# Sparse Representations of Solutions to a class of Random Boundary Value
Problems

Fang Yang  Fang Yang, Department of Mathematics
Zhejiang Normal University
China<EMAIL_ADDRESS>,  Min Chen  Min Chen, School of Mathematical
Sciences, Shanxi University.
Academy of Mathematics and Systems Science, Chinese Academy of Sciences
China<EMAIL_ADDRESS>,  Jiecheng Chen  Jiecheng Chen, Department of
Mathematics
Zhejiang Normal University
China<EMAIL_ADDRESS>,  Ying Zhang  Ying Zhang, Macau Institute of
Systems Engineering
Macau University of Science and Technology
China<EMAIL_ADDRESS>and  Tao Qian  Tao QIAN, Macao Center for
Mathematical Sciences
Macau University of Science and Technology
China<EMAIL_ADDRESS>
###### Abstract.

We introduce certain sparse representation methods, named as stochastic pre-
orthogonal adaptive Fourier decomposition 1 and 2 (SPOAFD1 and SPOAFD2) to
solve the Dirichlet boundary value problem and the Cauchy initial value
problem of random data. To solve the stochastic boundary value problems the
sparse representation is, as the initial step, applied to the random boundary
data. Due to the semigroup property of the Poisson and the heat kernel, each
entry of the expanding series can be lifted up to compose a solution of the
Dirichlet and the Cauchy initial value problem, respectively. The sparse
representation gives rise to analytic as well as numerical solutions to the
problems with high efficiency.

*Corresponding author.

MSC 2020 Classification: 41A99; 60H15

Keywords: Stochastic PDEs; Numerical PDEs; Sparse Representation; Reproducing
Kernel Hilbert Space; Stochastic Hardy space; Bochner space today

##  1\. Introduction

In many science and engineering problems, the stochastic partial differential
equations (SPDEs) are more realistic than deterministic PDEs. It is
particularly important to give effective numerical solutions of SPDEs ([26,
31, 32, 29, 4, 18, 63, 30, 20]). It is assumed that
$(\Omega,\mathcal{F},\mathbb{P})$ is a probability space and $D$ is a region
of $\mathbb{R}^{n}$ throughout this paper. In this paper we restrict ourselves
to deal with the cases where $D$ is either the unit ball or the upper-half
space. We will concern elliptic PDEs with random coefficients or random
forcing terms ([28, 65]), which have the form

(1.1) $\left\\{\begin{aligned} &\nabla(\kappa(x,\omega)\nabla
u(x,\omega))=f(x,\omega),\leavevmode\nobreak\ x\in D,{\rm
a.s.}\leavevmode\nobreak\ \omega,\\\
&u(x,\omega)=g(x,\omega),\leavevmode\nobreak\ x\in\partial
D,\leavevmode\nobreak\ {\rm a.s.}\leavevmode\nobreak\
\omega,\end{aligned}\right.$

where $\kappa,f$ and $g$ are $\mathbb{R}$-valued functions on $D\times\Omega$.
For such SPDEs a number of numerical methods have been developed, by using
methods including the Monte Carlo finite element method, Karhunen-Loève (K-L)
expansions, finite element methods (FEMs), stochastic Galerkin FEM and
stochastic collocation FEM ([28, 66, 33, 6]), etc.

Besides the above type SPDEs the method we will introduce may also be used to
the parabolic type PDEs with random boundary- or/and initial-values,
including, in particular, incompressible Navier Stokes equations ([55]). They
have the form

(1.2) $\left\\{\begin{aligned}
&\partial_{t}u(t,x,\omega)=\mathcal{L}[u(t,x,\omega),\omega],\leavevmode\nobreak\
\leavevmode\nobreak\ x\in D,\leavevmode\nobreak\ \leavevmode\nobreak\
t\in\mathbb{R}_{+},\leavevmode\nobreak\ {\rm a.s.}\leavevmode\nobreak\
\omega\in\Omega,\\\ &u(0,x,\omega)=u_{0}(x,\omega),\leavevmode\nobreak\
\leavevmode\nobreak\ x\in D,\leavevmode\nobreak\ {\rm
a.s.}\leavevmode\nobreak\ \omega\in\Omega,\\\
&\mathcal{B}[u(t,x,\omega)]=h(t,x,\omega),\leavevmode\nobreak\ x\in\partial
D,\leavevmode\nobreak\ {\rm a.s.}\leavevmode\nobreak\
\omega\in\Omega,\end{aligned}\right.$

where $\mathcal{L}$ is a linear or non-linear differential operator, $x\in D$
is the spatial coordinate, $t$ is the time variable, and $\mathcal{B}$ is a
linear differential operator. In the classical method the problem (1.2) can be
reduced to a certain type of deterministic PDEs and a system of stochastic
ordinary differential equations by the so called dynamically orthogonal
representation proposed in [55, 17].

Solutions of SPDEs are again random processes. Hence, it is meaningful to
study how to decompose a random process explicitly with efficient convergence.
Before we introduce stochastic adaptive Fourier decomposition (SAFD) type
sparse representations ([42]), we first recall two classical techniques for
representing random processes ([36]).

Among Wiener chaos spectral expansions there is Fourier-Hermite expansion
being a classical technique for Brownian motion ([22, 68, 25, 24, 27]). Assume
$\xi(\omega)\triangleq\\{\xi_{i}(\omega)\\}_{i=1}^{n}$ is a set of independent
standard Gaussian random variables. For a finite index
$\alpha=(\alpha_{1},\alpha_{2},\cdots,\alpha_{n})\in\mathbb{N}^{n}$ , the
multivariate Hermite polynomial of $x=(x_{1},\cdots,x_{n})\in\mathbb{R}^{n}$
can be written as:

(1.3) $H_{\alpha}(x)\triangleq\prod^{n}_{i=1}H_{\alpha_{i}}(x_{i}),$

where each $H_{\alpha_{i}}(x_{i})$ is a one dimensional Hermite polynomials of
$x_{i}.$ One dimensional Hermite polynomials form an orthonormal system

$\langle
h_{n},h_{m}\rangle_{\nu}\triangleq\int_{-\infty}^{+\infty}h_{n}h_{m}d\nu=\delta^{n}_{m}$

that induces the orthonormality of the higher dimensional systems. Suppose
that $u(\xi(\omega))$ is a function of the random variables $\xi(\omega)$ with
$\mathbb{E}_{\omega}|u(\xi(\omega))|^{2}<+\infty$, then the function $x\mapsto
u(x)$ belongs to $L^{2}(\mathbb{R}^{n},d\nu^{n})$ and $u(\xi(\omega))$ having
a Fourier-Hermite expansion converging in $L^{2}(\Omega,d\mathbb{P})$:

(1.4) $u(\xi(\omega))=\sum_{\alpha}u_{\alpha}H_{\alpha}(\xi(\omega)),$

where the Fourier coefficients
$u_{\alpha}\triangleq\mathbb{E}[u(\xi(\cdot))H_{\alpha}(\xi(\cdot))].$

For $x\in D$, assume $u(x,s,\omega)\triangleq f(x,W_{s}(\omega))$ is a
function of the Brownian motion $\\{W_{s}(\omega)\\}_{0\leq s\leq T}$ with
zero means. Analogous to the finite dimensional Fourier-Hermite expansion
(1.4), it admits a similar expansion. Before we state it we recall the
definition of the second order processes:

######  Definition 1.

([28]) A stochastic processes $\\{u(x,\omega)\\}_{x\in D}$ is called a second
order processes if for almost everywhere $x\in D$,

(1.5)
$\mathbb{E}|u(x,\cdot)|^{2}=\int_{\Omega}|u(x,\omega)|^{2}d\mathbb{P}(\omega)<\infty.$

Define the set of finite multi-indices as

$\mathcal{J}\triangleq\\{\alpha\in{\mathbb{N}}^{{\mathbb{N}}_{+}}:\leavevmode\nobreak\
|\alpha|=\sum_{i}\alpha_{i}<\infty\\}.$

For $\alpha\in\mathcal{J}$, define the Hermite polynomial of $\xi(\omega)$
similar to (1.3) by the tensor product

(1.6)
$T_{\alpha}(\xi(\omega))\triangleq\prod_{i=1}^{\infty}H_{\alpha_{i}}(\xi_{i}(\omega)),$

which is called a random Wick polynomial. The above product comprises only
finite number of factors since $|\alpha|<\infty$. Given a second order process
$u(x,s,\omega)\triangleq f(x,W_{s}(\omega)),$ assumed to be a function of the
Brownian motion $\\{W_{s}(\omega)\\}_{0\leq s\leq T}$, Cameron and Martin
([27, 13]) provided a Fourier expansion as the following,

(1.7)
$u(x,s,\omega)=\sum_{\alpha\in\mathcal{J}}u_{\alpha}(x,s)T_{\alpha}(\xi(\omega)),$

in $L^{2}(\Omega,d\mathbb{P})$, where $T_{\alpha}(\xi(\cdot))$ is defined by
(1.6) and
$u_{\alpha}(x,s)\triangleq\mathbb{E}[u(x,s,\cdot)T_{\alpha}(\xi(\cdot))]$. The
above expansion is usually called Wiener chaos expansion of $u(x,s,\omega)$
that represents the randomness of the solution analytically by a set of random
bases with deterministic coefficients. This method reduces the SPDEs to
solving the deterministic PDEs of the coefficients ([27, 69]).

