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A Multi Level Monte Carlo method with control variate for elliptic PDEs with log-normal coefficients

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Abstract

We consider the numerical approximation of the stochastic Darcy problem with log-normal permeability field and propose a novel Multi Level Monte Carlo (MLMC) approach with a control variate variance reduction technique on each level. We model the log-permeability as a stationary Gaussian random field with a covariance function belonging to the so called Matérn family, which includes both fields with very limited and very high spatial regularity. The control variate is obtained starting from the solution of an auxiliary problem with smoothed permeability coefficient and its expected value is effectively computed with a Stochastic Collocation method on the finest level in which the control variate is applied. We analyze the variance reduction induced by the control variate, and the total mean square error of the new estimator. To conclude we present some numerical examples and a comparison with the standard MLMC method, which shows the effectiveness of the proposed method.

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Acknowledgments

F. Nobile and F. Tesei have been partially supported by the Swiss National Science Foundation under the Project No. 140574 “Efficient numerical methods for flow and transport phenomena in heterogeneous random porous media” and by the Center for ADvanced MOdeling Science (CADMOS).

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Correspondence to Francesco Tesei.

Appendices

Appendix 1: Proof of Lemma 4

We consider a fixed \(\omega \in \varOmega \) and we will not specify the dependence on \(\omega \) in the proof. In order to prove Lemma 4 we will need the following three preliminary lemmas.

Lemma 7

Let \(\gamma (x)\) be a deterministic function in \({\mathrm {C}}^{\alpha }({\mathbb {R}}^d)\) and, \(\forall h\in {{\mathbb {R}}^d}\), let us define \(D_{h,\beta } \gamma (x)\) as

$$\begin{aligned} D_{h,\beta } \gamma (x) = \frac{\gamma (x+h)-\gamma (x)}{|h|^{\beta }}. \end{aligned}$$

Then, \(\forall \) \( 0<\beta \le \min (1,\alpha )\) it holds

$$\begin{aligned} \Vert D_{h,\beta }\gamma \Vert _{{\mathrm {C}}^{\alpha -\beta }({\mathbb {R}}^d)} \le (1+2\sqrt{d}) \Vert \gamma \Vert _{{\mathrm {C}}^{\alpha }({\mathbb {R}}^d)} . \end{aligned}$$

and in particular

$$\begin{aligned} | D_{h,\beta }\gamma |_{{\mathrm {C}}^{\alpha -\beta }({\mathbb {R}}^d)} \le 2\sqrt{d} |\gamma |_{{\mathrm {C}}^{\alpha }({\mathbb {R}}^d)} . \end{aligned}$$

Proof

Let us denote \(\alpha = A + s\), with \(A\in {\mathbb {N}}\) and \(s\in (0,1]\).

\(\bullet \) We first consider the case \(0\le \beta \le \alpha \le 1\) so that \(A=0\) and \(s=\alpha \). The norm we want to bound can be written as

$$\begin{aligned} \Vert D_{h,\beta }\gamma \Vert _{{\mathrm {C}}^{\alpha -\beta }({\mathbb {R}}^d)}= & {} \Vert D_{h,\beta }\gamma \Vert _{{\mathcal {C}}^{0}({\mathbb {R}}^d)} + | D_{h,\beta }\gamma |_{{\mathrm {C}}^{\alpha -\beta }({\mathbb {R}}^d)}=\\= & {} \underbrace{ \Vert D_{h,\beta }\gamma \Vert _{{\mathcal {C}}^{0}({\mathbb {R}}^d)}}_{(i)} + \underbrace{\sup _{x,t\in {\mathbb {R}}^d} \frac{|(D_{h,\beta }\gamma )(x+t) - (D_{h,\beta }\gamma )(x)|}{|t|^{\alpha -\beta }}}_{(ii)}. \end{aligned}$$

The first term can be bounded as

$$\begin{aligned} (i)&= \sup _{x \in {\mathbb {R}}^d} \frac{|\gamma (x+h)-\gamma (x)|}{|h|^{\beta }}\\&\le \sup _{x \in {\mathbb {R}}^d} \max \left\{ \sup _{|h|\ge 1} \frac{|\gamma (x+h)-\gamma (x)|}{|h|^{\beta }}, \sup _{|h|\le 1} \frac{|\gamma (x+h)-\gamma (x)|}{|h|^{\beta }} \right\} \\&\le \sup _{x \in {\mathbb {R}}^d}\max \left\{ \sup _{|h|\ge 1} \left( |\gamma (x+h)|+|\gamma (x)|\right) , \sup _{|h|\le 1} \frac{|\gamma (x+h)-\gamma (x)|}{|h|^{\alpha }}|h|^{\alpha -\beta } \right\} \\&\le 2\Vert \gamma \Vert _{{\mathcal {C}}^{0}({\mathbb {R}}^d)} + | \gamma |_{{\mathrm {C}}^{\alpha }({\mathbb {R}}^d)} . \end{aligned}$$

The second term can be bounded as

$$\begin{aligned} (ii)= & {} \max \left\{ \sup _{\begin{array}{c} x,t\in {\mathbb {R}}^d\\ |t|\ge |h| \end{array} } \frac{|\gamma (x+t+h)-\gamma (x+t) - \gamma (x+h)+\gamma (x)|}{|h|^{\beta }|t|^{\alpha -\beta }},\right. \\&\left. \sup _{\begin{array}{c} x,t\in {\mathbb {R}}^d\\ |t|\le |h| \end{array} } \frac{|\gamma (x+h+t)-\gamma (x+h) - \gamma (x+t)+\gamma (x)|}{|h|^{\beta }|t|^{\alpha -\beta }} \right\} \\\le & {} \max \left\{ \sup _{\begin{array}{c} x,t\in {\mathbb {R}}^d\\ |t|\ge |h| \end{array} } \frac{|\gamma (x+t+h)-\gamma (x+t)|+ |\gamma (x+h)-\gamma (x)|}{|h|^{\beta }|h|^{\alpha -\beta }},\right. \nonumber \\&\left. \sup _{\begin{array}{c} x,t\in {\mathbb {R}}^d\\ |t|\le |h| \end{array} } \frac{|\gamma (x+t+h)-\gamma (x+h)| + |\gamma (x+t)-\gamma (x)|}{|t|^{\beta }|t|^{\alpha -\beta }} \right\} \\\le & {} 2 | \gamma |_{{\mathrm {C}}^{\alpha }({\mathbb {R}}^d)} \end{aligned}$$

Hence we get \(\Vert D_{h,\beta }\gamma \Vert _{{\mathrm {C}}^{\alpha -\beta }({\mathbb {R}}^d)}\le 3 \Vert \gamma \Vert _{{\mathrm {C}}^{\alpha }({\mathbb {R}}^d)}\) and \(| D_{h,\beta }\gamma |_{{\mathrm {C}}^{\alpha -\beta }({\mathbb {R}}^d)}\le 2 |\gamma |_{{\mathrm {C}}^{\alpha }({\mathbb {R}}^d)}\).

\(\bullet \) Let us consider now the case \(0<\beta \le 1<\alpha \) so that \(\alpha =A+s\) with \(A\ge 1\). The proof can be further divided in two parts: \(s>\beta \) and \(s<\beta \) since for \(s=\beta \) the result is obvious. We start with the case \(s> \beta \). The norm that we want to bound can be written as

$$\begin{aligned} \Vert D_{h,\beta }\gamma \Vert _{{\mathrm {C}}^{\alpha -\beta }({\mathbb {R}}^d)} =&\Vert D_{h,\beta }\gamma \Vert _{{\mathcal {C}}^{0}({\mathbb {R}}^d)} + \sum _{k=1}^{A} |D_{h,\beta }\gamma |_{{\mathcal {C}}^{k}({\mathbb {R}}^d)} + | D_{h,\beta }\gamma |_{{\mathrm {C}}^{\alpha -\beta }({\mathbb {R}}^d)}\\ =&\underbrace{ \Vert D_{h,\beta }\gamma \Vert _{{\mathcal {C}}^{0}({\mathbb {R}}^d)}}_{(i)}+ \underbrace{ \sum _{k=1}^{A} \max _{|i|_1=k} \Vert D^{i}(D_{h,\beta }\gamma )\Vert _{{\mathcal {C}}^{0}({\mathbb {R}}^d)} }_{(ii)}\\&+ \underbrace{\max _{|i|_1=A}\sup _{x,t}\left\{ \dfrac{|D^{i}(D_{h,\beta }\gamma )(x+t) - D^{i}(D_{h,\beta }\gamma )(x)|}{|t|^{s-\beta }} \right\} }_{(iii)}. \end{aligned}$$

