Multi-index Stochastic Collocation Convergence Rates for Random PDEs with Parametric Regularity

Abstract

We analyze the recent Multi-index Stochastic Collocation (MISC) method for computing statistics of the solution of a partial differential equation (PDE) with random data, where the random coefficient is parametrized by means of a countable sequence of terms in a suitable expansion. MISC is a combination technique based on mixed differences of spatial approximations and quadratures over the space of random data, and naturally, the error analysis uses the joint regularity of the solution with respect to both the variables in the physical domain and parametric variables. In MISC, the number of problem solutions performed at each discretization level is not determined by balancing the spatial and stochastic components of the error, but rather by suitably extending the knapsack-problem approach employed in the construction of the quasi-optimal sparse-grids and Multi-index Monte Carlo methods, i.e., we use a greedy optimization procedure to select the most effective mixed differences to include in the MISC estimator. We apply our theoretical estimates to a linear elliptic PDE in which the log-diffusion coefficient is modeled as a random field, with a covariance similar to a Matérn model, whose realizations have spatial regularity determined by a scalar parameter. We conduct a complexity analysis based on a summability argument showing algebraic rates of convergence with respect to the overall computational work. The rate of convergence depends on the smoothness parameter, the physical dimensionality and the efficiency of the linear solver. Numerical experiments show the effectiveness of MISC in this infinite dimensional setting compared with the Multi-index Monte Carlo method and compare the convergence rate against the rates predicted in our theoretical analysis.

Appendices

Appendix 1: Summability of Series Expansion

We start by recalling a useful multivariate Faà di Bruno formula taken from [13, Theorem 2.1].

Lemma 11

Let ${\mathscr {B}}\subset {\mathbb {R}}^d$ be an open domain, $g:{\mathscr {B}}\rightarrow {\mathbb {R}}$ and $f:{\mathbb {R}}\rightarrow {\mathbb {R}}$ be functions of class $C^s(\mathscr {B})$ and denote $h=f\circ g : {\mathscr {B}}\rightarrow {\mathbb {R}}$. For any multi-index ${{\varvec{i}}}\in {\mathbb {N}}^d$, $|{{\varvec{i}}}| \le s$, and any ${{\varvec{x}}}\in {\mathscr {B}}$,

$$\begin{aligned} D^{{\varvec{i}}}h({{\varvec{x}}}) = {{\varvec{i}}}!\sum _{\lambda =1}^{|{{\varvec{i}}}|} f^{(\lambda )}(g({{\varvec{x}}})) \sum _{r=1}^{\lambda } \sum _{p_r({{\varvec{i}}},\lambda )} \prod _{j=1}^r \frac{(D^{{\varvec{\ell }}_j}g({{\varvec{x}}}))^{k_j}}{k_j! ({\varvec{\ell }}_j!)^{k_j}}, \end{aligned}$$

(47)

holds, where

$$\begin{aligned}&p_r({{\varvec{i}}},\lambda ) = \{ (k_j,{\varvec{\ell }}_j)\in {\mathbb {N}}\times {\mathbb {N}}^d_0, \; j=1,\ldots ,r: \;\; \varvec{0}\prec {\varvec{\ell }}_1\prec {\varvec{\ell }}_2 \prec \cdots \prec {\varvec{\ell }}_r, \\&\quad \sum _{j=1}^r k_j = \lambda , \;\;\sum _{j=1}^r k_j{\varvec{\ell }}_j = {{\varvec{i}}}\} \end{aligned}$$

and $\prec$ denotes the lexicographic ordering of multi-indices. The set $p_r({{\varvec{i}}},\lambda )$ denotes the set of possible decompositions of ${{\varvec{i}}}$ as a sum of $\lambda$ multi-indices with $r\le \lambda$ distinct multi-indices, ${\varvec{\ell }}_j$, taken with multiplicity $k_j$ such that $\sum _{j=1}^r k_j=\lambda$.