The other common approach for expanding random processes is K-L expansion
([28]), which applies to second order processes $u$ restricted to a space
$L^{2}(\Omega,L^{2}(D)).$

######  Definition 2.

([28, 42]) Suppose $D\subset\mathbb{R}^{n}.$ The space
$L^{2}(\Omega,L^{2}(D))$ is defined to be the Hilbert space consisting of all
$L^{2}(D)$-valued random variables $v:D\times\Omega\rightarrow\mathbb{R}$
satisfying

(1.8)
$\|v\|_{L^{2}(\Omega,L^{2}(D))}^{2}\triangleq\int_{\Omega}\int_{D}|v(x,\omega)|^{2}dxd\mathbb{P}<\infty.$

For brevity, we also write $\mathcal{N}=L^{2}(\Omega,L^{2}(D)).$ The reference
[42] uses the same type of spaces on the classical domains defined in terms of
Fourier expansions with random coefficients, being equivalent with the above
definition due to the Plancherel Theorem. [42] develops stochastic sparse
kernel-representations of stochastic reproducing kernel Hilbert spaces and
stochastic Hilbert spaces with a dictionary. The spaces
$L^{2}(\Omega,L^{2}(D))$ have also been studied as Bochner spaces which are
Hilbert spaces with an inner product defined through Bochner integrals. See
[23, 67, 57] and the references therein.

Let $u(x,\omega)$ be a process in $L^{2}(\Omega,L^{2}(D))$ and
$C(x,y)\triangleq{\rm Cov}(u(x,\cdot),u(y,\cdot))$ the covariance of the $u.$
By using $C(x,y)$ one defines the integral operator $\mathcal{C}$

(1.9)
$\mathcal{C}(\phi)(x)\triangleq\int_{D}C(x,y)\phi(y)dy,\leavevmode\nobreak\
x\in D,\leavevmode\nobreak\ \phi\in L^{2}(D).$

Setting $\tilde{u}(x)\triangleq\mathbb{E}(u(x,\cdot)),$ we have the K-L
expansion in the sense of $L^{2}(\Omega,L^{2}(D))$,

(1.10)
$u(x,\omega)=\tilde{u}(x)+\sum_{j=1}^{\infty}\sqrt{v_{j}}\phi_{j}(x)\xi_{j}(\omega),$

where the Fourier coefficients
$\xi_{j}(\omega)=\frac{1}{\sqrt{v_{j}}}\langle{u(\cdot,\omega)-\tilde{u}(\cdot),\phi_{j}(\cdot)}\rangle_{L^{2}(D)}$
, and $\\{v_{j},\phi_{j}\\}_{j}$ are the eigen-pairs of the operator
$\mathcal{C}$. Combined with the Galerkin method, the K-L Galerkin procedure
is a type of Galerkin method that employs, as basis functions, the empirical
eigenfunctions of the K-L decomposition ([38, 39]). It consequently reduces
the original PDE to a small set of ordinary differential equations or
algebraic equations. A boundary optimal control problem of the Navier-Stokes
equation is solved by this technique in [39]. Several complexity reduction
methods have been proposed, such as model reduction methods for SPDEs ([11,
34, 35]), sparse approximation methods for high-dimensional approximations
([12, 56, 37, 62]).

Both the above reviewed Fourier-Hermite expansion and the Karhunen-Loève
decomposition offer series expansions of the considered random signal or
random process into entries being of the tensor product form. Based on the
tensor forms one is able to solve the stochastic boundary value or initial
value problems with deterministic mathematical analysis methodology. The
decompositions are just general and not specially made for solving particular
SPDE problems. They, therefore, do not offer convenience to particular
questions, and exhibits low efficiency. In the present study, we propose an
easily implementable sparse representation method, based on optimal selections
of dictionary elements made of fundamental solutions of the problems. This,
through the $``$lifting up” method based on the semigroup property of the
fundamental solutions (see §4), offers remarkable convenience and efficiency
to solve the SPDE problems. In many cases, the dictionary elements are
parameterized reproducing kernels of a relevant analytic functional Hilbert
spaces.

The methodology of this paper is based on the theory and algorithms of Qian in
[42] in which SAFD for the Hardy space is first developed and then further
extended to become stochastic pre-orthogonal adaptive Fourier decomposition 1
and 2 (SPOAFD1, SPOAFD2) for general Hilbert spaces with a dictionary. The
SAFD type methods are further development of what is called adaptive Fourier
decomposition (AFD) whose comprehensive study includes a number of variations
including core adaptive Fourier decomposition (or just AFD, see [53, 41]),
pre-orthogonal adaptive Fourier decomposition (POAFD, available for general
Hilbert spaces with a dictionary in which a Takenaka-Malmquist system is
unavailable, see [52, 43, 47, 48]), unwinding Blaschke expansion (unwinding
AFD, independently developed also by Coifman et al., see [16, 14, 40, 50],
$n$-best reproducing kernel approximation ($n$-best AFD, see [51, 49]), and
also in higher dimensions for matrix-valued functions ([2, 3]). The study of
the AFD type algorithms was originated from signal decomposition into
meaningful positive instantaneous frequency in physics ([53]). The maximal
selection principle of the AFD methods shares the matching pursuit idea (or
greedy algorithms) but emphasises attainability of the global optimal
parameters that leads to the delicate analysis on multiple selection of
parameter and the multiple kernel concept. The contribution of the present
study is to initiate application of the SAFD methods that gives solutions of
the stochastic boundary value problems in the form of fast converging series
of lifted and shifted fundamental solutions of the related operators.

We note that the present paper deals with two types of random data on the
boundary. The first type is those expressible as a bi-variate-function,
$f(t,X),$ of which one is the boundary variable $t$ and the other is a random
variable $X$. The second type is that the random data on the boundary is
actually a random process. For the latter we take Brownian bridge $W_{t}$ as
an example. Detailed studies involving comparison of various types of
decomposition methods and applications of the SAFD methods are reported in our
separate papers ([45],[70]).

Denote by $B_{r}(a)$ the open ball centred at $a$ with radius $r$ in
$\mathbb{R}^{n+1},$ and $\partial B_{r}(a)$ its boundary, also called the
$r$-sphere. In particular, we write $B_{1}\triangleq B_{1}(0)$ in short. We
deal with the Dirichlet problem for Laplace equation with a random boundary
condition

(1.11) $\left\\{\begin{aligned} &\Delta_{x}u(x,\omega)=0,\leavevmode\nobreak\
\leavevmode\nobreak\ x\in B_{1}\subseteq\mathbb{R}^{n+1},\leavevmode\nobreak\
{\rm a.s.}\leavevmode\nobreak\ \omega\in\Omega,\\\
&u(x,\omega)=f(x,\omega),\leavevmode\nobreak\ \leavevmode\nobreak\
x\in{\partial B_{1}},\leavevmode\nobreak\ {\rm a.s.}\leavevmode\nobreak\
\omega\in\Omega,\end{aligned}\right.$

and the Cauchy initial problem of the heat equation with a random initial
condition

(1.12) $\left\\{\begin{aligned}
&(\partial_{t}-\Delta_{x})u(t,x,\omega)=0,\leavevmode\nobreak\
\leavevmode\nobreak\
(t,x)\in\mathbb{R}^{1}_{+}\times\mathbb{R}^{n},\leavevmode\nobreak\ {\rm
a.s.}\leavevmode\nobreak\ \omega\in\Omega,\\\
&u(0,x,\omega)=f(x,\omega),\leavevmode\nobreak\ \leavevmode\nobreak\
x\in\mathbb{R}^{n},\leavevmode\nobreak\ {\rm a.s.}\leavevmode\nobreak\
\omega\in\Omega.\end{aligned}\right.$

Becus and Cozzarelli defined the random generalized solutions of such types of
SPDE problems, and proved existence and uniqueness of the solutions ([10, 7,
8]). Becus also proposed an iterative scheme for solving such random heat
equation and established its convergence ([9]). Tasaka proposed the finite
element solutions of the one-dimensional heat equation with random initial
conditions ([60]).

In this paper, using the random signal analysis methodology theoretically
developed in [42], we give and analyse the solutions of (1.11) and (1.12).
Different from the existing methods, the proposed SAFD methods are easy to be
implemented and directly give solutions of the SPDE problems under study for
all random boundary or initial data. Our methods depend on the distribution of
$X$ for the boundary bi-variate-function $f(t,X)$ type, and the covariance
function $C(s,t)$ for random process type. For the latter our algorithm does
not require any information on the related eigenvalues, nor the
eigenfunctions, as it does for proceeding the K-L decomposition.

The paper is organized as follows. In §2 we review relevant preliminary
knowledge and define notations that will be used in the following sections. In
§3 we formulate the Dirichlet problem of Laplace equation with a random
boundary condition and the Cauchy problem of heat equation with a random
initial condition, and, respectively, introduce the stochastic harmonic Hardy
space and the stochastic heat Hardy space as the appropriate function spaces
for the solutions. In §4 we develop the SPOAFD methodology in obtaining sparse
representations of solutions of the concerned stochastic boundary/initial
problems. The stochastic boundary vanishing conditions for the given function
spaces are proved, which guarantee applicability of the SPOAFD algorithms. The
numerical experiments are presented in §5 to confirm the theoretical
estimates.

##  2\. Preliminaries

We will first set our notation for the deterministic case. For the Dirichlet
problem of Laplace equation we focus on the $(n+1)$-dimensional unit ball
$B_{1}(0)$ and the upper half-space $\mathbb{R}^{1+n}_{+}$ contexts, and for
the Cauchy initial problem of the heat equation on $\mathbb{R}_{+}^{1+n}$.

