.

An important mathematical ingredient in a number of applications, in particular in UQ, Bayesian PDE inversion, risk analysis, but also in statistical learning theory applied to input-output maps for PDEs, is the construction of measures on function spaces. A particular interest is in GMs on separable on Hilbert or Banach or, more generally, on locally convex spaces of uncertain input data for PDEs. Accordingly, we review constructions of such measures, in terms of suitable bases of the input spaces. This implies, in particular, separability of the spaces of admissible PDE inputs or, at least, the uncertain input data being a separably-valued random element of otherwise nonseparable spaces (such as, e.g., $L^{\infty}({{D}})$) of valid inputs for the PDE of interest.

Let $(\Omega,\mathcal{A},\mu)$ be a measure space and $1\leq p\leq\infty$. Recall that the normed space $L^{p}(\Omega,\mu)$ is defined as the space of all $\mu$-measurable functions $u$ from $\Omega$ to $\mathbb{R}$ such that the norm

$$\|u\|_{L^{p}(\Omega,\mu)}:=\ \left(\int_{\Omega}|u(x)|^{p}\,\,\mathrm{d}\mu(x)\right)^{1/p}<\infty.$$

When $p=\infty$ the norm of $u\in L^{\infty}(\Omega,\mu)$ is given by

$$\|u\|_{L^{\infty}(\Omega,\mu)}:=\underset{x\in\Omega}{\operatorname{ess\,sup}}|u(x)|.$$

If $\Omega\subset{\mathbb{R}}^{m}$ and $\mu$ is the Lebesgue measure, we simply denote these spaces by $L^{p}(\Omega)$.

Throughout this section, $X$ will denote a real separable and locally convex space with Borel $\sigma$-field $\mathcal{B}(X)$ and with dual space $X^{*}$.

We review and present constructions of GMs $\gamma$ on $X$.