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Uncertainty quantification for random domains using periodic random variables

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We consider uncertainty quantification for the Poisson problem subject to domain uncertainty. For the stochastic parameterization of the random domain, we use the model recently introduced by Kaarnioja et al. (SIAM J. Numer. Anal., 2020) in which a countably infinite number of independent random variables enter the random field as periodic functions. We develop lattice quasi-Monte Carlo (QMC) cubature rules for computing the expected value of the solution to the Poisson problem subject to domain uncertainty. These QMC rules can be shown to exhibit higher order cubature convergence rates permitted by the periodic setting independently of the stochastic dimension of the problem. In addition, we present a complete error analysis for the problem by taking into account the approximation errors incurred by truncating the input random field to a finite number of terms and discretizing the spatial domain using finite elements. The paper concludes with numerical experiments demonstrating the theoretical error estimates.
Error criterion (5.10) with increasing n. a Varying decay rate θ∈{2.1,2.5,3.0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta \in \{2.1,2.5, 3.0\}$$\end{document}, and fixed dimension s=10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=10$$\end{document}. b Varying dimensions s∈{10,40,100}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\in \{10,40,100\}$$\end{document}, and fixed decay rate θ=2.1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta =2.1$$\end{document}. c Detail of the case s=10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=10$$\end{document}, θ=2.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta =2.5$$\end{document}
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Numerische Mathematik (2024) 156:273–317
https://doi.org/10.1007/s00211-023-01392-6
Numerische
Mathemat ik
Uncertainty quantification for random domains using
periodic random variables
Harri Hakula1·Helmut Harbrecht2·Vesa Kaarnioja3·Frances Y. Kuo4·
Ian H. Sloan4
Received: 31 October 2022 / Revised: 13 October 2023 / Accepted: 13 December 2023 /
Published online: 12 January 2024
© The Author(s) 2024
Abstract
We consider uncertainty quantification for the Poisson problem subject to domain
uncertainty. For the stochastic parameterization of the random domain, we use the
model recently introduced by Kaarnioja et al. (SIAM J. Numer. Anal., 2020) in which
a countably infinite number of independent random variables enter the random field
as periodic functions. We develop lattice quasi-Monte Carlo (QMC) cubature rules
for computing the expected value of the solution to the Poisson problem subject to
domain uncertainty. These QMC rules can be shown to exhibit higher order cubature
convergence rates permitted by the periodic setting independently of the stochastic
dimension of the problem. In addition, we present a complete error analysis for the
problem by taking into account the approximation errors incurred by truncating the
input random field to a finite number of terms and discretizing the spatial domain using
BVesa Kaarnioja
vesa.kaarnioja@fu-berlin.de
Harri Hakula
harri.hakula@aalto.fi
Helmut Harbrecht
helmut.harbrecht@unibas.ch
Frances Y. Kuo
f.kuo@unsw.edu.au
Ian H. Sloan
i.sloan@unsw.edu.au
1Department of Mathematics and Systems Analysis, Aalto University School of Science, P.O. Box
11100, 00076 Aalto, Finland
2Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, 4051 Basel,
Switzerland
3Fachbereich Mathematik und Informatik, Freie Universität Berlin, Arnimallee 6, 14195 Berlin,
Germany
4School of Mathematics and Statistics, UNSW Sydney, Sydney, NSW 2052, Australia
123
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274 H. Hakula et al.
finite elements. The paper concludes with numerical experiments demonstrating the
theoretical error estimates.
Mathematics Subject Classification 65D30 ·65D32 ·35R60
1 Introduction
Modeling domain uncertainty is pertinent in many engineering applications, where
the shape of the object may not be perfectly known. For example, one might think of
manufacturing imperfections in the shape of products fabricated by line production,
or shapes which stem from inverse problems such as tomography.
If the domain perturbations are small, one can apply the perturbation technique
as in, e.g., [2,13,17]. In that approach, which is based on Eulerian coordinates, the
shape derivative is used to linearize the problem under consideration around a nominal
reference geometry. By Hadamard’s theorem, the resulting linearized equations for
the first order shape sensitivities are homogeneous equations posed on the nominal
geometry, with inhomogeneous boundary data only. By using a first order shape Taylor
expansion, one can derive tensor deterministic partial differential equations (PDEs)
for the statistical moments of the problem.