In what follows, we denote \(\xi _x^{y}\) a point of the segment \(\overline{xy}\), i.e. \(\xi _x^{y}=\theta x + (1-\theta )y\) for some \(\theta \in [0,1]\). The first term (i) can be bounded as

$$\begin{aligned} (i)= & {} \sup _{x \in {\mathbb {R}}^d} \frac{|\gamma (x+h)-\gamma (x)|}{|h|^{\beta }}\\\le & {} \sup _{x \in {\mathbb {R}}^d} \max \left\{ \sup _{|h|\ge 1} \frac{|\gamma (x+h)-\gamma (x)|}{|h|^{\beta }}, \sup _{|h|\le 1} \frac{|\gamma (x+h)-\gamma (x)|}{|h|^{\beta }} \right\} \\\le & {} \sup _{x \in {\mathbb {R}}^d} \max \left\{ \sup _{|h|\ge 1} \left( |\gamma (x+h)|+|\gamma (x)|\right) , \sup _{|h|\le 1} \frac{|\nabla \gamma (\xi _x^{x+h})\cdot h|}{|h|^{\beta }} \right\} \\\le & {} 2\Vert \gamma \Vert _{{\mathcal {C}}^{0}({\mathbb {R}}^d)} + \sqrt{d}| \gamma |_{{\mathcal {C}}^{1}({\mathbb {R}}^d)} . \end{aligned}$$

Each term of (ii), for \(k=1,\ldots ,A-1\), can be bounded as

$$\begin{aligned}&\max _{|i|_1=k} \left\{ \sup _{x\in {\mathbb {R}}^d} \frac{|D^i \gamma (x+h) - D^i \gamma (x)| }{|h|^{\beta }} \right\} \\&\quad \le \max _{|i|_1=k} \left\{ \sup _{x \in {\mathbb {R}}^d} \max \left\{ \sup _{|h|\ge 1} \left( |D^i \gamma (x+h)|+|D^i \gamma (x)|\right) , \sup _{|h|\le 1} \frac{|\nabla D^i \gamma (\xi _{i,x}^{x+h})\cdot h|}{|h|^{\beta }}\right\} \right\} \\&\quad \le 2| \gamma |_{{\mathcal {C}}^{k}({\mathbb {R}}^d)} + \sqrt{d}| \gamma |_{{\mathcal {C}}^{k+1}({\mathbb {R}}^d)} . \end{aligned}$$

The last term of (ii) for \(k=A\), analogously to what we did in the case \(0<\beta \le \alpha \le 1\), can be bounded as

$$\begin{aligned} |D_{h,\beta }\gamma |_{{\mathcal {C}}^{A}({\mathbb {R}}^d)} \le 2| \gamma |_{{\mathcal {C}}^{A}({\mathbb {R}}^d)} + | \gamma |_{{\mathrm {C}}^{\alpha }({\mathbb {R}}^d)}. \end{aligned}$$

Hence the term (ii) can be bounded as

$$\begin{aligned} (ii)\le 2| \gamma |_{{\mathcal {C}}^{1}({\mathbb {R}}^d)} +(2+\sqrt{d})\sum _{k=2}^A | \gamma |_{{\mathcal {C}}^{k}({\mathbb {R}}^d)}+ | \gamma |_{{\mathrm {C}}^{\alpha }({\mathbb {R}}^d)}. \end{aligned}$$

The last term (iii) can be bounded as

$$\begin{aligned} (iii)\le & {} \max _{|i|_1=A}\left\{ \max \left\{ \sup _{\begin{array}{c} x,t\in {\mathbb {R}}^d\\ |t|\ge |h| \end{array} } \frac{|D^i \gamma (x+t+h)-D^i \gamma (x+t)|+|D^i \gamma (x+h)-D^i \gamma (x)|}{|h|^{\beta }|t|^{s-\beta }},\right. \right. \\&\left. \left. \sup _{\begin{array}{c} x,t\in {\mathbb {R}}^d\\ |t|\le |h| \end{array} } \frac{|D^i \gamma (x+t+h)-D^i \gamma (x+t)| + |D^i \gamma (x+h)-D^i \gamma (x)|}{|h|^{\beta }|t|^{s-\beta }} \right\} \right\} \le \! 2 | \gamma |_{{\mathrm {C}}^{\alpha }({\mathbb {R}}^d)} . \end{aligned}$$

So the norm of \(\Vert D_{h,\beta }\gamma \Vert _{{\mathrm {C}}^{\alpha -\beta }({\mathbb {R}}^d)}\) can we bounded as

$$\begin{aligned}&\Vert D_{h,\beta }\gamma \Vert _{{\mathrm {C}}^{\alpha -\beta }({\mathbb {R}}^d)}\le (i)+(ii)+(iii)\le 2\Vert \gamma \Vert _{{\mathcal {C}}^{0}({\mathbb {R}}^d)} +(2+\sqrt{d})\sum _{k=1}^A | \gamma |_{{\mathcal {C}}^{k}({\mathbb {R}}^d)}\\&\quad +\, 3 | \gamma |_{{\mathrm {C}}^{\alpha }({\mathbb {R}}^d)}\le (2+\sqrt{d}) \Vert \gamma \Vert _{{\mathrm {C}}^{\alpha }({\mathbb {R}}^d)}. \end{aligned}$$

and \(| D_{h,\beta }\gamma |_{{\mathrm {C}}^{\alpha -\beta }({\mathbb {R}}^d)}\le 2 | \gamma |_{{\mathrm {C}}^{\alpha }({\mathbb {R}}^d)} \).

\(\bullet \) Let us consider now the case \(s<\beta \). The quantity we want to bound becomes

$$\begin{aligned} \Vert D_{h,\beta }\gamma \Vert _{{\mathrm {C}}^{\alpha -\beta }({\mathbb {R}}^d)} =&\Vert D_{h,\beta }\gamma \Vert _{{\mathcal {C}}^{0}({\mathbb {R}}^d)} + \sum _{k=1}^{A-1} |D_{h,\beta }\gamma |_{{\mathcal {C}}^{k}({\mathbb {R}}^d)} + | D_{h,\beta }\gamma |_{{\mathrm {C}}^{\alpha -\beta }({\mathbb {R}}^d)}\\ =&\underbrace{ \Vert D_{h,\beta }\gamma \Vert _{{\mathcal {C}}^{0}({\mathbb {R}}^d)}}_{(i)} + \underbrace{ \sum _{k=1}^{A-1} \max _{|i|_1=k} \Vert D^{i}(D_{h,\beta }\gamma )\Vert _{{\mathcal {C}}^{0}({\mathbb {R}}^d)} }_{(ii)}\\&+ \underbrace{\max _{|i|_1=A-1}\sup _{x,t}\left\{ \dfrac{|D^{i}(D_{h,\beta }\gamma )(x+t) - D^{i}(D_{h,\beta }\gamma )(x)|}{|t|^{1+s-\beta }} \right\} }_{(iii)}. \end{aligned}$$

We have already derived a bound for the terms (i) and (ii). The term (iii) can be bounded as follows:

$$\begin{aligned} (iii)=\max _{|i|_1=A-1} \max \left\{ \underbrace{\sup _{\begin{array}{c} x,t\in {\mathbb {R}}^d\\ |t|\ge |h| \end{array} } \dfrac{|D^{i}(D_{h,\beta }\gamma )(x+t) - D^{i}(D_{h,\beta }\gamma )(x)|}{|t|^{1+s-\beta }}}_{(I_i)}, \underbrace{\sup _{\begin{array}{c} x,t\in {\mathbb {R}}^d\\ |t|\le |h| \end{array} } \dfrac{|D^{i}(D_{h,\beta }\gamma )(x+t) - D^{i}(D_{h,\beta }\gamma )(x)|}{|t|^{1+s-\beta }}}_{II_i} \right\} \end{aligned}$$