Also from [13, Corollary 2.9], we have that, for any ${{\varvec{i}}}\in {\mathbb {N}}^d$,

$$\begin{aligned} {{\varvec{i}}}! \sum _{r=1}^{\lambda } \sum _{p_r({{\varvec{i}}},\lambda )} \prod _{j=1}^r \frac{1}{k_j! ({\varvec{\ell }}_j!)^{k_j}} = S_{|{{\varvec{i}}}|,\lambda }, \end{aligned}$$

where $S_{n,k}$ is the Stirling number of the second kind, which counts the number of ways to partition a set of n objects into k non-empty subsets. Similarly, the Bell number, $B_n=\sum _{k=0}^{n} S_{n,k}$, counts the number of partitions of a set of n objects, whereas the ordered Bell numbers are defined by $\tilde{B}_n = \sum _{k=0}^{n} k! S_{n,k}$ and satisfy the recursive relation $\tilde{B}_n = \sum _{k=0}^{n-1} \left( {\begin{array}{c}n\\ k\end{array}}\right) \tilde{B}_k$. Clearly, $B_n\le \tilde{B}_n$. Moreover, the bound

$$\begin{aligned} B_n\le \tilde{B}_n \le n!/(\log 2)^n \end{aligned}$$

(48)

was given in [3, Lemma A.3]. We now use these results to show the following result

Lemma 12

Let ${\mathscr {B}} \subset {\mathbb {R}}^d$ be an open-bounded domain and $\kappa \in C^s(\overline{{\mathscr {B}}})$ (real or complex valued) for $s \ge 0$. Then, $a = e^\kappa \in C^s(\overline{{\mathscr {B}}})$ and we have the estimate

$$\begin{aligned} \Vert a\Vert _{C^s(\overline{{\mathscr {B}}})} \le \frac{s!}{(\log 2)^s}\Vert a\Vert _{C^0(\overline{{\mathscr {B}}})}(1+\Vert \kappa \Vert _{C^s(\overline{{\mathscr {B}}})})^s. \end{aligned}$$

Proof

Using formula (47), we have for any ${{\varvec{i}}}\in {\mathbb {N}}^d$, $|{{\varvec{i}}}|\le s$ and any ${{\varvec{x}}}\in {\mathscr {B}}$

$$\begin{aligned} |D^{{\varvec{i}}}e^{\kappa ({{\varvec{x}}})}|&= {{\varvec{i}}}!\sum _{\lambda =1}^{|{{\varvec{i}}}|} e^{\kappa ({{\varvec{x}}})} \sum _{r=1}^{\lambda } \sum _{p_r({{\varvec{i}}},\lambda )} \prod _{j=1}^r \frac{|D^{{\varvec{\ell }}_j}\kappa ({{\varvec{x}}})|^{k_j}}{k_j! ({\varvec{\ell }}_j!)^{k_j}} \le \Vert a\Vert _{C^0(\overline{{\mathscr {B}}})} \sum _{\lambda =1}^{|{{\varvec{i}}}|} \Vert \kappa \Vert _{C^s(\overline{{\mathscr {B}}})}^\lambda S_{|{{\varvec{i}}}|,\lambda }\\&\le \Vert a\Vert _{C^0(\overline{{\mathscr {B}}})} (1+\Vert \kappa \Vert _{C^s(\overline{{\mathscr {B}}})})^{|{{\varvec{i}}}|} B_n. \end{aligned}$$

The result then follows from the bound on the Bell numbers in (48). $\square$

1.1 $L^p(\Gamma )$ Summability, Pointwise in Space

We now consider a diffusion coefficient as in Assumption A2:

$$\begin{aligned} a({{\varvec{x}}},{{\varvec{y}}}) = \exp \left\{ \sum _{j\ge 1} \psi _j({{\varvec{x}}})y_j\right\} = \prod _{j=1}^\infty e^{y_j\psi _j({{\varvec{x}}})}, \qquad {{\varvec{x}}}\in {\mathscr {B}}, \end{aligned}$$

with $y_j$, $j \ge 1$, independent random variables, all uniformly distributed in $[-1,1]$ and recall the definition of the sequence ${{\varvec{b}}}_{s}=\{b_{s,j}\}_{j\ge 1}$, for all $s\in {\mathbb {N}}$ in (6).

Lemma 13

If ${{\varvec{b}}}_0\in \ell ^2$ then ${\mathbb {E}}\left[ a({{\varvec{x}}})^p\right] <\infty$ for all $0<p<\infty$ and $\forall {{\varvec{x}}}\in {\mathscr {B}}$.