The approach we consider in this work is the domain mapping approach. It transfers
the shape uncertainty onto a fixed reference domain and hence reflects the Lagrangian
setting. Especially, it allows us to deal with large deformations, see [27,31] for exam-
ple. Analytic dependency of the solution on the random domain mapping in the case
of the Poisson equation was shown in [3,15] and in the case of linear elasticity in
[14]. Related shape holomorphy for acoustic and electromagnetic scattering problems
has been shown in [19,21] and for Stokes and Navier–Stokes equations in [4]. Mixed
spatial and stochastic regularity of the solutions to the Poisson equation on random
domains, required for multilevel cubature methods, has been recently proved in [16].
In order to introduce the domain mapping approach, let (Ω, F,P)be a probability
space, where Ωis the set of possible outcomes, the σ-algebra F2Ωis the set of all
events, and Pis a probability measure. We are interested in uncertainty quantification
for the Poisson problem
Δu(x)=f(x), xD(ω),
u(x)=0,xD(ω), for P-almost every ωΩ, (1.1)
where the domain D(ω) Rd,d∈{2,3}, is assumed to be uncertain. In the domain
mapping framework, one starts with a reference domain Dref Rd. For example, in a
manufacturing application, Dref might be the planned domain provided by computer-
aided design with no imperfections. It is assumed that a random perturbation field
V(ω) :Dref D(ω)
is prescribed. The Poisson problem (1.1) on the perturbed domain is then transformed
onto the fixed reference domain Dref by a change of variable. This results in a PDE on
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Uncertainty quantification for random domains... 275
the reference domain, equipped with a random coefficient and a random source term,
which are correlated by the random deformation, and complicated by the occurrence
of the Jacobian of the transformation. Note that the dependence on the random pertur-
bation field is nonlinear. Nonetheless, in the domain mapping approach, the reference
domain Dref needs only to be Lipschitz, while the random perturbations V(ω) have
to be C2-smooth. This is in contrast to the perturbation approach mentioned above,
where both the reference domain and the random perturbations need to be C2-smooth.
The approach adopted in this paper is first to truncate the initially infinite number
of scalar random variables to a finite (but possibly large) number; then to approxi-
mate the transformed and truncated PDE on the reference domain by a finite element
method; and finally to approximate the expected value of the solution (which is an inte-
gral, possibly high dimensional) by a carefully designed quasi-Monte Carlo (QMC)
method—i.e., an equal weight cubature rule.
In [3,15], the perturbation field was represented by an affine, vector-valued
Karhunen–Loève expansion, under the assumption of uniformly independent and iden-
tically distributed random variables. In this work, we instead expand the perturbation
field V(ω) using periodic random variables, following the recent work [23]. The
advantage of the periodic setting is that it permits higher order convergence of the
QMC approximation to the expectation of the solution to (1.1). The periodic repre-
sentation is equivalent in law to an affine representation of the input random field,
where the random variables entering the series are i.i.d. with the Chebyshev probabil-
ity density ρ(z)=1
π1z2,z(1,1). The density ρis associated with Chebyshev
polynomials of the first kind, which is a popular family of basis functions in the
method of generalized polynomial chaos (gPC). The choice of density is a modeling
assumption, and it might be argued that in many applications choosing the Chebyshev
density over the uniform density is a matter of taste rather than conviction, see [22,
Sections 1–2] for more discussion.
The paper is organized as follows. The problem is formulated in Sect.2,after
which Sect. 3establishes the regularity of the solution with respect to the stochastic
variables; a more difficult task than in most such applications because of the nonlinear
nature of the transformation, but a prerequisite for the design of good QMC rules.
Section 4considers error analysis, estimating separately the errors from dimension
truncation, finite element approximation and QMC cubature. In the latter case, the
error analysis leads to the design of appropriate weight parameters for the function
space, ones that allow rigorous error bounds for the QMC contribution to the error.
Numerical experiments that test the theoretical analysis are presented in Sect. 5, and
some conclusions are drawn in Sect. 6. The Appendix Acontains several technical
results required for the parametric regularity analysis.
2 Problem formulation
2.1 Notations
Let U:= [0,1]Ndenote a set of parameters. Let us fix a bounded domain Dref Rd
with Lipschitz boundary and d∈{2,3}as the reference domain.
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276 H. Hakula et al.
We use multi-index notation throughout this paper. The set of all finitely supported
multi-indices is denoted by F:= {νNN
0:|supp(ν)|<∞}, where supp(ν):=
{jN:νj= 0}is the support of a multi-index ν:= 1
2,...). For any sequence
b=(bj)
j=1of real numbers and any νF, we define
bν:=
jsupp(ν)
bνj
j,
where the product over the empty set is defined as 1 by convention. We also define the
multi-index notation
ν
y:=
jsupp(ν)
νj
yνj
j
for higher order partial derivatives with respect to variable y.