By bounding separately the two terms we get

$$\begin{aligned} I_i= & {} \sup _{\begin{array}{c} x,t\in {\mathbb {R}}^d\\ |t|\ge |h| \end{array} } \dfrac{|D^{i}\gamma (x+t+h) - D^{i}\gamma (x+t)-D^{i}\gamma (x+h) + D^{i}\gamma (x)|}{|h|^{\beta }|t|^{1+s-\beta }} \\= & {} \sup _{\begin{array}{c} x,t\in {\mathbb {R}}^d\\ |t|\ge |h| \end{array} } \dfrac{|(\nabla D^{i}\gamma (\xi ^{x+t+h}_{i,x+t}) - \nabla D^{i}\gamma (\xi ^{x+h}_{i,x}))\cdot h|}{|h|^{\beta }|t|^{1+s-\beta }}\\\le & {} \sup _{\begin{array}{c} x,t\in {\mathbb {R}}^d\\ |t|\ge |h| \end{array} } \dfrac{\max _{j=1,\ldots ,d}| \partial _{x_j} (D^{i}\gamma (\xi ^{x+t+h}_{i,x+t})) - \partial _{x_j} (D^{i}\gamma (\xi ^{x+h}_{i,x}))|}{|\xi ^{x+t+h}_{i,x+t}-\xi ^{x+h}_{i,x}|^s}\\&\times \dfrac{\sqrt{d}|h||\xi ^{x+t+h}_{i,x+t}-\xi ^{x+h}_{i,x}|^s}{|h|^{\beta }|t|^{1+s-\beta }}\\\le & {} \sqrt{d}| \gamma |_{{\mathrm {C}}^{\alpha }({\mathbb {R}}^d)} \sup _{\begin{array}{c} x,t\in {\mathbb {R}}^d\\ |t|\ge |h| \end{array} } \dfrac{|h|^{1-\beta }(|h|^s+|t|^s)}{|t|^{1+s-\beta }}\le 2\sqrt{d}| \gamma |_{{\mathrm {C}}^{\alpha }({\mathbb {R}}^d)} . \end{aligned}$$

Similarly for the term \(II_i\), we have:

$$\begin{aligned} II_i= & {} \sup _{\begin{array}{c} x,t\in {\mathbb {R}}^d\\ |t|\le |h| \end{array} } \dfrac{|D^{i}\gamma (x+t+h)-D^{i}\gamma (x+h) - D^{i}\gamma (x+t) + D^{i}\gamma (x)|}{|h|^{\beta }|t|^{1+s-\beta }} \\= & {} \sup _{\begin{array}{c} x,t\in {\mathbb {R}}^d\\ |t|\le |h| \end{array} } \dfrac{|(\nabla D^{i}\gamma (\xi ^{x+t+h}_{i,x+h}) - \nabla D^{i}\gamma (\xi ^{x+t}_{i,x}))\cdot t|}{|h|^{\beta }|t|^{1+s-\beta }}\\\le & {} \sup _{\begin{array}{c} x,t\in {\mathbb {R}}^d\\ |t|\le |h| \end{array} } \dfrac{\max _{j=1,\ldots ,d}| \partial _{x_j} (D^{i}\gamma (\xi ^{x+t+h}_{i,x+t})) - \partial _{x_j} (D^{i}\gamma (\xi ^{x+h}_{i,x}]))|}{|\xi ^{x+t+h}_{i,x+h}-\xi ^{x+t}_{i,x}|^s}\\&\times \dfrac{\sqrt{d}|t||\xi ^{x+t+h}_{i,x+h}-\xi ^{x+t}_{i,x}|^s}{|h|^{\beta }|t|^{1+s-\beta }}\\\le & {} \sqrt{d}| \gamma |_{{\mathrm {C}}^{\alpha }({\mathbb {R}}^d)} \sup _{\begin{array}{c} x,t\in {\mathbb {R}}^d\\ |t|\le |h| \end{array} } \dfrac{|t|^{\beta -s}(|h|^s+|t|^s)}{|h|^{\beta }}\le 2\sqrt{d}| \gamma |_{{\mathrm {C}}^{\alpha }({\mathbb {R}}^d)} . \end{aligned}$$

and then the term (iii) can be bounded as \((iii)\le 2\sqrt{d}| \gamma |_{{\mathrm {C}}^{\alpha }({\mathbb {R}}^d)} \). Finally the norm of \(\Vert D_{h,\beta }\gamma \Vert _{{\mathrm {C}}^{\alpha -\beta }({\mathbb {R}}^d)}\) can we bounded as

$$\begin{aligned}&\Vert D_{h,\beta }\gamma \Vert _{{\mathrm {C}}^{\alpha -\beta }({\mathbb {R}}^d)}\le (i)+(ii)+(iii)\le 2\Vert \gamma \Vert _{{\mathcal {C}}^{0}({\mathbb {R}}^d)} +(2+\sqrt{d})\sum _{k=1}^{A-1} | \gamma |_{{\mathcal {C}}^{k}({\mathbb {R}}^d)}\\&\quad +\, \sqrt{d}| \gamma |_{{\mathcal {C}}^{A}({\mathbb {R}}^d)}+\, (1+2\sqrt{d}) | \gamma |_{{\mathrm {C}}^{\alpha }({\mathbb {R}}^d)}. \end{aligned}$$

By comparing this expression with the one obtained in the previous case it is possible to conclude that \(\forall h\in {{\mathbb {R}}^d}\), \(0<\beta \le \min \{\alpha ,1\}\) it holds

$$\begin{aligned} \Vert D_{h,\beta }\gamma \Vert _{{\mathrm {C}}^{\alpha -\beta }({\mathbb {R}}^d)} \le (1+2\sqrt{d}) \Vert \gamma \Vert _{\mathrm {C}^{\alpha }({\mathbb {R}}^d)},\quad | D_{h,\beta }\gamma |_{{\mathrm {C}}^{\alpha -\beta }({\mathbb {R}}^d)} \le 2\sqrt{d} |\gamma |_{\mathrm {C}^{\alpha }({\mathbb {R}}^d)} . \end{aligned}$$

\(\square \)

Lemma 8

Let \(\kappa (x)\in {\mathrm {C}}^{\alpha }({\mathbb {R}}^d)\) be a deterministic function as in Lemma 7, and let \(\kappa ^{\epsilon }(x)\) be a smoothed version of \(\kappa (x)\) defined as \(\kappa ^{\epsilon }(x)=(\kappa *\phi _{\epsilon })(x)\) with \( \phi _{\epsilon }(x)\) as in (11). It holds:

$$\begin{aligned} \Vert \kappa - \kappa ^{\epsilon }\Vert _{{\mathcal {C}}^{0}({\mathbb {R}}^d)} \le C(\alpha ,d) |\kappa |_{{\mathrm {C}}^{\min (\alpha ,2)}({\mathbb {R}}^d)} \epsilon ^{\min (\alpha ,2)}, \end{aligned}$$

where \(C(\alpha ,d)=\frac{1}{\sqrt{2\pi }^d}\int _{{\mathbb {R}}^d}|y|^{\alpha } e^{-\frac{|y|^2}{2}} \ dy\).

Proof

By definition we have

$$\begin{aligned} |(\kappa -\kappa ^{\epsilon })(x)|=\left| \int _{{\mathbb {R}}^d} ( \kappa (x+y)-\kappa (x)) \phi _{\epsilon }(y)dy\right| \quad \forall x \in {\mathbb {R}}^d. \end{aligned}$$

\(\bullet \) Let us start with the case \({0<\alpha \le 1}\): if \(\kappa (x)\in {\mathrm {C}}^{\alpha }({\mathbb {R}}^d)\) then we obtain

$$\begin{aligned}&|(\kappa -\kappa ^{\epsilon })(x)|\le \int _{{\mathbb {R}}^d} \frac{|\kappa (x+y)-\kappa (x)|}{|y|^{\alpha }} |y|^{\alpha }\phi _{\epsilon }(y)dy\le |\kappa |_{{\mathrm {C}}^{\alpha }({\mathbb {R}}^d)} \int _{{\mathbb {R}}^d} |y|^{\alpha }\phi _{\epsilon }(y)dy\\&\quad \le C(\alpha ,d) |\kappa |_{{\mathrm {C}}^{\alpha }({\mathbb {R}}^d)} \epsilon ^{\alpha }. \end{aligned}$$