For deterministic Dirichlet problems in bounded domains we adopt the
formulation that $D\subset\mathbb{R}^{1+n}$ is a bounded domain with a smooth
boundary, and $f\in L^{2}(\partial D).$ We need to determine a function $u$
satisfying

(2.13) $\left\\{\begin{aligned} &\Delta u(x)=0,\leavevmode\nobreak\
\forall\leavevmode\nobreak\ x\in D,\\\ &u(x)_{\partial D}=f(x),\ {\rm for\
a.e.}\ x\in{\partial D}.\end{aligned}\right.$

We note that the second equation in (2.13) takes the non-tangential boundary
limit sense. The solution to (2.13) can be written as the boundary integral
([21])

(2.14) $u(x)=\int_{\partial D}\frac{\partial}{\partial
n_{y}}G(x,y)f(y)d\sigma(y),$

where $G(x,y)$ is the Green’s function of $D$. Restricting the question to the
unit ball $B_{1}=D$ , the function $\frac{\partial}{\partial n_{y}}G(x,y)$
becomes the Poisson kernel (for the unit ball) $P_{x}(y)$ defined as:

(2.15) $P_{x}(y^{\prime})\triangleq P(x,y^{\prime})\triangleq
c_{n}\frac{1-|x|^{2}}{|x-y^{\prime}|^{n}},\leavevmode\nobreak\ x\in
B_{1},\leavevmode\nobreak\ y^{\prime}\in\partial B_{1},$

where $c_{n}$ is a constant depending on $n$ that can be different from line
to line. Thus the solution of the problem can be written explicitly, as
Poisson integral of $f,$

(2.16) $u_{f}(x)\triangleq\int_{\partial
B_{1}}P(x,y^{\prime})f(y^{\prime})d\sigma(y^{\prime}).$

We can also obtain the explicit formula for the solution of (2.13) for the
upper half-space $\mathbb{R}^{1+n}_{+}=D$ and $f\in L^{2}(\mathbb{R}^{n}),$

(2.17)
$u_{f}(x)=\int_{\mathbb{R}^{n}}{P}(x,\underline{y})f(\underline{y})d\underline{y},$

where ${P}(x,\underline{y})$ is the counterpart Poisson kernel for the upper
half-space $\mathbb{R}_{+}^{1+n},$ where $x$ is often written as
$x=x_{0}+\underline{x},x_{0}\in\mathbb{R}^{1}_{+},\leavevmode\nobreak\
\underline{x}\in\mathbb{R}^{n},$ and

${P}(x,\underline{y})=c_{n}\frac{x_{0}}{(x_{0}^{2}+|\underline{x}-\underline{y}|^{2})^{\frac{n+1}{2}}}.$

For $D$ being either the unit ball $B_{1}$ or the upper-half space case there
exists a harmonic Hardy $h^{p}$ space theory. We give an account for the unit
ball case. For the purpose of this study, we restrict to $p=2.$ The harmonic
Hardy space $h^{2}(B_{1})$ is defined to be

######  Definition 3.

([1]) (The harmonic Hardy space on the ball)

(2.18) $h^{2}(B_{1})\triangleq\\{u:\Delta u|_{B_{1}}=0,\leavevmode\nobreak\
\sup_{0<r<1}\|u(r\cdot)\|_{2}<\infty\\}.$

The norm of $u\in h^{2}(B_{1})$ is defined as
$\|u\|_{h^{2}}=\sup\limits_{0<r<1}\|u(r\cdot)\|_{2}.$

It is a fundamental result that every $u\in h^{2}(B_{1})$ has a non-tangential
boundary limit function ([59]) and there holds

$\sup_{0<r<1}\|u(r\cdot)\|_{2}=\lim_{r\rightarrow
1}\|u(r\cdot)\|_{2}=\|\lim_{r\rightarrow
1}u(r\cdot)\|_{2},\leavevmode\nobreak\ \forall\leavevmode\nobreak\ u\in
h^{2}(B_{1}).$

The $h^{2}(B_{1})$ is a Hilbert space equipped with the inner product

(2.19) $\langle u,v\rangle\triangleq\int_{\partial
B_{1}}f(s)g(s)d\sigma(s),\leavevmode\nobreak\ \forall\leavevmode\nobreak\
u,v\in h^{2}(B_{1}),$

where the $f(s),\leavevmode\nobreak\ g(s)$ are the non-tangential limits of,
respectively, the Poisson integrals $u=P[f],\leavevmode\nobreak\ v=P[g],$ and
$d\sigma(s)$ is the normalized measure on the sphere. Thus, $h^{2}(B_{1})$ is
isometric to $L^{2}(\partial B_{1})$ , denoted

(2.20) $L^{2}(\partial B_{1})\cong h^{2}(B_{1}).$

The harmonic Hardy space $h^{2}(B_{1})$ is a special case of the Hardy space
introduced by Stein and Fefferman ([19]). Within the same category, another
Hardy space, $H^{2}_{heat},$ is defined in relation to the heat equation in
$\mathbb{R}_{+}^{1+n}.$ Denote by

$\varphi_{t}(x)\triangleq(\frac{1}{4t\pi})^{n/2}e^{-\frac{1}{4t}|x|^{2}},$

the heat kernel. Then a solution of the corresponding heat equation can be
written as

(2.21) $\displaystyle
u_{f}(t,x)\triangleq\int_{\mathbb{R}^{n}}\varphi_{t}(x-y)f(y)dy=(\varphi_{t}\ast
f)(x),\leavevmode\nobreak\ f\in L^{2}(\mathbb{R}^{n}),$

and $u=u_{f}$ belongs to heat-Hardy space $H^{2}_{heat}(\mathbb{R}_{+}^{1+n})$
defined as

(2.22) $\\{u:\leavevmode\nobreak\ \exists\leavevmode\nobreak\ f\in
L^{2}(\mathbb{R}^{n}),\leavevmode\nobreak\ u=\varphi_{t}\ast
f,\leavevmode\nobreak\ \varphi^{+}(f)\triangleq\sup_{t>0}|(\varphi_{t}\ast
f)|\in L^{2}(\mathbb{R}^{n})\\},$

with the norm
$\|u\|_{H^{2}}\triangleq\|\varphi^{+}(f)\|_{L^{2}}\simeq\|f\|_{L^{2}}$.
Similarly, $H^{2}_{heat}(\mathbb{R}_{+}^{1+n})$ is isometric to
$L^{2}(\mathbb{R}^{n})$. The point of such formulation is that the spaces
$h^{2}(B_{1})$ and $H^{2}_{heat}(\mathbb{R}_{+}^{1+n})$ are reproducing kernel
Hilbert spaces.

We recall the needed reproducing kernel structure applicable to both
$h^{2}(B_{1})$ and $H^{2}_{heat}(\mathbb{R}_{+}^{1+n}).$

######  Definition 4.

([58]) (Reproducing kernel Hilbert space) Let $D\neq\emptyset$. A reproducing
kernel Hilbert space (RKHS) on the set $D$ is a Hilbert space $\mathcal{H}(D)$
with a function

$K:D\times D\rightarrow\mathbb{C}$

possessing the reproducing property

(2.23) $\left\\{\begin{aligned} &K_{q}\triangleq
K(\cdot,q)\in\mathcal{H}(D),\leavevmode\nobreak\ \forall\leavevmode\nobreak\
q\in D,\\\ &\langle f,K_{q}\rangle=f(q),\leavevmode\nobreak\
\forall\leavevmode\nobreak\ q\in D,\leavevmode\nobreak\
\forall\leavevmode\nobreak\ f\in\mathcal{H}(D).\end{aligned}\right.$

The function $K_{q}(\cdot)$ in (2.23) is called the reproducing kernel of
$\mathcal{H}(D)$ at $q$.

For instance, denote by $P_{t}(s^{\prime})$ the Poisson kernel defined by
(2.15), where $t\triangleq|t|t^{\prime},\leavevmode\nobreak\ t^{\prime}\in
D=B_{1},$ and $s\triangleq|s|s^{\prime},\leavevmode\nobreak\ s^{\prime}\in
B_{1}.$ The harmonic Hardy space $h^{2}(B_{1})$ is a RKHS ([46]) with the
reproducing kernel

(2.24) $K(s,t)\triangleq\langle
P_{s},P_{t}\rangle=P_{|s||t|s^{\prime}}(t^{\prime})=P_{|s||t|t^{\prime}}(s^{\prime}).$

The reproducing kernel for the heat equation is

(2.25) $K(x,y)\triangleq\langle
h_{x},h_{y}\rangle=h_{s+x}(\underline{y})=h_{t+y}(\underline{x}),$

where $x=t+\underline{x},\leavevmode\nobreak\ y=s+\underline{y}$ both are in
$D=\mathbb{R}^{n+1}_{+},$ while $t,s$ are in $\mathbb{R}^{1}_{+}.$

##  3\. Dirichlet problem with stochastic boundary value and Cauchy problem
with stochastic initial value

Consider a Dirichlet problem with a random boundary data, i.e.

(3.26) $\left\\{\begin{aligned} &\Delta u(x,\omega)=0,\leavevmode\nobreak\
\forall\leavevmode\nobreak\ x\in
D\subseteq\mathbb{R}^{n+1},\leavevmode\nobreak\ {\rm a.s.}\leavevmode\nobreak\
\omega\in\Omega,\\\ &u(x,\omega)=f(x,\omega),\leavevmode\nobreak\ \ {\rm for\
a.e.}\ \leavevmode\nobreak\ x\in{\partial D},\leavevmode\nobreak\ {\rm
a.s.}\leavevmode\nobreak\ \omega\in\Omega,\end{aligned}\right.$

where $f\in L^{2}(\Omega,L^{2}({D})),\leavevmode\nobreak\ {D}=B_{1}.$ We also
write $f_{\omega}(\cdot)$ as $f(\cdot,\omega)$ when no ambiguity arises.
Babuska was among the first to study rigorously existence of solutions of
Dirichlet problems with random boundary condition ([5]). Becus and Cozzarelli
studied existence and properties of general solutions of (3.26), see [10, 7,
8]. Furthermore, explicit formulae for the correlation tensor of the
generalized solutions are obtained when the random boundary function is a
white noise or a homogeneous random field on a sphere ([54]).