For any matrix M,letσ(M)denote the set of all singular values of Mand let
M2:= max σ(M)denote the matrix spectral norm. For a function von Dref we
define
vL(Dref ):=
ess supxDref |v(x)|if v:Dref R,
ess supxDref v(x)2if v:Dref Rd,
ess supxDref v(x)2if v:Dref Rd×d,
where we apply the vector 2-norm (Euclidean norm) or matrix 2-norm (spectral norm)
depending on whether v(x)is a vector or a matrix. Similarly, we define
vW1,(Dref ):=
maxess sup
xDref |v(x)|,ess sup
xDref ∇v(x)2if v:Dref R,
maxess sup
xDref v(x)2,ess sup
xDref v(x)2if v:Dref Rd,
where v(x)is the gradient vector if v(x)is a scalar, and v(x)is the Jacobian matrix
if v(x)is a vector. We will also need the standard Sobolev norm
vH1
0(Dref ):= ∇vL2(Dref )=Dref ∇v(x)2
2dx1/2
for a scalar function v:Dref Rin H1
0(Dref ). We also define the norm
vCk(D):= max
|ν|≤ksup
xD|ν
xv(x)|for vCk(D),
where DRdis a nonempty domain.
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Uncertainty quantification for random domains... 277
In this paper we will make use of Stirling numbers of the second kind defined by
S(n,m):= 1
m!
m
j=0
(1)mjm
jjn
for integers nm0, except for S(0,0):= 1.
2.2 Parameterization of domain uncertainty
Let V:Dref ×URdbe a vector field such that
V(x,y):= x+1
6
i=1
sin(2πyi)ψi(x), xDref ,yU,(2.1)
with stochastic fluctuations ψi:Dref Rd. Denoting the Jacobian matrix of ψiby
ψ
i, the Jacobian matrix J:Dref Rd×dof vector field Vis
J(x,y):= I+1
6
i=1
sin(2πyi)ψ
i(x), xDref ,yU.
We assume that the family of admissible domains {D(y)}yUis parameterized by
D(y):= V(Dref ,y), yU,
and define the hold-all domain by setting
D:=
yU
D(y).
The convention of explicitly writing down the factor 1/6in(2.1) is not a crucial
part of the analysis, but it ensures that the mean and covariance of the vector field V
match those of an affine and uniform parameterization for representing domain uncer-
tainty using the same sequence of stochastic fluctuations (ψi)
i=1, with the uniform
random variables supported on [−1/2,1/2]. See also the discussion in [23].
In order to ensure that the aforementioned domain parameterization is well-posed
and to enable our subsequent analysis of dimension truncation and finite element
errors, we state the following assumptions for later use:
(A1) For each yU,V(·,y):Dref Rdis an invertible, twice continuously
differentiable vector field.
(A2) For some C>0, there holds
V(·,y)C2(Dref )Cand V1(·,y)C2(D(y)) C
for all yU.
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278 H. Hakula et al.
(A3) There exist constants 0
min 1σmax <such that
σmin min σ(J(x,y)) max σ(J(x,y)) σmax for all xDref ,yU,
where σ(J(x,y)) denotes the set of all singular values of the Jacobian matrix.
(A4) There holds ψiW1,(Dref )<for all iN, and
i=1ψiW1,(Dref )<.
(A5) For some p(0,1), there holds
i=1ψip
W1,(Dref )<.
(A6) ψ1W1,(Dref )≥ψ2W1,(Dref )···.
(A7) The reference domain Dref Rdis a convex, bounded polyhedron.
For later convenience we define the sequence b=(bi)
i=1and the constant ξb0
by setting
bi:= 1
6ψiW1,(Dref )and ξb:=
i=1
bi<.(2.2)
It follows that we can take σmax := 1+ξb.
Remark. Under assumption (A3), there holds
det J(x,y)>0 for all xDref ,yU.(2.3)
This follows from the continuity of the determinant and det J(x,0)=1.
2.3 The variational formulation on the reference domain
The variational formulation of the model problem (1.1) can be stated as follows: for
yU, find u(·,y)H1
0(D(y)) such that
D(y)u(x,y)·∇v(x)dx=D(y)
f(x)v(x)dxfor all vH1
0(D(y)), (2.4)
where fC(D)is assumed to be an analytic function.