Proof

For any ${{\varvec{x}}}\in {\mathscr {B}}$, we estimate the p-th moment of $a({{\varvec{x}}},{{\varvec{y}}})$, exploiting the independence of the random variables, $y_j$:

$$\begin{aligned} {\mathbb {E}}\left[ a({{\varvec{x}}})^p\right]&= {\mathbb {E}}\left[ \prod _{j=1}^\infty e^{py_j\psi _j({{\varvec{x}}})}\right] = \prod _{j=1}^\infty {\mathbb {E}}\left[ e^{py_j\psi _j({{\varvec{x}}})}\right] = \prod _{j=1}^\infty \frac{\sinh (p\psi _j({{\varvec{x}}}))}{p\psi _j({{\varvec{x}}})} \\&= \exp \left\{ \sum _{j=1}^\infty \log \left( \frac{\sinh (p\psi _j({{\varvec{x}}}))}{p\psi _j({{\varvec{x}}})}\right) \right\} \end{aligned}$$

where in the last two equalities we have implicitly assumed that $\sinh (z)/z=1$ for $z=0$. Setting $\theta _0(p;{{\varvec{x}}}) = \prod _{j=1}^\infty \frac{\sinh (p\psi _j({{\varvec{x}}}))}{p\psi _j({{\varvec{x}}})}$ and observing that $\log (\sinh (z)/z) \sim z^2/6$, we have

$$\begin{aligned} {\mathbb {E}}\left[ a({{\varvec{x}}})^p\right] = \theta _0(p;{{\varvec{x}}})<\infty \quad \forall {{\varvec{x}}}\in {\mathscr {B}}, \;\; 0<p<\infty \qquad \Longleftrightarrow \qquad \sum _{j=1}^\infty \psi _j({{\varvec{x}}})^2 <\infty . \end{aligned}$$

Since $\sum _{j=1}^\infty b_{0,j}^2 <\infty$ implies $\sum _{j=1}^\infty \psi _j({{\varvec{x}}})^2 <\infty$ for any ${{\varvec{x}}}\in {\mathscr {B}}$, this concludes the proof. $\square$

A similar result holds for higher-order derivatives of a.

Lemma 14

Let $s\in {\mathbb {N}}_+$. If ${{\varvec{b}}}_s\in \ell ^2$, then for any ${{\varvec{i}}}\in {\mathbb {N}}^d$, $|{{\varvec{i}}}|=s$, ${\mathbb {E}}\left[ (D^{{\varvec{i}}}a({{\varvec{x}}}))^{2p}\right] <\infty$ for all $0<p<\infty$ and $\forall {{\varvec{x}}}\in {\mathscr {B}}$.

Proof

Since the calculations are tedious, we prove the result here for $s=1$ only. Using the chain rule, we have

$$\begin{aligned} (\partial _{x_i}a({{\varvec{x}}},{{\varvec{y}}}))^{2p}&= \left( \sum _{j \ge 1} a({{\varvec{x}}},{{\varvec{y}}})\partial _{x_i}\psi _j({{\varvec{x}}})y_j\right) ^{2p} \\&= a({{\varvec{x}}},{{\varvec{y}}})^{2p}{\mathop {\mathop {\sum }\limits _{{{\varvec{q}}}\in {\mathbb {N}}^{{\mathbb {N}}}}}\limits _{|{{\varvec{q}}}|=2p}} (2p)! \prod _{j=1}^\infty \frac{1}{q_j!}(\partial _{x_i}\psi _j({{\varvec{x}}})y_j)^{q_j}\\&={\mathop {\mathop {\sum }\limits _{{{\varvec{q}}}\in {\mathbb {N}}^{{\mathbb {N}}}}}\limits _{|{{\varvec{q}}}|=2p}} (2p)! \prod _{j=1}^\infty \frac{1}{q_j!}(\partial _{x_i}\psi _j({{\varvec{x}}})y_j)^{q_j}e^{2py_j\psi _j({{\varvec{x}}})}. \end{aligned}$$

Hence,

$$\begin{aligned} {\mathbb {E}}\left[ (\partial _{x_i}a({{\varvec{x}}},{{\varvec{y}}}))^{2p}\right] = {\mathop {\mathop {\sum }\limits _{{{\varvec{q}}}\in {\mathbb {N}}^{{\mathbb {N}}}}}\limits _{|{{\varvec{q}}}|=2p}} (2p)! \prod _{j=1}^\infty (\partial _{x_i}\psi _j({{\varvec{x}}}))^{q_j}{\mathbb {E}}\left[ \frac{1}{q_j!}y_j^{q_j}e^{2py_j\psi _j({{\varvec{x}}})}\right] . \end{aligned}$$