\(\bullet \) If \(1<\alpha \le 2\) we consider a Taylor expansion of \(\kappa (x+y)\) around x and set \(\alpha = 1 +s\) with \(s\in (0,1]\). Since odd moments of a normal distribution vanish, we get:

$$\begin{aligned} |(\kappa -\kappa ^{\epsilon })(x)|= & {} \left| \int _{{\mathbb {R}}^d} \left( \nabla \kappa (x)\cdot y +(\nabla \kappa (\xi _{x}^{x+y})-\nabla \kappa (x))\cdot y \right) \phi _{\epsilon }(y)dy \right| \\\le & {} \int _{{\mathbb {R}}^d} \max _{|i|_1=1} \frac{|D^i \kappa (\xi _{x}^{x+y})-D^i \kappa (x)|}{|\xi _{x}^{x+y}-x|^s} \sqrt{d} |y|^{1+s} \phi _{\epsilon }(y)dy\\\le & {} C(\alpha ,d) \sqrt{d} |\kappa |_{{\mathrm {C}}^{\alpha }({\mathbb {R}}^d)} \epsilon ^{\alpha }. \end{aligned}$$

\(\bullet \) Finally by considering \(2<\alpha \) and by expanding further the function \(\kappa \), since the second moment of a normal distribution does not vanish, we get:

$$\begin{aligned} |(\kappa -\kappa ^{\epsilon })(x)|= & {} \left| \int _{{\mathbb {R}}^d} \left( \nabla \kappa (x)\cdot y + \sum _{j,k=1}^{d}\frac{\partial ^2 \gamma }{\partial x_j\partial x_k}(\xi _{x}^{x+y}) y_jy_k \right) \phi _{\epsilon }(y)dy \right| \\\le & {} C(\alpha ,d) |\kappa |_{{\mathcal {C}}^{2}({\mathbb {R}}^d)} \epsilon ^{2} . \end{aligned}$$

\(\square \)

Lemma 9

Let \(\gamma (x)\in {\mathrm {C}}^{\alpha }(\overline{D})\) and \(\gamma ^{\epsilon }(x)\in {\mathrm {C}}^{\alpha }(\overline{D})\) be two deterministic functions and let \(a(x)=e^{\gamma (x)}\) and \(a^{\epsilon }(x)=e^{\gamma ^{\epsilon }(x)}\). For any \(0<\beta \le \min (1,\alpha )\) it holds

$$\begin{aligned} \Vert a- a^{\epsilon }\Vert _{{\mathrm {C}}^{\beta }( \overline{D} )} \le \Vert a \Vert _{{\mathcal {C}}^{0}(\overline{D})} \Vert 1+\frac{a^{\epsilon }}{a} \Vert _{{\mathcal {C}}^{0}(\overline{D})}(1+|\gamma |_{\mathrm {C}^{\beta }(\overline{D})}) \Vert \gamma -\gamma ^{\epsilon }\Vert _{{\mathrm {C}}^{\beta }( \overline{D} )}. \end{aligned}$$

Proof

We bound separately the terms coming from the definition of the \({\mathrm {C}}^{\beta }\) norm, namely \(\Vert a- a^{\epsilon }\Vert _{{\mathrm {C}}^{\beta }( \overline{D} )}= \Vert a- a^{\epsilon }\Vert _{{\mathcal {C}}^{0}( \overline{D} )} +| a- a^{\epsilon }|_{{\mathrm {C}}^{\beta }( \overline{D} )}\). For the first one we simply observe that

$$\begin{aligned} \Vert a- a^{\epsilon }\Vert _{{\mathcal {C}}^{0}( \overline{D} )}\le \Vert e^\gamma + e^{\gamma ^{\epsilon }}\Vert _{{\mathcal {C}}^{0}( \overline{D} )} \Vert \gamma - \gamma ^{\epsilon }\Vert _{{\mathcal {C}}^{0}( \overline{D} )}. \end{aligned}$$

For the second term we start by considering the inequality

$$\begin{aligned} | a- a^{\epsilon }|_{{\mathrm {C}}^{\beta }( \overline{D} )}\le \Vert e^{\gamma }\Vert _{{\mathcal {C}}^{0}( \overline{D} )} \underbrace{\left| 1- e^{\gamma -\gamma ^{\epsilon }}\right| _{{\mathrm {C}}^{\beta }( \overline{D} )}}_{(i)} + \underbrace{ | e^{\gamma }|_{{\mathrm {C}}^{\beta }( \overline{D} )} }_{(ii)} \underbrace{\left\| 1- e^{\gamma -\gamma ^{\epsilon }}\right\| _{{\mathcal {C}}^{0}( \overline{D} )} }_{(iii)}. \end{aligned}$$

The terms in the above equation can be bounded as follows:

$$\begin{aligned} (i)\le & {} \left\| e^{\gamma ^{\epsilon }-\gamma }\right\| _{{\mathcal {C}}^{0}( \overline{D} )} |\gamma ^{\epsilon } -\gamma |_{{\mathrm {C}}^{\beta }( \overline{D} )},\\ (ii)\le & {} \left\| e^{\gamma }\right\| _{{\mathcal {C}}^{0}( \overline{D} )} |\gamma |_{{\mathrm {C}}^{\beta }( \overline{D} )},\\ (iii)\le & {} \left\| 1+ e^{\gamma ^{\epsilon }-\gamma }\right\| _{{\mathcal {C}}^{0}( \overline{D} )} \Vert \gamma ^{\epsilon } -\gamma \Vert _{{\mathcal {C}}^{0}( \overline{D} )}. \end{aligned}$$

By putting everything together we obtain

$$\begin{aligned} \Vert a- a^{\epsilon }\Vert _{{\mathrm {C}}^{\beta }( \overline{D} )}\le&\left( \Vert e^{\gamma }+ e^{\gamma ^{\epsilon }}\Vert _{{\mathcal {C}}^{0}( \overline{D} )} + \left\| e^{\gamma }\right\| _{{\mathcal {C}}^{0}( \overline{D} )} |\gamma |_{{\mathrm {C}}^{\beta }( \overline{D} )} \left\| 1+ e^{\gamma ^{\epsilon }-\gamma }\right\| _{{\mathcal {C}}^{0}( \overline{D} )} \right) \\&\times \Vert \gamma -\gamma ^{\epsilon }\Vert _{{\mathcal {C}}^{0}( \overline{D} )} \\&+\, \left\| e^{\gamma } \right\| _{{\mathcal {C}}^{0}( \overline{D} )} \left\| e^{\gamma ^{\epsilon }-\gamma }\right\| _{{\mathcal {C}}^{0}( \overline{D} )} |\gamma ^{\epsilon } -\gamma |_{{\mathrm {C}}^{\beta }( \overline{D} )}\\ \le&\Vert e^{\gamma } \Vert _{{\mathcal {C}}^{0}(\overline{D})} \Vert 1+e^{\gamma ^{\epsilon }-\gamma } \Vert _{{\mathcal {C}}^{0}(\overline{D})}(1+|\gamma |_{\mathrm {C}^{\beta }(\overline{D})}) \Vert \gamma ^{\epsilon } -\gamma \Vert _{{\mathrm {C}}^{\beta }( \overline{D} )} \end{aligned}$$

which is the desired result. \(\square \)

Thanks to these results we can prove Lemma 4.

Proof

(of Lemma 4.) From lemma 7 we have that \(\tilde{\gamma } \in {\mathrm {C}}^{\alpha }({\mathbb {R}}^d)\) implies \(D_{h,\beta } \tilde{\gamma } \in {\mathrm {C}}^{\alpha -\beta }({\mathbb {R}}^d)\) \(\forall \beta \le \min (\alpha ,1)\). By using the definitions given in (17), thanks to Lemmas 8 and 7 we get

$$\begin{aligned} \left\| D_{h,\beta } \tilde{\gamma } - \left( D_{h,\beta } \tilde{\gamma } \right) ^{\epsilon } \right\| _{{\mathcal {C}}^{0}({\mathbb {R}}^d)}\lesssim & {} \left| D_{h,\beta } \tilde{\gamma } \right| _{C^{\min (\alpha -\beta ,2)}({\mathbb {R}}^d)}\epsilon ^{\min (\alpha -\beta ,2)} \\\lesssim & {} \left| \tilde{ \gamma } \right| _{C^{\min (\alpha ,2+\beta )}({\mathbb {R}}^d)} \epsilon ^{\min (\alpha -\beta ,2)} \end{aligned}$$