We can transport the variational formulation (2.4) to the reference domain by a
change of variable. Let us first define the matrix-valued function
B(x,y):= J(x,y)J(x,y), xDref ,yU,(2.5)
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Uncertainty quantification for random domains... 279
the transport coefficient
A(x,y):= B1(x,y)det J(x,y), xDref ,yU,(2.6)
and the transported source term
fref (x,y):=
f(x,y)det J(x,y),
f(x,y):= fV(x,y),xDref ,yU.
Then we can recast the problem (2.4) on the reference domain as follows: for yU,
find u(·,y)H1
0(Dref )such that
Dref A(x,y)u(x,y)·∇v(x)dx=Dref
fref (x,y)v(x)dx(2.7)
for all vH1
0(Dref ).
The solutions to problems (2.4) and (2.7) are connected to one another by
u(·,y)=uV1(·,y), y⇐⇒ u(·,y)=uV(·,y), y,yU.
In the sequel, we focus on analyzing the problem (2.7).
3 Parametric regularity of the solution
In order to develop higher order rank-1 lattice cubature rules for the purpose of inte-
grating the solution y→ u(·,y)of (2.7) with respect to the parametric variable,
we need to derive bounds on the partial derivatives ν
yu(·,y)in the Sobolev norm
·H1
0(Dref ).
The analysis presented in this section is based on [15] and proceeds as follows:
We derive bounds for ν
yB1(x,y)and ν
ydet J(x,y)for all νFin Lem-
mata 3.1 and 3.2. These results are then used to derive a bound on the partial
derivatives of the coefficient ν
yA(x,y)in Lemma 3.3.
Since the right-hand side of (2.7) depends on the parametric variable yU,we
develop the derivative bounds on ν
yfref (x,y)in Lemma 3.5, with the aid of the
derivative bound developed for ν
y
f(x,y)in Lemma 3.4.
–Anaprioribound is developed for the the solution u(x,y)of (2.7) in Lemma 3.6.
Finally, the main result of this section is stated in Theorem 3.7 which contains the
partial derivative bound for ν
yu(x,y),νF, making use of the aforementioned
results.
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280 H. Hakula et al.
3.1 Parametric regularity of the transport coefficient
Lemma 3.1 Let yU and νF. Under assumptions (A1)(A4), the matrix-valued
function B defined by (2.5)satisfies
ν
yB1(·,y)
L(Dref )
1
σ2
min 4π(1+ξb)
σ2
min |ν|
mν|m|!a|m|m!bm
i1
Si,mi),
with the sequence
ak:= (1+3)k+1(13)k+1
2k+13,k0.(3.1)
Proof We will first derive a regularity bound for ν
yB(·,y)L(Dref ),νF.The
bound for ν
yB(·,y)1L(Dref )follows by applying implicit differentiation to the
identity B1B=I.
From the definition of Jit is easy to see that for any multi-index mFand any
xDref and yUwe have
m
yJ(x,y)=
I+1
6i1sin(2πyi)ψ
i(x)if m=0,
1
6(2π)ksin(2πyj+kπ
2)ψj(x)if m=kej,k1,
0 otherwise.
(3.2)
Taking the matrix 2-norm, we obtain
m
yJ(x,y)2
1+1
6i1ψ
i(x)2if m=0,
1
6(2π)kψj(x)2if m=kej,k1,
0 otherwise,
and hence with (2.2) we obtain
m
yJ(·,y)L(Dref )
1+ξbif m=0,
(2π)kbjif m=kej,k1,
0 otherwise.
(3.3)
Now the Leibniz product rule yields for νF,
ν
yB(·,y)=
mνν
mm
yJ(x,y)νm
yJ(x,y),
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Uncertainty quantification for random domains... 281
and after taking the matrix 2-norm on both sides and then the L-norm over xwe
obtain
ν
yB(·,y)
L(Dref )
mνν
m
m
yJ(·,y)
L(Dref )
νm
yJ(·,y)
L(Dref ).
The bounds in (3.3) indicate that only the derivatives with |supp(ν)|≤2 will survive.
Indeed, for ν=kejwith k1, only m=ejfor =0,...,kremain, while for
ν=kej+kejwith k,k1 and j= j, only m=kejand m=kejremain. On
the other hand, for νwith |supp(ν)|≥3, in each term at least one of the factors must
vanish. We conclude that
ν
yB(·,y)
L(Dref )
(1+ξb)2if ν=0,
2(1+ξb)(2π)kbj+(2k2)(2π)kb2
jif ν=kej,k1,
2(2π)k+kbjbjif ν=kej+kej,k,k1,j= j
0if|supp(ν)|≥3,
(1+ξb)2if ν=0,
(4π)|ν|(1+ξb)jsupp(ν)bjif |supp(ν)|∈{1,2},
0if|supp(ν)|≥3,
(3.4)
where we used bjξbfor all jN.