In fact, the equation (3.26) can be regarded as a family of equations labeled
with $\omega\in\Omega$. For each $\omega\in\Omega$ possibly excluding an event
of zero probability,

(3.27) $\left\\{\begin{aligned} &\Delta u(x,\omega)=0,\leavevmode\nobreak\
\leavevmode\nobreak\ x\in B_{1},\\\ &u(x)=f_{\omega}(x),\leavevmode\nobreak\
\leavevmode\nobreak\ {\rm a.e.}\leavevmode\nobreak\ x\in\partial
B_{1},\end{aligned}\right.$

where $f\in L^{2}(\Omega,L^{2}(D)).$ Hence (3.27) becomes the classical
Dirichlet boundary problem. Thus we can consider a stochastic process formed
by the classical Poisson integral

(3.28) $u_{f_{\omega}}(x)\triangleq\int_{\partial
B_{1}}p(x,t^{\prime})f_{\omega}(t^{\prime})d\sigma(t^{\prime}),\leavevmode\nobreak\
x\in B_{1},$

where $p(x,t^{\prime})$ is the Poisson kernel in $B_{1}$. For a.s.
$\omega\in\Omega$, it follows that $u_{f_{\omega}}(x)$ solves the (3.27), and

$\lim_{\rho\rightarrow 1}u_{f_{\omega}}(\rho
t^{\prime})=f_{\omega}(t^{\prime})$

in both the $L^{2}$-norm and the a.e. pointwise sense.

The stochastic process $u_{f_{\omega}}(x)$ defined in (3.28) solves the
problem (3.26) in the $\mathbb{P}-$ a.s. sense. The solution of (3.27) is a.s.
in $h^{2}(B_{1})$ and, space-wise, it is isometrically identical with the
boundary value $f_{\omega}\in L^{2}(\partial B_{1}).$

Similarly, the Cauchy problem with a stochastic initial value is proposed in
$\mathbb{R}_{+}^{n+1},$

(3.29) $\left\\{\begin{aligned}
&(\partial_{t}-\Delta_{x})u(t,x,\omega)=0,\leavevmode\nobreak\
\leavevmode\nobreak\ (t,x)\in\mathbb{R}^{1}_{+}\times\mathbb{R}^{n},\quad{\rm
a.s.}\leavevmode\nobreak\ \omega,\\\
&u(0,x,\omega)=f(x,\omega),\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm
a.e.\ }\ x\in\mathbb{R}^{n},\quad{\rm a.s.}\leavevmode\nobreak\
\omega,\end{aligned}\right.$

with $f\in L^{2}(\Omega,L^{2}(\mathbb{R}^{n})).$ For a.s. $\omega,$
$f_{\omega}\in L^{2}(\mathbb{R}^{n})$, the convolution

(3.30) $\displaystyle
u_{f_{\omega}}(t,x)=\int_{\mathbb{R}^{n}}\varphi_{t}(x-y)f_{\omega}(y)dy\triangleq(\varphi_{t}\ast
f_{\omega})(x)$

gives the solution of the classical heat equation with initial condition $f\in
L^{2}(\Omega,L^{2}(\mathbb{R}^{n})),$ and for each valid $\omega$ the solution
$u_{f_{\omega}}$ belongs to heat-Hardy space
$H^{2}_{heat}(\mathbb{R}_{+}^{1+n})$ whose space norm is defined as the
$L^{2}(\mathbb{R}^{n})$ norm of the non-tangential maximal function
$\sup_{(t,y)\in\Gamma^{\alpha}_{x}}|u_{f_{\omega}}(t,y)|,$ where
$\Gamma^{\alpha}_{x}$ is the orthogonal $\alpha$-cone in
$\mathbb{R}_{+}^{1+n}$ with tip at $x\in\mathbb{R}^{n}.$ The associated
stochastic Hardy spaces may be defined.

######  Definition 5.

(Stochastic harmonic Hardy space) The Hilbert space
$L^{2}(\Omega,h^{2}(B_{1}))$ is a function space that contains all
$h^{2}(B_{1})$-valued random variables
$u:B_{1}\times\Omega\rightarrow\mathbb{R}$ satisfying

(3.31) $\displaystyle u(\cdot,\omega)\in h^{2}(B_{1}),\leavevmode\nobreak\
{\rm a.s.}\leavevmode\nobreak\ \omega\in\Omega,$
$\displaystyle\mathbb{E}_{\omega}\|u(\cdot,\omega)\|^{2}_{h^{2}(B_{1})}<\infty.$

Analogously, the boundary stochastic Hilbert space
$L^{2}(\Omega,L^{2}(\partial B_{1}))$ is defined as:

(3.32) $\displaystyle L^{2}(\Omega,L^{2}(\partial B_{1}))\triangleq\\{$
$\displaystyle f:\leavevmode\nobreak\ \partial
B_{1}\times\Omega\rightarrow\mathbb{R},$ $\displaystyle f(\cdot,\omega)\in
L^{2}(\partial B_{1}),\leavevmode\nobreak\ {\rm a.s.}\leavevmode\nobreak\
\omega\in\Omega,$
$\displaystyle\mathbb{E}_{\omega}\|f(\cdot,\omega)\|^{2}_{L^{2}(\partial
B_{1})}<\infty.\\}.$

Stochastic harmonic Hardy space $L^{2}(\Omega,h^{2}(B_{1}))$ is a Hilbert
space with the inner product

(3.33) $\langle
f,g\rangle_{L^{2}(\Omega,h^{2}(B_{1}))}\triangleq\int_{\Omega}\int_{\partial
B_{1}}f(t,\omega)g(t,\omega)d\sigma(t)d\mathbb{P},\leavevmode\nobreak\
\forall\leavevmode\nobreak\ f,\leavevmode\nobreak\ g\in
L^{2}(\Omega,h^{2}(B_{1})).$

Corresponding to the heat equation case a pair of stochastic Hilbert spaces
are similarly defined.

######  Definition 6.

(stochastic heat-Hardy space) The Hilbert space
$L^{2}(\Omega,H^{2}_{heat}(\mathbb{R}_{+}^{1+n}))$ is a function space that
contains all $H^{2}_{heat}(\mathbb{R}_{+}^{1+n})$-valued random variables
$u:\mathbb{R}_{+}^{1+n}\times\Omega\rightarrow\mathbb{R}$ satisfying

(3.34) $\displaystyle u(\cdot,\omega)\in
H^{2}_{heat}(\mathbb{R}_{+}^{1+n}),\leavevmode\nobreak\ {\rm
a.s.}\leavevmode\nobreak\ \omega\in\Omega,$
$\displaystyle\mathbb{E}_{\omega}\|u(\cdot,\omega)\|^{2}_{H^{2}_{heat}(\mathbb{R}_{+}^{1+n})}<\infty.$

Analogously, the boundary Hilbert space $L^{2}(\Omega,L^{2}(\mathbb{R}^{n}))$
is defined as:

(3.35) $\displaystyle L^{2}(\Omega,L^{2}(\mathbb{R}^{n}))\triangleq\\{$
$\displaystyle f:\leavevmode\nobreak\
\mathbb{R}^{n}\times\Omega\rightarrow\mathbb{R},$ $\displaystyle
f(\cdot,\omega)\in L^{2}(\mathbb{R}^{n}),\leavevmode\nobreak\ {\rm
a.s.}\leavevmode\nobreak\ \omega\in\Omega,$
$\displaystyle\mathbb{E}_{\omega}\|f(\cdot,\omega)\|^{2}_{L^{2}(\mathbb{R}^{n})}<\infty.\\}.$

As shown in the above contexts the defined stochastic Hardy space are RKHSs.
In spite of the fact that they are isometric, however, the corresponding
stochastic $L^{2}$ spaces are not RKHSs. The mechanism we take for advantage
in solving the random boundary value and initial value problems is that in a
RKHS, $\mathcal{H}(D),$ the span of the kernels $\\{K(\cdot,q)\\}|_{q\in D}$
is dense in $\mathcal{H}(D),$ and the boundary limit functions of the same
class $\\{K(\cdot,q)\\}|_{q\in D}$ is dense in $L^{2}(D),$ as well as with
equivalent norms. This suggests that we in the $L^{2}(D)$ do sparse
approximation by using the dictionary $\\{K(\cdot,q)\\}|_{q\in D}$ and once
this is done we $``$lift up” the approximating series in the boundary
stochastic $L^{2}$ space to give the stochastic Hardy space approximation in
$\mathcal{H}(D).$

We note that the above defined spaces $L^{2}(\Omega,L^{2}(\partial B_{1}))$
and $L^{2}(\Omega,L^{2}({\mathbb{R}}^{n}))$ in Definitions 5 and 6 are
boundary stochastic $L^{2}$-spaces, and $L^{2}(\Omega,h^{2}(B_{1}))$ and
$L^{2}(\Omega,H^{2}_{heat}(\mathbb{R}_{+}^{1+n}))$ are the corresponding
stochastic solution spaces. We note that the solution spaces $h^{2}(B_{1})$
and $H^{2}_{heat}(\mathbb{R}_{+}^{1+n})$ are RKHSs ([58]), briefly denoted as
$H^{2}$ in the sequel.