Since \(D_{h,\beta } \gamma ^{\epsilon } = \left( D_{h,\beta } \tilde{\gamma } \right) ^{\epsilon }\) and thanks to the fact that the previous estimate is valid uniformly in h, we can take the supremum of \(\left\| D_{h,\beta } \gamma - \left( D_{h,\beta } \gamma \right) ^{\epsilon } \right\| _{{\mathcal {C}}^{0}({\mathbb {R}}^d)} \) with respect to h. By doing this we get

$$\begin{aligned} | \tilde{\gamma } -\gamma ^{\epsilon }|_{{\mathrm {C}}^{\beta }( {\mathbb {R}}^d )} \le \sup _{h\in {\mathbb {R}}^d} \left\| D_{h,\beta } \gamma - \left( D_{h,\beta } \gamma \right) ^{\epsilon } \right\| _{{\mathcal {C}}^{0}({\mathbb {R}}^d)} \lesssim \left| \tilde{ \gamma } \right| _{C^{\min (\alpha ,2+\beta )}({\mathbb {R}}^d)} \epsilon ^{\min (\alpha -\beta ,2)} . \end{aligned}$$

Now we get the desired result by observing that \( \left| \tilde{ \gamma } \right| _{C^{\alpha }({\mathbb {R}}^d)}\le \Vert \varphi \Vert _{C^{\alpha }(\overline{D}_{1})} \Vert \gamma \Vert _{C^{\alpha }(\overline{D}_{1})}\) . In fact, since \(\varphi \) vanishes on \(D_1^c\), we obtain

$$\begin{aligned} \Big | \tilde{ \gamma } \Big |_{C^{\alpha }({\mathbb {R}}^d)}= & {} \max \left\{ \sup _{\begin{array}{c} x\in \overline{D}_1\\ y\in {\mathbb {R}}^d \end{array} } \dfrac{| \gamma (x)(\varphi (x)-\varphi (y)) + \varphi (y)(\gamma (x)-\gamma (y)) |}{|x-y|^{\alpha }},\right. \\&\left. \sup _{\begin{array}{c} y\in \overline{D}_1\\ x\in {\mathbb {R}}^d \end{array} } \dfrac{| \gamma (y)(\varphi (y)-\varphi (x)) + \varphi (x)(\gamma (y)-\gamma (x)) |}{|x-y|^{\alpha }} \right\} \\&\le \max \left\{ \Vert \gamma \Vert _{{\mathcal {C}}^0(\overline{D}_1)} | \varphi |_{{\mathrm {C}}^{\alpha }({\mathbb {R}}^d)} + \sup _{ x,y\in \overline{D}_1 } \dfrac{| \varphi (y)(\gamma (x)-\gamma (y)) |}{|x-y|^{\alpha }}, \right. \\&\left. \Vert \gamma \Vert _{{\mathcal {C}}^0(\overline{D}_1)} | \varphi |_{{\mathrm {C}}^{\alpha }({\mathbb {R}}^d)}+ \sup _{ x,y\in \overline{D}_1 } \dfrac{| \varphi (x)(\gamma (y)-\gamma (x)) |}{|x-y|^{\alpha }} \right\} \\&\le \Vert \gamma \Vert _{{\mathcal {C}}^0(\overline{D}_1)} | \varphi |_{{\mathrm {C}}^{\alpha }({\mathbb {R}}^d)} + \Vert \varphi \Vert _{{\mathcal {C}}^0(\overline{D}_1)} | \gamma |_{{\mathrm {C}}^{\alpha }(\overline{D}_1)} =\Vert \gamma \Vert _{{\mathcal {C}}^0(\overline{D}_1)} | \varphi |_{{\mathrm {C}}^{\alpha }(\overline{D}_1)}\\&\quad + \, \Vert \varphi \Vert _{{\mathcal {C}}^0(\overline{D}_1)} | \gamma |_{{\mathrm {C}}^{\alpha }(\overline{D}_1)} \le \Vert \varphi \Vert _{C^{\alpha }(\overline{D}_{1})} \Vert \gamma \Vert _{C^{\alpha }(\overline{D}_{1})}. \end{aligned}$$

Hence, by considering the inequality given in Lemma 9:

$$\begin{aligned} \Vert a- a^{\epsilon }\Vert _{{\mathrm {C}}^{\beta }( \overline{D} )} \le \! \Vert a \Vert _{{\mathcal {C}}^{0}(\overline{D})} \Vert 1+\frac{a^{\epsilon }}{a} \Vert _{{\mathcal {C}}^{0}(\overline{D})} (1+|\log (a)|_{\mathrm {C}^{\beta }(\overline{D})}) \Vert \gamma - \gamma ^{\epsilon } \Vert _{{\mathrm {C}}^{\beta }( \overline{D} )} \quad \text { a.s. in } \varOmega ; \end{aligned}$$

since in D it holds \(\gamma =\tilde{\gamma }\) we get \(\Vert \gamma - \gamma ^{\epsilon } \Vert _{{\mathrm {C}}^{\beta }( \overline{D} )}=\Vert \tilde{\gamma } - \gamma ^{\epsilon } \Vert _{{\mathrm {C}}^{\beta }( \overline{D} )}\le \Vert \tilde{\gamma } - \gamma ^{\epsilon } \Vert _{{\mathrm {C}}^{\beta }( {\mathbb {R}}^d )}\) and we can conclude that

$$\begin{aligned}&\Vert a- a^{\epsilon }\Vert _{{\mathrm {C}}^{\beta }( \overline{D} )}\\&\quad \lesssim \Vert a \Vert _{{\mathcal {C}}^{0}(\overline{D})} \Vert 1+\frac{a^{\epsilon }}{a} \Vert _{{\mathcal {C}}^{0}(\overline{D})} (1+|\gamma |_{{\mathrm {C}}^{\beta }(\overline{D})}) \Vert \varphi \Vert _{C^{\min (\alpha ,2+\beta )}(\overline{D}_{1})}\\&\qquad \times \Vert \gamma \Vert _{C^{\min (\alpha ,2+\beta )}(\overline{D}_{1})}\epsilon ^{\min (\alpha -\beta ,2)} \lesssim C_{4}(\omega ,\alpha ) \epsilon ^{\min (\alpha -\beta ,2)}. \end{aligned}$$

To prove the second bound concerning the \({\mathcal {C}}^1\) norm when \(\alpha >1\) we start again from the definition:

$$\begin{aligned} \Vert a- a^{\epsilon } \Vert _{{\mathcal {C}}^{1}(\overline{D})} = \Vert a- a^{\epsilon } \Vert _{{\mathcal {C}}^0(\overline{D})} + \max _{|i|_1= 1}\Vert D^i (a- a^{\epsilon }) \Vert _{{\mathcal {C}}^0(\overline{D})}. \end{aligned}$$

The first term, thanks to Lemma 8, can be bounded as

$$\begin{aligned}&\Vert a- a^{\epsilon } \Vert _{{\mathcal {C}}^0(\overline{D})}\le a_{max}\Vert 1+e^{\gamma ^{\epsilon }-\gamma } \Vert _{{\mathcal {C}}^0(\overline{D})}\Vert \gamma -\gamma ^{\epsilon } \Vert _{{\mathcal {C}}^0(\overline{D})}\\&\quad \lesssim a_{max}\Vert 1+e^{\gamma ^{\epsilon }-\gamma } \Vert _{{\mathcal {C}}^0(\overline{D})} \Vert \varphi \Vert _{{\mathrm {C}}^{\min (\alpha ,2)}(\overline{D}_1)} \Vert \gamma \Vert _{{\mathrm {C}}^{\min (\alpha ,2)}(\overline{D}_1)} \epsilon ^{\min (\alpha ,2)}. \end{aligned}$$