We now distinguish between even or odd $q_j$. For even $q_j$, we have

$$\begin{aligned} {\mathbb {E}}\left[ \frac{1}{q_j!}y_j^{q_j}e^{2py_j\psi _j({{\varvec{x}}})}\right] \le {\mathbb {E}}\left[ \frac{1}{q_j!}e^{2py_j\psi _j({{\varvec{x}}})}\right] = \frac{1}{q_j!} \frac{\sinh (2p\psi _j({{\varvec{x}}}))}{2p\psi _j({{\varvec{x}}})}, \end{aligned}$$

while for $q_j$ odd we have

$$\begin{aligned} {\mathbb {E}}\left[ \frac{1}{q_j!}y_j^{q_j}e^{2py_j\psi _j({{\varvec{x}}})}\right]&=\! \frac{1}{q_j!}\int _{-1}^1 \frac{1}{2}y^{q_j}e^{2py\psi _j({{\varvec{x}}})}dy \!=\! \frac{1}{q_j!}\int _{0}^1 y^{q_j}\sinh (2py\psi _j({{\varvec{x}}}))dy \\&=\frac{1}{q_j!}\sum _{n=0}^\infty \frac{(2p\psi _j({{\varvec{x}}}))^{2n+1}}{(2n+1)!}\int _0^1y^{2n+1+q_j}dy \\&= \frac{1}{q_j!}\sum _{n=0}^\infty \frac{(2p\psi _j({{\varvec{x}}}))^{2n+1}}{(2n+1)!(2n+2+q_j)} \\&\le \frac{1}{(q_j+1)!}\sinh (2p|\psi _j({{\varvec{x}}})|) \le \frac{2pb_{1,j}}{(q_j+1)!} \frac{\sinh (2p\psi _j({{\varvec{x}}}))}{2p\psi _j({{\varvec{x}}})}. \end{aligned}$$

Hence, defining the function

$$\begin{aligned} f(q_j) = {\left\{ \begin{array}{ll} \frac{1}{q_j!} &{} {\text {for }}q_j {\text { even}},\\ \frac{2pb_{1,j}}{(q_j+1)!} &{} {\text {for }}q_j {\text { odd}}, \end{array}\right. } \end{aligned}$$

we have

$$\begin{aligned} {\mathbb {E}}\left[ (\partial _{x_i}a({{\varvec{x}}},{{\varvec{y}}}))^{2p}\right]&\le {\mathop {\mathop {\sum }\limits _{{{\varvec{q}}}\in {\mathbb {N}}^{{\mathbb {N}}}}}\limits _{|{{\varvec{q}}}|=2p}} (2p)! \prod _{j=1}^\infty b_{1,j}^{q_j} f(q_j) \frac{\sinh (2p\psi _j({{\varvec{x}}}))}{2p\psi _j({{\varvec{x}}})} \\&= \theta _0(2p;{{\varvec{x}}}){\mathop {\mathop {\sum }\limits _{{{\varvec{q}}}\in {\mathbb {N}}^{{\mathbb {N}}}}}\limits _{|{{\varvec{q}}}|=2p}} (2p)! \prod _{j=1}^\infty b_{1,j}^{q_j} f(q_j)\\&\le \theta _0(2p;{{\varvec{x}}}){\mathop {\mathop {\sum }\limits _{{{\varvec{q}}}\in {\mathbb {N}}^{{\mathbb {N}}}}}\limits _{|{{\varvec{q}}}|=2p, {{\varvec{q}}}\text { even}}} (2p)! (1+2p)^{|{{\varvec{q}}}|_0} \prod _{j=1}^\infty \frac{b_{1,j}^{q_j}}{q_j!} \\&\le (1+2p)^{p} \theta _0(2p;{{\varvec{x}}}){\mathop {\mathop {\sum }\limits _{{{\varvec{q}}}\in {\mathbb {N}}^{{\mathbb {N}}}}}\limits _{|{{\varvec{q}}}|=p}} (2p)! \prod _{j=1}^\infty \frac{b_{1,j}^{2q_j}}{(2q_j)!} \\&\le (1+2p)^{p} (2p)^p \theta _0(2p;{{\varvec{x}}}){\mathop {\mathop {\sum }\limits _{{{\varvec{q}}}\in {\mathbb {N}}^{{\mathbb {N}}}}}\limits _{|{{\varvec{q}}}|=p}} p! \prod _{j=1}^\infty \frac{(b_{1,j}^2)^{q_j}}{q_j!} \\&= (1+2p)^{p} (2p)^p \theta _0(2p;{{\varvec{x}}}) \left( \sum _{j\ge 1} b_{1,j}^2\right) \end{aligned}$$

from which we see that ${\mathbb {E}}\left[ (\partial _{x_i}a({{\varvec{x}}},{{\varvec{y}}}))^{2p}\right]$ is bounded for any $0\le p<\infty$ and any ${{\varvec{x}}}\in {\mathscr {B}}$ if ${{\varvec{b}}}_1\in \ell ^2$. $\square$