The idea of POAFD algorithm was initiated in [52] and further formulated in
[43] and [44] (also see [15]). The work [47] and [48] carry out the POAFD
algorithm to the weighted Bergman spaces in the unit disc, and the work [46]
carries out it to the $h^{2}(B_{1})$ and $H^{2}_{heat}(\mathbb{R}_{+}^{1+n})$
spaces which are all for deterministic signals. The work [42] generalizes the
AFD and the POAFD methods to random signals. There are two types of
generalizations, called SPOAFD1 and SPOAFD2. Each of them can be applied to
the above defined stochastic Hardy spaces. In this paper, we are based on the
SPOAFD methods to solve SPDEs. In the following section we will first outline
the deterministic POAFD, and then the SPOAFD algorithms for the self-
containing purpose.

##  4\.  POAFD and SPOAFD Algorithms

For the self-containing purpose we give a revision of POAFD and SPOAFD
Algorithms (see [44, 15], or [47]).

###  4.1. POAFD

In order to perform POAFD the underlying Hilbert space $L^{2}(\partial D)$ is
assumed to have a subset indexed by elements in $D,$ denoted $K_{q},q\in D,$
whose span is dense (dense-span property) in $L^{2}(\partial D).$ The
collection of the reproducing kernels of any RKHS, in particular, has such
dense-span property. By this reason, with a little abuse of terminology we
call the functions in such dense-span collection by $``$kernels”. Besides the
density we further assume the _boundary vanishing condition_ (BVC): For any
function $G\in L^{2}(\partial D)$ there holds

(4.36) $\displaystyle\lim_{q\to\partial D}|\langle G,E_{q}\rangle|=0,$

where $E_{q}$ is the normalized kernel function:

(4.37) $\displaystyle E_{q}=\frac{K_{q}}{\|K_{q}\|}.$

It is necessary to introduce the notion _multiple kernels_. Let
$(q_{1},\cdots,q_{n})$ be an $n$-tuple of parameters in $D.$ Denote by $l(k)$
the multiplicity of $q_{k}$ in the $k$-tuple $(q_{1},\cdots,q_{k}),1\leq k\leq
n.$ With a little abuse of notation, we define the $k$-th multiple kernel as

$\tilde{K}_{q_{k}}=\left[\left(\frac{\partial}{\partial\overline{q}}\right)^{(l(k)-1)}K_{q}\right]_{q=q_{k}},\
k=1,2,\cdots,n,$

where $\frac{\partial}{\partial\overline{q}}$ is interpreted as a directional
derivative. Multiple kernels are generated from performing the _pre-orthogonal
maximal selection principle_ : Suppose we already have an $n$-tuple
$\\{q_{1},\cdots,q_{n}\\},$ possibly with multiplicity. We correspondingly
have the $n$-tuple of multiple kernels,
$\\{\tilde{K}_{q_{1}},\cdots,\tilde{K}_{q_{n}}\\}.$ Applying the G-S
orthonormalization process consecutively, we obtain an equivalent
$n$-orthonormal system, $\\{E_{1},\cdots,E_{n}\\},$ where $E_{1}=E_{q_{1}}.$
For any given $G$ in the Hilbert space we now investigate whether there exists
a $q_{n+1}\in D$ that gives rise to the supreme value

$|\langle G,E_{n+1}\rangle|=\sup\\{|\langle G,E_{n+1}^{q}\rangle|\ :\
q\in{D},\leavevmode\nobreak\ q\neq q_{1},\cdots,q_{n}\\},$

where the finiteness of the supreme is guaranteed by the Cauchy-Schwarz
inequality, and $E_{n+1}^{q}$ be such that
$\\{E_{1},\cdots,E_{n},E_{n+1}^{q}\\}$ is the G-S orthonormalization of
$\\{\tilde{K}_{q_{1}},\cdots,\tilde{K}_{q_{n}},K_{q}\\},$ and $E_{n+1}$ be
such that $\\{E_{1},\cdots,E_{n},E_{n+1}\\}$ is the G-S orthonormalization of
$\\{\tilde{K}_{q_{1}},\cdots,\tilde{K}_{q_{n}},\tilde{K}_{q_{n+1}}\\}.$ In
fact, since $q$ is distinct from the proceeding $q_{1},\cdots,q_{n},$
$E_{n+1}^{q}$ is given by

(4.38) $\displaystyle E_{n+1}^{q}=\frac{K_{q}-\sum_{k=1}^{n}\langle
K_{q},E_{k}\rangle E_{k}}{\|K_{q}-\sum_{k=1}^{n}\langle K_{q},E_{k}\rangle
E_{k}\|}.$

Under BVC a compact argument concludes that there exists a point $q_{n+1}\in
D$ and $q^{(l)},l=1,2,\cdots,$ such that $q^{(l)}$ are all different from
$q_{1},\cdots,q_{n},$ $\lim_{l\to\infty}q^{(l)}=q_{n+1},$ and

(4.39) $\displaystyle\lim_{l\to\infty}|\langle
G,E_{n+1}^{q^{(l)}}\rangle|=\sup\\{|\langle G,E_{n+1}^{q}\rangle|\ :\
q\in{D},q\neq q_{1},\cdots,q_{n}\\}=|\langle G,E_{n+1}\rangle|,$

where, by a Taylor expansion argument involving the Lagrange remainder, we
have

(4.40) $\displaystyle
E_{n+1}=\frac{\tilde{K}_{q_{n+1}}-\sum_{k=1}^{n}\langle\tilde{K}_{q_{n+1}},E_{k}\rangle
E_{k}}{\sqrt{\|\tilde{K}_{q_{n}}\|^{2}-\sum_{k=1}^{n}|\langle\tilde{K}_{q_{n+1}},E_{k}\rangle|^{2}}}$

(see [44, 15], or [47]). Iteratively applying the above process to $G,$ we
obtain a sequence of maximally selected $\\{q_{k}\\}_{k=1}^{\infty},$ and has

(4.41) $\displaystyle G=\sum_{k=1}^{\infty}\langle G,E_{k}\rangle E_{k}.$

We will call $\\{E_{k}\\}$ the consecutive Gram-Schmidt orthogonalization of
the multiple kernels corresponding to maximally selected $\\{q_{k}\\},$ the
latter being possibly with multiples. It is noted that the outcome system
$\\{E_{k}\\}$ is not necessarily a basis, it, however, is made for, and can
well express the given signal. For the POAFD expansion we have in the space
norm sense the same convergent rate $O(n^{-1/2})$ as that for the K-L
expansion ([43, 44, 15, 28]).

Next, we incorporate probability and formulate stochastic POAFD 1 and 2
(SPOAFD1 and 2). For the stochastic Dirichlet problem the set $D$ stands for
the unit ball $B_{1}\in{\mathbb{R}}^{1+n}$ or the upper half space
$\mathbb{R}_{+}^{1+n},$ while for the stochastic Cauchy initial value case $D$
stands for the half space $\mathbb{R}_{+}^{1+n}.$ By involving such domain $D$
we adopt the setting $\mathcal{N}=L^{2}(\Omega,L^{2}(\partial D))$ with any
underlying probability space $(\Omega,\mathcal{F},\mathbb{P}).$ We will be
related to the stochastic Hardy spaces $L^{2}(\Omega,H^{2}),$ where $H^{2}$
stands for either $h^{2}(B_{1}),$ or $h^{2}(\mathbb{R}_{+}^{1+n}),$ and
$H^{2}_{heat}(\mathbb{R}_{+}^{1+n}),$ depending on the individual problem.
Their reproducing kernels are uniformly denoted as $K(\cdot,q)=K_{q}(\cdot),$
whose span is dense in the concerned $L^{2}(\Omega,H^{2}),$ as well in
$L^{2}(\Omega,L^{2}(D)).$ The stochastic Hardy spaces are respectively
imbedded, through the non-tangential boundary limits, into the $\mathcal{N}$
spaces, namely $\mathcal{N}=L^{2}(\Omega,L^{2}(\partial B_{1}))$ or
$\mathcal{N}=L^{2}(\Omega,L^{2}({\mathbb{R}}^{n})).$

###  4.2. SPOAFD1

The SPOAFD1 algorithm is designed for a stochastic signal
$f(t,\omega)\triangleq f_{\omega}(t),$ being expressible as
$f_{\omega}(t)\triangleq\tilde{f}(t)+r_{\omega}(t),$ where
$\tilde{f}(t)\triangleq\mathbb{E}(f_{\omega}(t))$ and $r_{\omega}(t)\triangleq
f_{\omega}(t)-\tilde{f}(t),$ and the error random signal $r_{\omega}(t)$ is
assumed to be of small $\mathcal{N}$-norm. Obviously,
$\mathbb{E}(r_{\omega}(t))=0$ for a.e. $t\in D$. SPOAFD1 is designed to
analyze a given stochastic signal $f$ by means of the POAFD expanding
orthonormal system $\\{E_{k}\\}_{k}$ of the deterministic signal $\tilde{f}.$

The SPOAFD1 method to solve the stochastic boundary value or initial value
problems consists of three steps. Below as examples we work on the stochastic
Dirichlet problem on the unit ball and on the stochastic Cauchy problem on the
upper-half space. The first step is to compute out $\tilde{f}.$ The second
step is to apply POAFD to the deterministic signal $\tilde{f}$ with the
expansion

(4.42)
$\displaystyle\tilde{f}=\sum_{k=1}^{\infty}\langle\tilde{f},E_{k}\rangle
E_{k}$

valid on the boundary, where $\\{E_{k}\\}$ is consecutively the multiple G-S
orthogonalizations of the multiple kernels corresponding to $\\{q_{k}\\}.$ The
third step is to term by term solve the problems and add them together by
invoking boundedness of the Poisson kernel convolution operators. The
semigroup property of the spherical Poisson kernel then can $``$lift up” the
expansion on the sphere and, owing to the isometry between the $h^{2}(D)$ and
the $L^{2}(\partial D)$ spaces, give (see §4 of [46])