For the second term, since the derivatives and the convolution commute, we obtain

$$\begin{aligned}&D^i \left( e^{\gamma }(1- e^{\gamma ^{\epsilon }-\gamma })\right) \le |e^{\gamma }D^i (\gamma )(1- e^{\gamma ^{\epsilon }-\gamma })|+|e^{\gamma } e^{\gamma ^{\epsilon }-\gamma }D^i (\gamma ^{\epsilon }-\gamma )|\\&\quad \le e^{\gamma } \left( 1+e^{\gamma ^{\epsilon }-\gamma }\right) \left( |D^i (\gamma )||\gamma ^{\epsilon }-\gamma |+|D^i\gamma -(D^i\gamma )^{\epsilon }|\right) \end{aligned}$$

therefore we get

$$\begin{aligned} \Vert D^i (a- a^{\epsilon }) \Vert _{{\mathcal {C}}^0(\overline{D})}\le & {} a_{max} \left\| 1+e^{\gamma ^{\epsilon }-\gamma } \right\| _{{\mathcal {C}}^{0}(\overline{D})} \big ( \Vert D^i \gamma \Vert _{{\mathcal {C}}^0(\overline{D})}\Vert \underbrace{\gamma - \gamma ^{\epsilon }}_{\in {\mathrm {C}}^{\alpha }(\overline{D})} \Vert _{{\mathcal {C}}^0(\overline{D})}\\&+\, \Vert \underbrace{D^i\gamma -(D^i\gamma )^{\epsilon }}_{ \in {\mathrm {C}}^{\alpha -1}(\overline{D})} \Vert _{{\mathcal {C}}^0(\overline{D})} \big ) \\\le & {} a_{max} \left\| 1+e^{\gamma ^{\epsilon }-\gamma } \right\| _{{\mathcal {C}}^{0}(\overline{D})} \big ( \Vert D^i \gamma \Vert _{{\mathcal {C}}^0(\overline{D})} \Vert \tilde{\gamma }\Vert _{{\mathrm {C}}^{\min (\alpha ,2)}({\mathbb {R}}^d)} \epsilon ^{\min (\alpha ,2)}\\&+\, \Vert D^i\tilde{\gamma }\Vert _{{\mathrm {C}}^{\min (\alpha -1,2)}({\mathbb {R}}^d)} \epsilon ^{\min (\alpha -1,2)}\big ) \end{aligned}$$

which implies

$$\begin{aligned} \max _{|i|_1= 1}\Vert D^i (a- a^{\epsilon }) \Vert _{{\mathcal {C}}^0(\overline{D})}&\lesssim a_{max} \left\| 1+e^{\gamma ^{\epsilon }-\gamma } \right\| _{{\mathcal {C}}^{0}(\overline{D})} \big ( | \gamma |_{{\mathcal {C}}^1(\overline{D})} \Vert \tilde{\gamma }\Vert _{ {\mathrm {C}}^{\min (\alpha ,2)}({\mathbb {R}}^d)}\\&\quad +\,\Vert \tilde{\gamma }\Vert _{{\mathrm {C}}^{\min (\alpha ,3)}({\mathbb {R}}^d)} \big )\epsilon ^{\min (\alpha -1,2)}. \end{aligned}$$

Finally, by putting everything together, and by recalling that \( \left| \tilde{ \gamma } \right| _{C^{\beta }({\mathbb {R}}^d)}\le \Vert \varphi \Vert _{C^{\beta }(\overline{D}_{1})} \Vert \gamma \Vert _{C^{\beta }(\overline{D}_{1})}\) \(\forall \beta \in {\mathbb {R}}_+\), we get the desired result:

$$\begin{aligned}&\Vert a - a^{\epsilon }\Vert _{{\mathcal {C}}^{1}(\overline{D})}\\&\quad \lesssim a_{max} \left\| 1+e^{\gamma ^{\epsilon }-\gamma } \right\| _{{\mathcal {C}}^{0}(\overline{D})} \left( 1+|\gamma |_{{\mathcal {C}}^{1}(\overline{D})}\right) \\&\qquad \Vert \gamma \Vert _{\mathrm {C}^{\min (\alpha ,3 )}(\overline{D}_{1})} \Vert \varphi \Vert _{\mathrm {C}^{\min (\alpha ,3)}(\overline{D}_{1})} \epsilon ^{\min (\alpha -1,2)} ,\quad \text { a.s. in }\varOmega . \end{aligned}$$

\(\square \)

Appendix 2: on products of Hölder and Sobolev functions

Lemma 10

Let \(b\in {\mathrm {C}}^{\alpha }(\overline{D})\) and let v be a function in \(H^{\beta }(D)\) for some \(0<\beta <\alpha \le 1\). It holds

$$\begin{aligned} \Vert bv\Vert _{H^{\beta }(D)} \lesssim \frac{1}{\sqrt{\eta }} \Vert b\Vert _{{\mathrm {C}}^{\beta +\eta }} \Vert v\Vert _{H^{\beta }(D)}\quad \forall 0<\eta \le \alpha -\beta . \end{aligned}$$

Proof

By definition the \(H^{\beta }\) norm of the function bv is

$$\begin{aligned} \Vert bv\Vert _{H^{\beta }(D)}^2= & {} \Vert bv\Vert _{L^2(D)}^2 + |bv|_{H^{\beta }(D)}^2 = \Vert bv\Vert _{L^2(D)}^2\\&+ \int _{D\times D} \frac{ \left| b(x)v(x) - b(y)v(y)\right| ^2}{|x-y|^{d+2\beta }} dxdy. \end{aligned}$$

The first term can be easily bounded as \(\Vert bv\Vert _{L^2(D)}^2\le \Vert b\Vert _{{\mathcal {C}}^0(\overline{D})}^2\Vert v\Vert _{L^2(D)}^2\). For the second term we obtain

$$\begin{aligned}&\int _{D\times D} \frac{ \left| b(x)v(x) - b(y)v(y)\right| ^2}{|x-y|^{d+2\beta }} dxdy \le 2 \Vert b\Vert _{{\mathcal {C}}^0(\overline{D})}^2 |v|_{H^{\beta }(D)}^2\\&\qquad +\, 2\int _{D\times D} \frac{ \left| b(x) - b(y)\right| ^2}{|x-y|^{2(\beta +\eta )}}\frac{v(y)^2}{|x-y|^{d-2\eta }}dxdy\\&\quad \le 2 \Vert b\Vert _{{\mathcal {C}}^0(\overline{D})}^2 |v|_{H^{\beta }(D)}^2 + 2 \Vert b\Vert _{{\mathrm {C}}^{\beta +\eta }(\overline{D})}^2 \int _{D\times D} \frac{v(y)^2}{|x-y|^{d-2\eta }}dxdy. \end{aligned}$$

If we extend v by 0 in \({\mathbb {R}}^d \setminus D\), and denote \(\tilde{v}\) this extension and \(\rho =\max _{x\in D} |x|\), the integral appearing in the right hand side of the above inequality can be bounded as

$$\begin{aligned} \int _{D\times D} \frac{v(y)^2}{|x-y|^{d-2\eta }}dxdy\le & {} \int _{{\mathbb {R}}^d\times {\mathbb {R}}^d} \frac{\tilde{v}(y)^2}{|x-y|^{d-2\eta }}\mathbbm {1}_{\{|x-y|\le 2\rho \}}dxdy\\\le & {} \left\| \tilde{v}^2(x) *\frac{\mathbbm {1}_{\{|x|\le 2\rho \}}}{|x|^{d-2\eta }}\right\| _{L^1({\mathbb {R}}^d)}\\\le & {} \left\| \tilde{v}^2 \right\| _{L^1({\mathbb {R}}^d)} \left\| \frac{\mathbbm {1}_{\{|x|\le 2\rho \}}}{|x|^{d-2\eta }}\right\| _{L^1({\mathbb {R}}^d)} \lesssim \frac{\rho ^{2\eta }}{\eta } \left\| v \right\| _{L^2(D)}^2. \end{aligned}$$

By putting everything together we obtain

$$\begin{aligned} \Vert bv\Vert _{H^{\beta }(D)} \lesssim \frac{1}{\sqrt{\eta }} \Vert b\Vert _{{\mathrm {C}}^{\beta +\eta }(\overline{D})} \Vert v\Vert _{H^{\beta }(D)}, \end{aligned}$$

which is the desired result. \(\square \)

Appendix 3: regularity of Gaussian random fields with Matérn covariance

Lemma 11

Let \(\text {cov}_{\gamma }\) be a covariance function belonging to the Matérn family defined in (5) on an open bounded convex domain D. Then, if \(\nu \) is not an integer, \(\text {cov}_{\gamma }\in {\mathrm {C}}^{2\nu }(\bar{D}\times \bar{D})\), otherwise \(\text {cov}_{\gamma }\in {\mathrm {C}}^{\alpha }(\bar{D}\times \bar{D})\) for any \(\alpha <2\nu \) if \(\nu \in {\mathbb {N}}_+\) .