Our assumption (A3) immediately yields for all yUthat
B1(·,y)L(Dref )=ess sup
xDref max σJ1(x,y)2
1
σ2
min
,
which proves the lemma for the case ν=0. Now we consider νF\{0}. To obtain
the derivative bounds on B1we start with B1B=Iand apply the Leibniz product
rule to obtain
ν
yB1(x,y)B(x,y)=
mνν
mνm
yB1(x,y)m
yB(x,y)=0.
Separating out the m=0term, post-multiplying both sides by B1(x,y), taking the
matrix 2-norm followed by the L-norm over x, and applying (3.4), we obtain
ν
yB1(·,y)
L(Dref )
≤B1(·,y)L(Dref )
0=mν
ν
m
νm
yB1(·,y)
L(Dref )
m
yB(·,y)
L(Dref )
1+ξb
σ2
min
0=mν
|supp(m)|≤2
(4π)|m|ν
m
νm
yB1(·,y)
L(Dref )
jsupp(m)
bj.
123
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282 H. Hakula et al.
The assertion follows by applying Lemmata A.1 and A.2 together with Υν=
ν
yB1(·,y)
L(Dref ),c0=1 2
min,c=4π(1+ξb)/σ 2
min,q=2, and βj=bj.
Lemma 3.2 Under assumptions (A1)(A4), there holds for all yU and νF
that
ν
ydet J(·,y)L(Dref )d!(2+ξb)d(2π)|ν|
wν|w|!bw
i1
Si,w
i).
Proof Let yU. The proof is carried out by induction over all the minors (subde-
terminants) of the matrix J(x,y)=: [J,(x,y)]d
,=1,xDref . For any qd,
let J,(x,y):= [Jt,
t(x,y)]q
t,t=1denote the q×qsubmatrix specified by the
indices := {1,...,
q}and := {
1,...,
q}with 1 1<··· <
qdand
1
1<···<
qd. We will prove by induction on qthat
ν
ydet J,(·,y)L(Dref )q!(2+ξb)q(2π)|ν|
wν|w|!bw
i1
Si,w
i). (3.5)
Similarlyto(3.2), we have for the derivatives of matrix elements
m
yJt,
t(x,y)=
1+1
6i1sin(2πyi)[ψ
i(x)]t,
tif m=0,
1
6(2π)ksin(2πyj+kπ
2)[ψj(x)]t,
tif m=kej,k1,
0 otherwise,
(3.6)
for t,t∈{1,...,q}.Using[ψ
i(x)]t,
t≤ψ
i(x)2and (2.2), we obtain for the
(1×1)-minors
m
ydet Jt,
t(·,y)L(Dref )=m
yJt,t(·,y)L(Dref )
1+ξbif m=0,
(2π)kbjif m=kej,k1,
0 otherwise
Suppose (3.5) holds for all submatrices of size up to (q1)×(q1), and now
we consider the case for a q×qsubmatrix. For arbitrary tq, the Laplace cofactor
expansion yields
det J,(x,y)=
q
t=1
(1)t+tJt,
t(x,y)det J\{t},\{
t}(x,y).
The Leibniz product rule then gives
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Uncertainty quantification for random domains... 283
ν
ydet J,(x,y)
=
q
t=1
(1)t+t
mν
ν
mm
yJt,
t(x,y)νm
ydet J\{t},\{
t}(x,y).
Together with (3.6) and the induction hypothesis (3.5) we obtain
ν
ydet J,(·,y)L(Dref )
q
t=1Jt,
t(·,y)L(Dref )ν
ydet J\{t},\{
t}(·,y)L(Dref )
+
j1
νj
k=1νj
kkejJt,
t(·,y)L(Dref )
×νkejdet J\{t},\{
t}(·,y)L(Dref )
q
t=1(1+ξb)(q1)!(2+ξb)q1(2π)|ν|
wν|w|!bw
i1
Si,w
i)
+
j1
νj
k=1νj
k(2π)kbj(q1)!(2+ξb)q1(2π)|ν|−k
×
wνkej|w|!bwS jk,wj)
i1
i=j
Si,w
i).(3.7)
To simplify the last term, we obtain in complete analogy with the derivation presented
in [23, Lemma 2.2] the identity
j1
νj
k=1νj
kbj
wνkej|w|!bwS jk,wj)
i1
i=j
Si,w
i)
=
wν|w|!bw
i1
Si,w
i).