$u_{\tilde{f}}(x)=\sum_{k=1}^{\infty}\langle\tilde{f},E_{k}\rangle
u_{E_{k}}(x)\triangleq\sum_{k=1}^{\infty}\langle\tilde{f},E_{k}\rangle
P(E_{k})(x)=\sum_{k=1}^{\infty}c_{k}P_{|x|q_{k}}(x^{\prime}),\qquad x\in
B_{1},$

where $P(E_{k})$ is the classical solution of the Dirichelet problem with
boundary data $E_{k},$ $x=|x|x^{\prime},\leavevmode\nobreak\
q_{k}=|q_{k}|q_{k}^{\prime}$ and $\\{c_{k}\\}_{k}$ are certain constants (see
the later half of the proof of Theorem 4 below). Similarly, for the stochastic
Cauchy initial value problem, the SPOAFD1 expansion together with the lifting
up based on the semigroup property gives rise to the solution (see §3.2 of
[46])

$u_{\tilde{f}}(x)=\sum_{k=1}^{\infty}\langle\tilde{f},E_{k}\rangle
u_{E_{k}}(x)\triangleq\sum_{k=1}^{\infty}\langle\tilde{f},E_{k}\rangle
P_{t}\ast
E_{k}(\underline{x})=\sum_{k=1}^{\infty}d_{k}h_{t+q_{k}}(\underline{x}),\qquad
x\in\mathbb{R}^{n+1}_{+},$

where $x=t+\underline{x},q_{k}=s_{k}+\underline{q_{k}},$ and $\\{c_{k}\\}_{k}$
are certain constants. Note that the convergence rates of the solution series
are as the same as those expanding the stochastic boundary or the initial
values. Below, like $u_{\tilde{f}},$ we will use the self-explanatory
notations $u_{f}$ and $u_{r}.$ Denote

(4.43) $\displaystyle
d_{u_{f_{\omega}}}(x)=u_{f_{\omega}}(x)-\sum_{k=1}^{\infty}\langle{f_{\omega}},E_{k}\rangle
u_{E_{k}}(x).$

Since $\|r\|^{2}_{\mathcal{N}}=\|{\rm var}f\|_{L^{1}(\partial D)},$ the
results of [42] imply

######  Theorem 1.

$\|u_{f_{\omega}}-u_{\tilde{f}}\|_{\mathcal{N}}\leq\|{\rm var}f\|_{L^{1}(D)},$

and

$\|d_{u}\|_{\mathcal{N}}\leq\|{\rm var}f\|_{L^{1}(\partial
D)}-\sum_{j=1}^{\infty}\mathbb{E}|\langle r_{\omega},E_{j}\rangle|^{2},$

and, in particular,

$\|d_{u}\|_{\mathcal{N}}\leq\|{\rm var}f\|_{L^{1}(\partial D)}.$

The theorem concludes that SPOAFD1 solves the stochastic boundary and initial
problems up to an error dominated by the $L^{1}(\partial D)$-average of the
variation of $f.$

###  4.3. SPOAFD2

SPOAFD2, or, abbreviated as SPOAFD, is the main method introduced by this
paper, because through it we gain a.s. convergence of the involved series
expansions in the $H^{2}(D)$ and the $L^{2}(\partial D)$ spaces. SPOAFD is
built upon the following _statistical maximal selection principle_ (_SMSP_).

######  Theorem 2.

Assume that $f\in L^{2}(\Omega,L^{2}(\partial D)),$ where with respect to the
dictionary $\\{K_{q}\\}$ the space $L^{2}(\partial D)$ satisfies BVC. There
exists $q_{1}\in D$ such that

(4.44) $q_{1}=\arg\max_{q\in D}\mathbb{E}|\langle
f_{\omega},E_{q}\rangle|^{2},$

where $E_{q}$ is the normalized reproducing kernel of the Hardy space $H^{2}$
defined as in (4.37) for either the Dirichlet boundary value or the Cauchy
initial value problem.

Proof: By invoking the main result of [42] it is sufficient to prove the
_statistical boundary vanishing condition_ property, i.e.

(4.45) $\lim_{q\rightarrow\partial D}\mathbb{E}|\langle
f_{\omega},E_{q}\rangle|^{2}=0.$

Since

$f_{\omega}\in L^{2}(\partial D),\leavevmode\nobreak\ {\rm
a.s.}\leavevmode\nobreak\ \omega\in\Omega,$

we have

(4.46) $\lim_{q\rightarrow\partial D}|\langle
f_{\omega},E_{q}\rangle|^{2}=0,\leavevmode\nobreak\ {\rm
a.s.}\leavevmode\nobreak\ \omega\in\Omega,$

as proved in [46]. Noting that, for $q\to\partial D$ the functions $|\langle
f_{\omega},E_{q}\rangle|^{2}$ in $\omega$ is dominated uniformly in $q$ by an
integrable function in $L^{1}(\Omega,d\mathbb{P}):$

(4.47) $\displaystyle|\langle f_{\omega},E_{q}\rangle|^{2}$
$\displaystyle\leq\|f_{\omega}\|^{2}_{H^{2}}\|E_{q}\|^{2}_{H^{2}}$
$\displaystyle=\|f_{\omega}\|^{2}_{H^{2}}\in L^{1}(\Omega,d\mathbb{P}),$

by Lebesgue’s dominated convergence theorem, we have

(4.48) $\lim_{|q|\rightarrow\partial D}\mathbb{E}|\langle
f_{\omega},E_{q}\rangle|^{2}=0.$

By employing the usual compact argument we obtain a maximal selection of $q\in
D.$ The proof is complete.$\hfill\square$

The availability of SMSP implies that SPOAFD2 can be performed in
$L^{2}(\Omega,L^{2}(B_{1}))$ and $L^{2}(\Omega,L^{2}(\mathbb{R}^{n})).$ That
is,

######  Theorem 3.

([42]) Suppose $f\in\mathcal{N}=L^{2}(\Omega,L^{2}(D)).$ Assume that
$\\{q_{j}\\}_{j}$ are selected under SMSP,

(4.49) $q_{l}=\arg\max_{q\in D}\mathbb{E}|\langle
f_{\omega},E_{l}\rangle|^{2},$

where $\\{E_{k}\\}$ is the consecutive G-S orthogonalization of the multiple
kernels corresponding to the maximally selected $\\{q_{k}\\}.$ Then there
holds

(4.50)
${f_{\omega}}\stackrel{{\scriptstyle\mathcal{N}}}{{=}}\sum_{k=1}^{\infty}\langle
f_{\omega},E_{k}\rangle{E_{k}}.$

As a consequence of the above theorem, we have

######  Theorem 4.

Under the SPOAFD2 expansion (4.50) there holds

(4.51)
$u_{f_{\omega}}\stackrel{{\scriptstyle\mathcal{N}}}{{=}}\sum_{k=1}^{\infty}\langle
f_{\omega},E_{k}\rangle
u_{E_{k}}=\sum_{k=1}^{\infty}c_{k}(\omega)\tilde{K}_{q_{k}},$

where $\\{\tilde{K}_{q_{k}}\\}$ are the multiple kernels whose consecutive G-S
orthogonalizations are $\\{E_{k}\\}$ computed as in (4.40), $\\{q_{k}\\}$ are
the parameter sequence consecutively obtained through applying SMSP to
$f_{\omega},$ $\\{c_{k}(\omega)\\}$ are accordingly asserted coefficients
depending on $\omega.$

Proof: The POAFD convergence result amounts, for each $\omega,$ possibly
excluding an event of probability zero,

$f_{\omega}\stackrel{{\scriptstyle L^{2}(\partial
D)}}{{=}}\sum_{k=1}^{\infty}\langle f_{\omega},E_{k}\rangle E_{k}.$

Since the lifting up operators $f_{\omega}\mapsto u_{f_{\omega}}$ for solving
the classical problems is either isometric or bounded (for the heat kernel
case), we have, for a.s. $\omega\in\Omega,$

$u_{f_{\omega}}\stackrel{{\scriptstyle
H^{2}(D)}}{{=}}\sum_{k=1}^{\infty}\langle f_{\omega},E_{k}\rangle u_{E_{k}}.$

Since the Hardy space projection
$u_{f}=P_{H^{2}}(f):L^{2}(\Omega,L^{2}(\partial D))\to L^{2}(\Omega,H^{2}(D))$
is linear and isometric, we have, for any fixed $n,$

$\displaystyle\|u_{f_{\omega}}-\sum_{k=1}^{n}\langle f_{\omega},E_{k}\rangle
u_{E_{k}}\|_{L^{2}(\Omega,H^{2}(D))}$ $\displaystyle\leq$
$\displaystyle\|f_{\omega}-\sum_{k=1}^{n}\langle f_{\omega},E_{k}\rangle
E_{k}\|_{L^{2}(\Omega,L^{2}(\partial D))}.$

By invoking the SPOAFD2 convergence result in [42] the last quantity tends to
zero as $n\to\infty.$ Therefore, (4.51) holds. Next, we deduce the
coefficients $c_{k}(\omega).$

Denote, for every positive integer $n$,
$\mathcal{E}_{n}=(E_{1},\cdots,E_{n}),\leavevmode\nobreak\
\mathcal{K}_{n}=(\tilde{K}_{q_{1}},\cdots,\leavevmode\nobreak\
\tilde{K}_{q_{n}}),\leavevmode\nobreak\
\mathcal{C}_{n}=(c_{1}(\omega),\cdots,c_{n}(\omega)),$ and
$\mathcal{F}_{n}=(\langle f_{\omega},E_{1}\rangle,\cdots,\langle
f_{\omega},E_{n}\rangle).$ Then there follows