Proof

By definition we have

$$\begin{aligned} \text {cov}_{\gamma }(x_1,x_2)=\widetilde{\text {cov}}_{\gamma }(|x_1-x_2|)=\frac{\sigma ^2}{\varGamma (\nu )2^{\nu -1}}\left( \sqrt{2\nu }\frac{|x_1-x_2|}{L_c}\right) ^{\nu } K_{\nu }\left( \sqrt{2\nu }\frac{|x_1-x_2|}{L_c}\right) ; \end{aligned}$$

\(K_{\nu }:{\mathbb {R}}_{+}\rightarrow {\mathbb {R}}_+\) is given by \(K_{\nu }(\rho )=\frac{\pi }{2\sin {\pi \nu }}(I_{-\nu }(\rho )-I_{\nu }(\rho ))\) where \(I_{\alpha }(\rho )=\sum _{m=0}^{\infty }\frac{1}{m! \varGamma (m+\alpha +1)}\left( \frac{\rho }{2}\right) ^{2m+\alpha }\). This formula is valid when \(\nu \) is not an integer, i.e. \(\nu =n+s\) with \(n\in {\mathbb {N}}\) and \(s\in (0,1)\). Since \(\forall \epsilon >0\) the function \(K_{\nu }\in {\mathcal {C}}^{\infty }[\epsilon ,+\infty )\), and consequently \(\widetilde{\text {cov}}_{\gamma }\) as well, in order to prove the result we focus on the asymptotic behavior of the function \(\widetilde{\text {cov}}_{\gamma }(|x-y|)\) in a neighborhood of \(|x-y|=0\). By denoting \(\lambda _{\nu }=\frac{\sqrt{2\nu }}{2L_c}\) and by recalling that, for any \(x\in {\mathbb {R}}\setminus {\mathbb {Z}}\) it holds \(\varGamma (-x)=\frac{-\pi }{\sin \pi x \varGamma (x+1)}\), it is possible to obtain

$$\begin{aligned}&\widetilde{\text {cov}}_{\gamma }(|x_1-x_2|)\\&\quad =\frac{\sigma ^2\pi }{\varGamma (\nu )\sin {\pi \nu }}\left( \sum _{m=0}^{\infty }\frac{\lambda _{\nu }^{2m}|x_1-x_2|^{2m}}{m! \varGamma (m-\nu +1)} - \sum _{m=0}^{\infty }\frac{\lambda _{\nu }^{2(m+\nu )}|x_1-x_2|^{2(m+\nu )}}{m! \varGamma (m+\nu +1)} \right) \\&\quad =\sigma ^2\left( \sum _{m=0}^{n} \frac{(-1)^m\varGamma (\nu -m)\lambda _{\nu }^{2m}}{m!\varGamma (\nu )}|x_1-x_2|^{2m} - \frac{\lambda _{\nu }^{2\nu }|x_1-x_2|^{2\nu }}{\varGamma (\nu )\varGamma (\nu +1)\sin (\pi \nu )} \right) \\&\qquad +\frac{\sigma ^2}{\varGamma (\nu )\sin {\pi \nu }}\left( \sum _{m=1}^{\infty }\frac{\lambda _{\nu }^{2(m+n)}|x_1-x_2|^{2(m+n)}}{(m+n)! \varGamma (m+n-\nu +1)} -\frac{\lambda _{\nu }^{2(m+\nu )}|x_1-x_2|^{2(m+\nu )}}{m! \varGamma (m+\nu +1)} \right) . \end{aligned}$$

Hence, the asymptotic behavior is

$$\begin{aligned}&\widetilde{\text {cov}}_{\gamma }(|x_1-x_2|) \sim \sigma ^2\nonumber \\&\quad \left\{ \sum _{m=0}^{n} \frac{(-1)^m\varGamma (\nu -m)\lambda _{\nu }^{2m}}{m!\varGamma (\nu )}|x_1-x_2|^{2m} - \frac{\lambda _{\nu }^{2\nu }|x_1-x_2|^{2\nu }}{\varGamma (\nu )\varGamma (\nu +1)\sin (\pi \nu )} \right\} . \end{aligned}$$
(29)

Since the function \(f(z)=|z|^{2\nu }:{\mathbb {R}}^d\rightarrow {\mathbb {R}}\) belongs to the space \({\mathrm {C}}^{2\nu }(A)\) for any bounded set \(A\in {\mathbb {R}}^d\) we can conclude that \( \text {cov}_{\gamma }\in {\mathrm {C}}^{2\nu }(\bar{D}\times \bar{D})\).

When \(\nu =n\in {\mathbb {N}}_+\) the previous definition gives removable indeterminate values of the form \(\frac{0}{0}\); in this case the Bessel function \(K_{\nu }\) can be defined through the limit \(K_{n}(\rho )=\lim _{\nu \rightarrow n} K_{\nu }(\rho )\). The covariance function becomes:

Again we focus on the asymptotic behavior of the function \(\widetilde{\text {cov}}_{\gamma }(|x_1-x_2|)\) in a neighborhood of \(|x_1-x_2|=0\). We obtain

$$\begin{aligned} \widetilde{\text {cov}}_{\gamma }(|x_1-x_2|) \sim&\sigma ^2 \left\{ \sum _{m=0}^{n-1} \frac{(-1)^m(n-m-1)! \lambda _{n}^{2m}}{m!(n-1)!}|x_1-x_2|^{2m}\right. \nonumber \\&\left. - \frac{(-1)^{n} 2\lambda _{n}^{2n}}{n!(n-1)!}|x_1-x_2|^{2n}\log \left( \lambda _n|x_1 -x_2|\right) \right\} \end{aligned}$$
(30)

Since the function \(f(z)=|z|^{2n}\log (|z|):{\mathbb {R}}^d\rightarrow {\mathbb {R}}\) belongs to the space \({\mathrm {C}}^{\alpha }(A)\) for any \(\alpha <2\nu \) and for any bounded set \(A\in {\mathbb {R}}^d\) we can conclude that \({\mathrm{cov}}_{\gamma }\in {\mathrm {C}}^{\alpha }(\bar{D}\times \bar{D})\) for any \(\alpha <2\nu \). \(\square \)

Remark 7

Let \(\widetilde{\text {cov}}_{\gamma }(|x-y|)\) be a covariance function belonging to the Matérn family defined in (5) and let \(\gamma (x,\omega )\) be a centered Gaussian random field defined on \(\bar{D}\). Denote \(\nu =n+\alpha \) with \(n\in {\mathbb {N}}\) and \(\alpha \in (0,1]\). Then, for any multi-index \(i\in {\mathbb {N}}^d\) such that \(|i|_1\le n\), it holds:

$$\begin{aligned} {\mathbb {E}}[D^{i}\gamma (x,\cdot )D^{i}\gamma (y,\cdot )]= \frac{\partial ^{2|i|_1}}{\partial x_1^{i_1}\cdots \partial x_d^{i_d} \partial y_1^{i_1}\cdots \partial y_d^{i_d}}\widetilde{\text {cov}}_{\gamma }(|x-y|). \end{aligned}$$

Lemma 12

Let \(\gamma (x,\omega )\) be a centered Gaussian random field with covariance function \(\text {cov}_{\gamma }\) as in Lemma 11. Then \(\gamma \) admits a version with trajectories a.s. in \({\mathrm {C}}^{\alpha }(\overline{D})\) for any \(0<\alpha < \nu \).

Proof

Let us start with the case in which \(\nu \) is not an integer. Lemma 11 tells us that \(\widetilde{\text {cov}}_{\gamma }\in {\mathrm {C}}^{2\nu }(A)\) for any bounded set \(A\in {\mathbb {R}}^d\). Therefore, thanks to (30), by writing \(\nu =n+s\) with \(n\in {\mathbb {N}}\) and \(s\in (0,1)\), for any multi-index \(i\in {\mathbb {N}}^d\) such that \(|i|_1= n\), we obtain

$$\begin{aligned} {\mathbb {E}}[(D^i\gamma (x,\cdot )-D^i\gamma (y,\cdot ))^2] =&{\mathbb {E}}[(D^i\gamma (x,\cdot ))^2] +{\mathbb {E}}[(D^i\gamma (y,\cdot ))^2]\\&-\, 2{\mathbb {E}}[D^i\gamma (x,\cdot )D^i\gamma (y,\cdot )]\\ =&2\left( \frac{\partial ^{2|i|_1}\widetilde{\text {cov}}_{\gamma } }{\partial x_1^{i_1}\cdots \partial x_d^{i_d} \partial y_1^{i_1}\cdots \partial y_d^{i_d}} (0)\right. \\&-\,\left. \frac{\partial ^{2|i|_1}\widetilde{\text {cov}}_{\gamma }}{\partial x_1^{i_1}\cdots \partial x_d^{i_d} \partial y_1^{i_1}\cdots \partial y_d^{i_d}}(|x-y|) \right) \\ \le&C(\nu ) |x-y|^{2s}, \end{aligned}$$

where the last inequality comes from the fact that the coefficients appearing in the covariance function decay sufficiently fast. Since for any positive integer p it holds \({\mathbb {E}}[(D^i\gamma (x,\cdot )-D^i\gamma (y,\cdot ))^{2p}] \le c_p{\mathbb {E}}[(D^i\gamma (x,\cdot )-D^i\gamma (y,\cdot ))^2] ^p\) with \(c_p=\frac{1}{\sqrt{2\pi }}\int _{{\mathbb {R}}}x^{2p} e^{-\frac{x^2}{2}}dx\) we have

$$\begin{aligned} {\mathbb {E}}\big [(D^i\gamma (x,\cdot )-D^i\gamma (y,\cdot ))^{2p}\big ] \le c_p C(\nu )^p |x-y|^{2ps}. \end{aligned}$$