Plugging this expression into (3.7) and collecting terms yields (3.5), which in turn
completes the proof of the lemma.
Lemma 3.3 Under assumptions (A1)(A4), there holds for all yU and νFthat
ν
yA(·,y)L(Dref )
d!(2+ξb)d
σ2
min 4π(1+ξb)
σ2
min |ν|
mν
(|m|+1)!a|m|m!bm
i1
Si,mi),
123
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284 H. Hakula et al.
where the sequence (ak)
k=0is defined in (3.1).
Proof Let νF. By the Leibniz product rule, we have from (2.6) that
ν
yA(x,y)=
mνν
mm
yB1(x,y)νm
ydet J(x,y),xDref ,yU.
Taking the matrix 2-norm followed by the L-norm over x, and then applying Lem-
mata 3.1 and 3.2, we obtain
ν
yA(·,y)L(Dref )
mνν
m
m
yB1(·,y)
L(Dref )
νm
ydet J(·,y)
L(Dref )
c|ν|
mν
wm
μνmν
mAwBμ
i1S(mi,w
i)Simi
i)
=c|ν|
mν
wmm
wAwBmw
i1
Si,mi),
where Aw:= |w|!a|w|w!bwand Bμ:= |μ|! bμ, and we overestimated some multi-
plying factors to get
ck:= d!(2+ξb)d
σ2
min 4π(1+ξb)
σ2
min k
,k0.
The last equality is due to Lemma A.3.Usinga|w|a|m|and w!≤m!,wehave
wmm
wAwBmwa|m|m!bm
wmm
w|w|!|mw|!,
where the last sum over wcan be rewritten as
|m|
j=0
j!(|m|−j)!
wm
|w|=jm
w=|m|
j=0
j!(|m|−j)!|m|
j=(|m|+1)!,
thus completing the proof.
3.2 Parametric regularity of the transported source term
Lemma 3.4 Let f C(D)be analytic. Then there exist constants C f>0and
ρ1such that ν
xfL(D)Cfν!ρ|ν|for all νNd
0. Furthermore, under the
123
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Uncertainty quantification for random domains... 285
assumptions (A1)(A4), there holds for all yU and νFthat the derivatives of
f(x,y)=f(V(x,y)) are bounded by
ν
y
f(·,y)
L(Dref )Cf(2π)|ν|
mν|m|+d1
d1|m|!ρ|m|bm
i1
Si,mi).
Proof The bound ν
xfL(D)Cfν!ρ|ν|is a consequence of the Cauchy integral
formula for analytic functions of several variables (cf., e.g., [20, Theorem 2.2.1]).
Let yU. Trivially, we see that
f(·,y)L(Dref )Cf, so we may assume in
the following that νF\{0}. We will make use of the multivariate Faà di Bruno
formula [29, Theorem 3.1 and Remark 3.3]
ν
y
f(x,y):=
λNd
0
1≤|λ|≤|ν|
(∂λ
xf)(V(x,y)) κν,λ(x,y), (3.8)
where we define κν,λ(x,y)for general νFand λZdin a recursive manner as
follows: κν,0(x,y):= δν,0,κν,λ(x,y):= 0if|ν|<|λ|or λ0(i.e., λcontains
negative entries), and otherwise
κν+ej,λ(x,y):=
d
=1
0mνν
mm+ej
y[V(x,y)]κνm,λe(x,y). (3.9)
From the definition of Vin (2.1) it is easy to see that
m
y[V(x,y)]=
x+1
6i1sin(2πyi)[ψi(x)]if m=0,
1
6(2π)ksin(2πyj+kπ
2)[ψj(x)]if m=kej,k1,
0 otherwise,
which yields
m+ej
y[V(·,y)]
L(Dref )(2π)k+1bjif m=kej,k0,
0 otherwise.
Thus we obtain from (3.9) the recursion
κν+ej,λ(·,y)L(Dref )bj
νj
k=0
(2π)k+1νj
kd
=1
λ>0
κνkej,λe(·,y)L(Dref ).
123
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286 H. Hakula et al.
By Lemma A.4 with c=2π, we have for all νFand λNd
0that
κν,λ(·,y)L(Dref )(2π)|ν||λ|!
λ!
mν
|m|=|λ|
bm
i1
Si,mi),
which together with (3.8) yields for νF\{0}that
ν