${\mathcal{K}_{n}}^{t}=\mathcal{A}_{n}{\mathcal{E}_{n}}^{t},$

where the superscript $``$t” stands for transpose of the matrix, and
$\mathcal{A}_{n}=(a_{ij})_{1\leq i,j\leq n}$ is a lower triangle $n\times n$
matrix, where

$a_{ij}=\left\\{\begin{array}[]{ll}E_{q_{i}}(q_{j})=\frac{K(q_{j},q_{i})}{\|K_{q_{i}}\|},\leavevmode\nobreak\
i\geq j,\\\ 0,\leavevmode\nobreak\ i<j.\end{array}\right.$

Hence,

${\mathcal{E}_{n}}^{t}=\mathcal{A}_{n}^{-1}{\mathcal{K}_{n}}^{t},\quad{\rm
and}\quad\mathcal{C}_{n}=\mathcal{F}_{n}\mathcal{A}_{n}^{-1}.$

The proof is complete.$\hfill\square$

Theorem 4 shows that with $f$ being in the Bochner space
$L^{2}(\Omega,L^{2}(\partial D))$ the stochastic Dirichlet boundary value
problem and the stochastic Cauchy initial value problem are solvable in their
corresponding Bochner type stochastic Hardy spaces $L^{2}(\Omega,H^{2})$ with
explicit sparse representations.

We note that theoretically the introduced method has the standard convergence
rate $\frac{M}{\sqrt{n}}$ (see [44, 15] or [45]).

##  5\. Examples of Random Boundary Value Problem

In this section we first present the algorithm of SPOAFD2 (SPOAFD in short),
which is applied to numerical simulations of two types of random PDEs
described by the Laplace equation and the heat equation. Denote by
$\\{B_{1},\cdots,B_{j}\\}$ the G-S orthonormalization of the maximally
selected parameterized multiple kernels under the SPOAFD program.

Algorithm. SPOAFD
---
Input: original function $f(t,\omega)$
Output: solution $u_{n}$ of random equation
1: Initialize $u_{n}=0,\leavevmode\nobreak\ j=0,$ error=10, ${\rm
energy}_{\omega}=0.$
2: While error $\geq$ $10^{-4}$ do
3: $j\leftarrow j+1$
4: $a_{j}\leftarrow$ $\arg\max\limits_{b\in D}\mathbb{E}_{\omega}|\langle
f_{\omega},B_{j}^{b}\rangle|^{2}$
5: $u_{n}\leftarrow u_{n}+\langle f_{\omega},B_{j}^{a_{j}}\rangle
B_{j}^{a_{j}}$
6: ${\rm energy}_{\omega}\leftarrow{\rm energy}_{\omega}+|\langle
f_{\omega},B_{j}^{a_{j}}\rangle|^{2}$
7: error $\leftarrow\frac{\|f\|_{\mathcal{N}}^{2}-\mathbb{E}({\rm
energy}_{\omega})}{\|f\|_{\mathcal{N}}^{2}}$
8: End while

As already noted that the expansions using the parameterised fundamental
solutions have the convenience of lifting up to get the virtual solutions of
the problems. For such purpose below we only expand the random boundary data
by the respective dictionaries containing the needed fundamental solutions.
Without such convenience random boundary data can also be expanded by other
basis or dictionary elements, such as the Fourier-Hermite polynomials and the
eigenfunctions of K-L decomposition. A separate paper, [45], will devote to
comparison between the existing stochastic expansions and SPOAFD, without
involving applications to SPDEs.

######  Example 1.

Consider the Laplace equation in the unit disk
$\mathbb{D}\triangleq\\{z\in\mathbb{C}:\leavevmode\nobreak\ |z|<1\\}$ with the
random Dirichlet boundary condition:

(5.52) $\left\\{\begin{aligned} &\Delta u(z,\omega)=0,\leavevmode\nobreak\
\forall\leavevmode\nobreak\ z\in\mathbb{D},\leavevmode\nobreak\ {\rm
a.s.}\leavevmode\nobreak\ \omega\in\Omega,\\\
&u(e^{it},\omega)=1/\sqrt{5+(\sin t-X)^{2}},\leavevmode\nobreak\
\leavevmode\nobreak\ e^{it}\in\partial\mathbb{D},\leavevmode\nobreak\ {\rm
a.s.},\end{aligned}\right.$

where $t\in[0,2\pi]$ and $X=X(\omega)$ is a random variable with the density
function $p(s)=\frac{1}{m}\alpha(s),$ where

$\alpha(s)=\begin{cases}e^{-\frac{1}{(s-\pi)^{2}+1}},&0\leq s\leq\pi,\\\
e^{-\frac{1}{(s+\pi)^{2}+1}},&-\pi\leq s<0,\end{cases}$

and $m=\int_{-\pi}^{\pi}\alpha(s)ds.$

Since the boundary data is a bi-variate function of $t$ and $X,$ we can make
use the distribution of the random variable $X.$ We have three groups of
figures respectively for visual effects of the experiments at those $\omega$
such that $X(\omega)=0,-\pi,$ and $2.4504,$ which can be shown in Figure 1, 2
and 3. The relative errors in Table 1, 2, 3 are computed according to the
formulas given in the relevant algorithms.

2 partial sum

3 partial sum

4 partial sum

Figure 1. $X=0$ SPOAFD |  3.6902$\times 10^{-4}$ |  2.2177$\times 10^{-5}$ |  1.8945$\times 10^{-7}$
---|---|---|---
Table 1. relative error

2 partial sum

3 partial sum

4 partial sum

Figure 2. $X=-\pi$ SPOAFD |  6.7418$\times 10^{-5}$ |  6.2401$\times 10^{-8}$ | $6.7669\times 10^{-9}$
---|---|---|---
Table 2. relative error

2 partial sum

3 partial sum

4 partial sum

Figure 3. $X=2.4504$ SPOAFD |  1.4236$\times 10^{-4}$ |  6.5114$\times 10^{-7}$ | $6.0066\times 10^{-8}$
---|---|---|---
Table 3. relative error

######  Example 2.

The other case is the heat equation with the random initial condition:

(5.53) $\left\\{\begin{aligned}
&(\partial_{t}-\Delta_{x})u(t,x,\omega)=0,\leavevmode\nobreak\
\leavevmode\nobreak\
(t,x)\in\mathbb{R}^{1}_{+}\times\mathbb{R}^{1},\leavevmode\nobreak\ {\rm
a.s.}\leavevmode\nobreak\ \omega\in\Omega,\\\
&u(0,x,\omega)=\frac{1}{2+(\frac{x}{3}-X)^{2}}},\leavevmode\nobreak\
\leavevmode\nobreak\ x\in\mathbb{R}^{1},\leavevmode\nobreak\ {\rm
a.s.},{\end{aligned}\right.$

where $X$ is the evaluation of the random variable with the density function
$p(s)=\frac{1}{\sqrt{2\pi}}e^{-\frac{s^{2}}{2}}.$

Again for this example we can make direct use of the distribution of $X.$ We
have three groups of figures respectively for visual effects of the
experiments at those $\omega$ for which $X=0,-3.7,$ and $3.1$, which can be
shown in Figure 4, 5 and 6. The relative errors in Table 4,5,6 are computed
according to the formulas given in the relevant algorithms.

5 partial sum

10 partial sum

15 partial sum

Figure 4. $X=0$ SPOAFD | 0.0014 | $9.0613\times 10^{-6}$ | $7.1549\times 10^{-7}$
---|---|---|---
Table 4. relative error

5 partial sum

10 partial sum

15 partial sum

Figure 5. $X=-3.7$ SPOAFD | 0.1759 |  4.8514$\times 10^{-4}$ | $2.7590\times 10^{-5}$
---|---|---|---
Table 5. relative error

5 partial sum

10 partial sum

15 partial sum

Figure 6. $X=3.1$ SPOAFD | 0.1064 | 0.0016 | $2.5211\times 10^{-5}$
---|---|---|---
Table 6. relative error

######  Example 3.

In this example we decompose Brownian bridge. We are considering the
stochastic Dirichelet problem

(5.54) $\left\\{\begin{aligned} &\Delta u(z,\omega)=0,\leavevmode\nobreak\
\leavevmode\nobreak\ z\in B_{1},\\\ &u(e^{it})=W(t),\leavevmode\nobreak\
\leavevmode\nobreak\ t\in[0,2\pi],\end{aligned}\right.$

where $B_{1}={\bf D}$ is the unit disc in the complex plane, and $W(t)$ is the
Brownian bridge. By using SPOAFD for the Poisson kernel in the unit disc we
obtain an orthonormal system uniformly applicable for all Brownian bridges. We
arbitrarily chose two sample pathes to test effectiveness of the obtained
SPOAFD system. As shown in Figure 7 and 8, the results are quite promising.

SPOAFD: 25 partial sum

SPOAFD: 50 partial sum

SPOAFD: 75 partial sum

SPOAFD: 100 partial sum

SPOAFD2: 125 partial sum

relative error

Figure 7. sample path I of Brownian bridge

SPOAFD: 26 partial sum

SPOAFD: 52 partial sum

SPOAFD: 78 partial sum

SPOAFD: 104 partial sum

SPOAFD: 130 partial sum

relative error

Figure 8. sample path II of Brownian bridge

##  6\. Conclusion

Using the recently developed stochastic pre-orthogonal adaptive Fourier
decomposition (SPOAFD) to the dictionaries of parameterised Poisson and heat
kernels we obtain sparse representations of solutions of the Dirichlet
boundary value and the Cauchy initial value problems with random data,
including stochastic process cases. The numerical examples show promising
convergence speed that implies fast convergence of the SPOAFD solutions series
to the stochastic boundary and initial value problems. Theoretical and
technical aspects of SPOAFD in comparison with existing major decomposition
methods for stochastic processes will be studied in separate papers ([45,
70]).