1.2 $L^p(\Gamma )$ Summability, Uniform in Space

Assuming now that ${{\varvec{b}}}_s\in \ell ^2$ so that the random field, a, is s-times differentiable in an $L^p(\Gamma )$ sense according to Lemma 14, we show that this implies some uniform $L^p(\Gamma )$ summability as detailed in the next lemma.

Lemma 15

Let $s\in {\mathbb {N}}_+$. If ${{\varvec{b}}}_s\in \ell ^2$ then ${\mathbb {E}}\left[ \Vert a\Vert ^p_{W^{\upsilon ,\infty }({\mathscr {B}})}\right] <\infty$ for all $1\le p<\infty$ and $\upsilon <s$.

Proof

We exploit the Sobolev embedding, $W^{\upsilon +\frac{d}{2q},2q}({\mathscr {B}}) \subseteq W^{\upsilon ,\infty }({\mathscr {B}})$, for all $\upsilon \ge 0$ and $q\ge 1$. For $q\ge \max \{d/2(s-\upsilon ),p/2\}$, we have

$$\begin{aligned}&{\mathbb {E}}\left[ \Vert a\Vert ^{p}_{W^{\upsilon ,\infty }({\mathscr {B}})}\right] \le {\mathbb {E}}\left[ \Vert a\Vert ^{2q}_{W^{s-\frac{d}{2q},\infty }({\mathscr {B}})}\right] \lesssim {\mathbb {E}}\left[ \Vert a\Vert ^{2q}_{W^{s,2q}({\mathscr {B}})}\right] \\&\quad = {\mathbb {E}}\left[ \sum _{|{{\varvec{i}}}|\le s} \int _{{\mathscr {B}}} (D^{{\varvec{i}}}a({{\varvec{x}}}))^{2q}d{{\varvec{x}}}\right] = \sum _{|{{\varvec{i}}}|\le s} \int _{{\mathscr {B}}} {\mathbb {E}}\left[ (D^{{\varvec{i}}}a({{\varvec{x}}}))^{2q}\right] d{{\varvec{x}}}< \infty , \end{aligned}$$

where the last term is bounded from Lemma 14. $\square$

Now, we directly observe by taking $\upsilon =0$ in the previous result that $a_{\max } = \Vert a\Vert _{L^\infty ({\mathscr {B}})}$ has bounded moments,

$$\begin{aligned}{\mathbb {E}}\left[ a_{\max }^p\right] <\infty , \end{aligned}$$

for all $1\le p<\infty$ and $0<s$. Finally, by observing that, due to (2), in we have that $a_{\min } = \frac{1}{\Vert a^{-1}\Vert _{L^\infty ({\mathscr {B}})}}$ has the same distribution as $a_{\max }$. As a consequence, $a_{\min }$ has bounded moments as well. This implies in turn that (3) holds and thus problem (1) is well posed in the Bochner space, $L^p\left( \Gamma ;H^1_0({\mathscr {B}})\right)$. That is,

Corollary 16

(Well-posedness with log uniform coefficient) We have for $0<\nu$ that the problem in Example 1 is well posed in the Bochner space $L^p\left( \Gamma ;H^1_0({\mathscr {B}})\right)$. The corresponding solution, u, satisfies

$$\begin{aligned} \Vert u\Vert _{L^p(\Gamma ;H^1_0({\mathscr {B}}))} \le C {\mathbb {E}}\left[ \frac{1}{a_{\min }^p}\right] ^{1/p} \Vert \varsigma \Vert _{H^{-1}({\mathscr {B}})}. \end{aligned}$$

We observe that higher regularity of the solution, u, can be obtained by using larger values of s in Lemma 15. This in turn yields control on moments of $W^{\upsilon ,\infty }({\mathscr {B}})$ norms of the coefficient, a, and following, for instance, estimates similar to (2.10) in [18, Theorem 2.4], we can estimate moments of the $H^{1+s}({\mathscr {B}})$ norm of the solution, u. These regularity estimates, once combined with pathwise error estimates for the combination technique, can be further used to show the corresponding $\nu$-dependent convergence rates of MIMC [23], for Example 1, similar to what was presented in Sect. 5 for MISC in the current work.