## Acknowledgments

The study is supported by the Science and Technology Development Fund, Macau
SAR (File no. 0123/2018/A3) and NSFC 12071437.

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provided more specific details on the methodologies used to assess climate-
related risks and how the organization integrates climate risk data and
insights into its investment management processes.

Score7: 90

Q8: How are the processes for identifying, assessing, and managing climate-
related risks integrated into the organisation’s overall risk management?

A8: UBS AG integrates sustainability and climate risk considerations into its
internal and external reporting, risk appetite setting, and management and
control processes. The company has a climate risk program with defined
responsibilities, processes, and tools applicable to business divisions and
group functions. However, the report lacks concrete data on the integration of
climate-related risks into the organization’s overall risk management.

Analysis8: The disclosure provides a comprehensive description of how the
organization integrates processes for identifying, assessing, and managing
climate-related risks into its overall risk management. The report describes
how sustainability and climate risk exposures are monitored and metrics
reported internally to enable risk appetite setting. The report also describes
how management and control processes ensure that material sustainability and
climate risks are identified, measured, monitored, and escalated in a timely
manner. However, the report lacks specific details on how the organization
integrates climate-related risks into its overall risk management. For
instance, the report does not provide specific examples of how sustainability
and climate risk considerations are included in internal and external
reporting. Overall, the disclosure meets most of the requirements and scores
85 out of 100.

Score8: 85

Metrics and targets:

Q9: What metrics does the organisation use to assess climate-related risks and
opportunities? How do the metrics help ensure that the performance is in line
with its strategy and risk management process?

A9: UBS AG uses periodic climate-related credit and market risk metrics,
climate risk heatmaps, climate scenario analysis and stress test exercises,
and a sustainability and climate risk scorecard to assess climate-related
risks and opportunities. These metrics help ensure that the performance is in
line with its strategy and risk management process by integrating risk data
and insights into its investment management processes and enabling risk
appetite setting.

Analysis9: The given disclosure provides some information on the metrics used
to assess climate-related risks and opportunities. The report mentions that
key sustainability and climate risk considerations are included in internal
and external reporting, and that sustainability and climate risk exposures are
monitored and metrics reported internally to enable risk appetite setting.
However, the report lacks specific details on the metrics used to measure and
manage climate-related risks and opportunities. The report does not provide
metrics on climate-related risks associated with water, energy, land use, and
waste management. The report also does not provide forward-looking metrics for
the cross-industry, consistent with their business or strategic planning time
horizons. Overall, the disclosure is insufficient in meeting the TCFD
guidelines.

Score9: 40

Q10: Does the organisation disclose its Scope 1, Scope 2, and, if appropriate,
Scope 3 greenhouse gas (GHG) emissions? What are the related risks and do they
differ depending on the scope?

A10: Yes, UBS AG discloses its Scope 1, Scope 2, and Scope 3 greenhouse gas
emissions. The report identifies direct emissions from gas, oil and fuel
consumption, indirect emissions from electricity and district heating and
other indirect emissions from leased assets, paper, waste and business travel.
The report also mentions that they are looking to quantify any relevant scope
3 emissions from categories 1 to 14. The related risks are not explicitly
stated.

Analysis10: The sustainability report provides a comprehensive disclosure of
Scope 1 and Scope 2 GHG emissions, and a partial disclosure of Scope 3 GHG
emissions. The report mentions that the organization has made a first
estimation of Scope 3 emissions and identified the top contributors. However,
it does not provide a detailed breakdown of Scope 3 emissions by category,
which is a requirement for high-quality disclosure. The report also lacks
information on the related risks and how they differ depending on the scope.
Overall, the report partially meets the requirements for high-quality
disclosure of GHG emissions and related risks, but there is room for
improvement.

Score10: 70

Q11: What targets does the organisation use to understand/quantify/benchmark
climate-related risks and opportunities? How is the organization performing
against these targets?

A11: UBS AG uses quantitative climate risk appetite, integrates sustainability
and climate risk into policies and processes, builds in-house capacity,
centralizes and executes ESG data strategy to understand/quantify/benchmark
climate-related risks and opportunities. The report does not provide
sufficient information to evaluate the organization’s performance against
these targets.

Analysis11: The sustainability report provides some information on the targets
used by the organization to manage climate-related risks and opportunities and
performance against targets. The report mentions that the organization has
developed methodologies to set climate-related targets and identify climate-
related risks, and that sustainability and climate risk exposures are
monitored and metrics reported internally to enable risk appetite setting.
However, the report lacks specific details on the key climate-related targets,
time frames, base year, and key performance indicators used to assess progress
against targets. The report also does not provide a description of the
methodologies used to calculate targets and measures. Therefore, while the
report provides some information on the TCFD recommendation point, it does not
fully satisfy the TCFD guidelines.

Score11: 50

Average score: 70.0

##  Appendix I Comparison with Liu et al. (2023)

Instead of hallucination-free rates, Liu et al. (2023) measures citation
recall and precision to evaluate the attribution quality. Citation recall is
similar to our content-dimension hallucination-free rate. It measures the
proportion of verification-worthy statements that are fully supported by the
associated citations. However, citation recall is statement-level while our
metric is paragraph-level: an answer paragraph is hallucination-free only when
all its statements are grounded by evidence. citation precision is the portion
of citations that support their statements, while our source-dimension
hallucination-free rate measures the proportion of answers that honestly
include all supportive sources. It is important to note that Liu et al. (2023)
includes queries of various domains to evaluate the trustworthiness of
general-purpose search engines, while our human evaluation focus on
chatReport.

##  Appendix J Answers Are Easy to Trace

The following example showcases the traceability of an answer produced by
ChatGPT. There are three sentences in the answer. Searching for the underlined
keywords, we easily identify the first and second sentence’s evidence sentence
in Source 215 and 166 respectively.

⬇

Answer: Assurant conducted a preliminary scenario analysis in 2021 to identify
and analyze climate-related risks faced by 10 of its critical and/or highly
vulnerable facilities across the globe. The company’s longer-term strategic
planning process prioritized climate as a multi-year ESG area of focus.
However, the report lacks concrete data on the resilience of the
organization’s strategy in different climate-related scenarios.

Source 215: Preliminary Scenario Analysis: The process of screening for
climate-related risks and opportunities across operations and business units
informed our preliminary scenario analysis activities. We conducted an initial
climate scenario analysis in 2021, with third-party support, to identify and
analyze the various climate-related risks faced by 10 of Assurant’s critical
and/or highly vulnerable facilities across the globe ...

Source 166: ... Assurant’s longer-term strategic planning process, overseen by
our Board, prioritized climate as a multi-year ESG area of focus ...

##  Appendix K Answer Comparison

An example of ChatGPT and GPT-4 outputs comparison. ChatGPT answers the
question extractively and adheres to the authentic information: asserting that
it is “the executive leadership team” that reviews Walmart’s climate strategy
at least annually. However, GPT-4 mistakenly summarizes that the board
oversees ESG issues through “its Nominating and Governance Committee (NGC)”.
It asserts that “the NGC reviews Walmart’s climate strategy at least
annually”, which is not accurate. GPT-4 conducts unnecessary summarizations
and inferences. This hallucinates the company’s different committees.

⬇

Question:

How does the company’s board oversee climate-related risks and opportunities?

ChatGPT Answer:

Walmart’s board oversees the overall ESG strategy and certain priority issues,
including climate-related risks and opportunities. The Chief Sustainability
Officer provides updates on the ESG agenda and progress to the Nominating and
Governance Committee. The executive leadership team reviews Walmart’s climate
strategy at least annually. The Nominating and Governance Committee discussed
Walmart’s ESG strategies and progress, including climate and public policy
strategies and practices in 2021.

GPT-4 Answer:

The Walmart Board of Directors oversees the company’s climate-related risks
and opportunities through its Nominating and Governance Committee, which
reviews the climate strategy at least annually. The Board also engages with
stakeholders and advocates for an enabling policy environment. The company’s
executive leadership team and ESG Steering Committee also have oversight
responsibility for particular ESG issues.

##  Appendix L TCFD Introduction

The TCDF is an internationally recognized initiative established by the
Financial Stability Board in 2015. Its general structure - covering
governance, strategy, risk management, and metrics and targets - emerges as
the premier candidate of voluntary reporting frameworks to be adopted for
regulatory reporting and international standards888The main emerging standards
in this context are the draft disclosure standards provided by the
International Financial Reporting Standards’ International Sustainability
Standards Board (ISSB).. The purpose of the TCFD is to provide a voluntary
reporting framework to encourage companies to disclose information on climate-
related risks and opportunities. This information is crucial for stakeholders
and investors to make informed decisions that account for climate-related
risks. Companies are asked to disclose this information in their annual
reports and financial filings, within their sustainability reports, or as
stand-alone documents. To date, most companies have included the information
as part of their sustainability reports, due to no or limited assurance and
legal liability concerns if it was included in the annual reports. This is why
our analysis focuses on content in sustainability reports from the perspective
of a reporting standard such as the TCFD, but it can also be extended to any
other corporate report. However, it is important to keep in mind that our
analysis primarily evaluates the disclosed information rather than directly
measuring the genuine implementation of tangible actions by the company.