Appendix 2: Shift Theorem for Problem (1)

Here, we seek to establish a shift theorem for the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} - {\text {div}}(a({{\varvec{x}}})\,\nabla u({{\varvec{x}}})) = \varsigma ({{\varvec{x}}}) &{} \text {in}\quad {\mathscr {B}}=[0,1]^D \\ u({{\varvec{x}}}) \,=\, 0 &{} \text {on}\quad \partial {\mathscr {B}}, \end{array}\right. } \end{aligned}$$

(49)

under suitable assumptions on a and $\varsigma$.

With respect to problem (1), for convenience, we drop the dependence on the parameter vector, ${{\varvec{y}}}$. We consider an odd periodic extension of $\varsigma$, on $[-1,1]^D$, and an even periodic extension of the coefficient a on $[-1,1]^D$, named, respectively, $\tilde{\varsigma }$ and $\tilde{a}$. More precisely, for ${{\varvec{j}}}=\{0,1\}^D$, we denote by ${{\varvec{x}}}_{{\varvec{j}}}= ((-1)^{j_1}x_1,\ldots ,(-1)^{j_D}x_D)$ and

$$\begin{aligned}&\tilde{\varsigma }({{\varvec{x}}}_{{\varvec{j}}}+2{{\varvec{k}}}) = (-1)^{|{{\varvec{j}}}|}\varsigma ({{\varvec{x}}}), \quad \tilde{a}({{\varvec{x}}}_{{\varvec{j}}}+2{{\varvec{k}}})=a({{\varvec{x}}}), \quad \forall {{\varvec{x}}}\in [0,1]^2, \;\; {{\varvec{j}}}\in \{0,1\}^D, \quad {{\varvec{k}}}\in {\mathbb {N}}^D. \end{aligned}$$

The following Shift theorem holds for problem (49).

Lemma 17

If the coefficient a is such that its periodic extension satisfies $\tilde{a}\in W^{s,\infty }({\mathbb {R}}^D)$, $s\ge 0$ and $\varsigma \in C^\infty _0({\mathscr {B}})$ then $u\in H^{s+1}({\mathscr {B}})$.

Proof

We define the extended problem

$$\begin{aligned} {\left\{ \begin{array}{ll} - {\text {div}}(\tilde{a}({{\varvec{x}}})\,\nabla \tilde{u}({{\varvec{x}}})) = \tilde{\varsigma }({{\varvec{x}}}) &{} \text {in}\quad \tilde{{\mathscr {B}}}=[-1,1]^D \\ \int _{\tilde{{\mathscr {B}}}} u({{\varvec{x}}}) \,=\, 0 \\ {\text {periodic boundary conditions on }}\partial \tilde{{\mathscr {B}}}. \end{array}\right. } \end{aligned}$$

Since by assumption $\tilde{a}\in L^\infty ({\mathbb {R}}^D)$ and $\tilde{\varsigma }\in L^2(\tilde{{\mathscr {B}}})$, this problem has a unique solution, $\tilde{u}\in H^1_{per}(\tilde{{\mathscr {B}}})\setminus {\mathbb {R}}$, where we denote with $H^s_{per}(\tilde{{\mathscr {B}}})$ the space of periodic functions with (periodic) square integrable derivatives up to order s. It is easy to check that the solution $\tilde{u}$ is odd, that is $\tilde{u}({{\varvec{x}}}_{{\varvec{j}}})=(-1)^{{{\varvec{j}}}}\tilde{u}({{\varvec{x}}})$, $\forall {{\varvec{x}}}\in [0,1]^D$, hence $\tilde{u}=0$ (in the sense of traces) on $\partial {\mathscr {B}}$ and it coincides with the (unique) solution of (49) on ${\mathscr {B}}$. Moreover, standard elliptic regularity arguments allow us to say that $\tilde{u}\in H^s_{per}(\tilde{{\mathscr {B}}})$, hence $u\in H^s({\mathscr {B}})$. $\square$