Thanks to the Kolmogorov continuity theorem (see e.g. [29]) we can deduce that there exists a version of \(D^i\gamma \) which belongs to \({\mathrm {C}}^{a}(\bar{D})\) for any \(a<\frac{2ps-d}{2p}\); by taking the limit for \(p\rightarrow +\infty \) we can conclude that there exist a version of \(D^i\gamma \) which belongs to \({\mathrm {C}}^{a}(\bar{D})\) for any a strictly smaller than s. Consequently, since this reasoning can be repeated for every \(k<n\), \(k\in {\mathbb {N}}\), by picking 1 instead of s, we deduce that there exist a version of \(\gamma \) which belongs to \({\mathrm {C}}^{\alpha }(\bar{D})\) for any \(\alpha \) strictly smaller than \(\nu \). The proof in the case \(\nu \in {\mathbb {N}}\) is similar. By writing \(\nu =n+1\) in this case, thanks to (30), for any \(\epsilon >0\) and for any multi-index \(i\in {\mathbb {N}}^d\) such that \(|i|_1= n\) we obtain

$$\begin{aligned} {\mathbb {E}}[(D^i\gamma (x,\cdot )-D^i\gamma (y,\cdot ))^2] \le \tilde{C}(\nu ) \frac{\partial ^{2|i|_1}|x-y|^{2n}\log (|x-y|)}{\partial x_1^{i_1}\cdots \partial x_d^{i_d} \partial y_1^{i_1}\cdots \partial y_d^{i_d}} \le C_{\epsilon }(\nu ) |x-y|^{2-\epsilon }. \end{aligned}$$

Again, thanks to the Kolmogorov continuity theorem we can deduce that there exists a version of \(D^i\gamma \) which belongs to \({\mathrm {C}}^{a}(\bar{D})\) for any \(a<\frac{p(2-\epsilon )-d}{2p}\); thanks to the arbitrariness of \(\epsilon \) by taking the limit for \(p\rightarrow +\infty \) we can conclude that there exist a version of \(D^i\gamma \) which belongs to \({\mathrm {C}}^{a}(\bar{D})\) for any a strictly smaller than 1. Consequently we deduce that there exist a version of \(\gamma \) which belongs to \({\mathrm {C}}^{\alpha }(\bar{D})\) for any \(\alpha \) strictly smaller than \(\nu \). \(\square \)

Appendix 4: optimal rates in Theorem 1

Here we present a sharper bound for the infimum \(\inf _{\begin{array}{c} 0\le \beta \le \min (\alpha ,1)\\ 0<\eta +\beta \le \alpha \end{array} } \frac{h^{\beta } \epsilon ^{\min (\alpha -\beta -\eta ,2)}}{(\alpha -\beta )^2\sqrt{\eta }} \) than the one presented in Proposition 1 that can be obtained with very tedious calculations.

Lemma 13

Let \(\epsilon \le e^{-\frac{1}{2\alpha }}\); the following bounds hold:

  • \(1/2\le \alpha \le 1\):

    $$\begin{aligned} \inf _{\begin{array}{c} 0\le \beta <\alpha \\ 0<\eta +\beta \le \alpha \end{array} } \frac{h^{\beta } \epsilon ^{\alpha -\beta -\eta }}{(\alpha -\beta )^2\sqrt{\eta }} \lesssim \,{\left\{ \begin{array}{ll} \,{\left\{ \begin{array}{ll} h^{\alpha }\left| \log \dfrac{h}{\epsilon } \right| ^2|\log \epsilon |^{\frac{1}{2}}, &{} \quad h\ge \epsilon ^5,\\ h^{\alpha }|\log h|^{\frac{5}{2}}, &{}\quad h\le \epsilon ^5\ \end{array}\right. } , &{}\quad h\le e^{-\frac{2}{\alpha }}\epsilon ,\\ \epsilon ^{\alpha }|\log \epsilon |^{\frac{1}{2}}\,&{} \quad h\ge e^{-\frac{2}{\alpha }}\epsilon .\ \end{array}\right. } \end{aligned}$$
  • \(1< \alpha \le 2\):

    $$\begin{aligned}&\inf _{\begin{array}{c} 0\le \beta \le 1 \\ 0<\eta +\beta \le \alpha \end{array} } \frac{h^{\beta } \epsilon ^{\alpha -\beta -\eta }}{(\alpha -\beta )^2\sqrt{\eta }} \\&\lesssim {\left\{ \begin{array}{ll} h\epsilon ^{\alpha -1}|\log h|^{\frac{1}{2}}, \, \quad \quad h\le \min (e^{-\frac{5}{2(\alpha -1)}},e^{-\frac{2}{\alpha -1}}\epsilon ),\\ h^{\alpha }|\log h|^{\frac{5}{2}}, \, \quad \quad \min (e^{-\frac{5}{2(\alpha -1)}},e^{-\frac{2}{\alpha -1}}\epsilon )\le h\le e^{-\frac{2}{\alpha -1}}\epsilon ,\\ \,{\left\{ \begin{array}{ll} \,{\left\{ \begin{array}{ll} h^{\alpha }|\log h|^{\frac{5}{2}}, &{}\quad h\le \epsilon ^5,\\ h^{\alpha }\left| \log \dfrac{h}{\epsilon } \right| ^2|\log \epsilon |^{\frac{1}{2}}, &{}\quad h\ge \epsilon ^5,\ \end{array}\right. } , &{}\quad \text { if } e^{-\frac{2}{\alpha -1}}\le \epsilon ^5,\\ h^{\alpha }\left| \log \dfrac{h}{\epsilon } \right| ^2|\log \epsilon |^{\frac{1}{2}}, &{}\quad \text { if } e^{-\frac{2}{\alpha -1}}\ge \epsilon ^5,\ \end{array}\right. } \quad e^{-\frac{2}{\alpha -1}}\epsilon \le h\le e^{-\frac{2}{\alpha -1}}\epsilon ,\\ \epsilon ^{\alpha }|\log \epsilon |^{\frac{1}{2}}, \quad h\ge \! e^{-\frac{2}{\alpha }}\epsilon .\ \end{array}\right. } \end{aligned}$$
  • \(2< \alpha \le 3\):

    $$\begin{aligned}&\inf _{\begin{array}{c} 0\le \beta \le 1 \\ 0<\eta +\beta \le \alpha \end{array} } \frac{h^{\beta } \epsilon ^{\alpha -\beta -\eta }}{(\alpha -\beta )^2\sqrt{\eta }} \\&\lesssim {\left\{ \begin{array}{ll} h\epsilon ^{\alpha -1}|\log h|^{\frac{1}{2}}, \, \quad \quad h\le \min (e^{-\frac{5}{2(\alpha -1)}},e^{-\frac{2}{\alpha -1}}\epsilon ),\\ h^{\alpha }|\log h|^{\frac{5}{2}}, \, \quad \quad \min (e^{-\frac{5}{2(\alpha -1)}},e^{-\frac{2}{\alpha -1}}\epsilon )\le h\le e^{-\frac{2}{\alpha -1}}\epsilon ,\\ \,{\left\{ \begin{array}{ll} \,{\left\{ \begin{array}{ll} h^{\alpha }|\log h|^{\frac{5}{2}}, &{}\quad h\le \epsilon ^5,\\ h^{\alpha }\left| \log \dfrac{h}{\epsilon } \right| ^2|\log \epsilon |^{\frac{1}{2}}, &{}\quad h\ge \epsilon ^5,\ \end{array}\right. } , &{}\quad \text { if } e^{-\frac{2}{\alpha -1}}\le \epsilon ^5,\\ h^{\alpha }\left| \log \dfrac{h}{\epsilon } \right| ^2|\log \epsilon |^{\frac{1}{2}}, &{}\quad \text { if } e^{-\frac{2}{\alpha -1}}\ge \epsilon ^5,\ \end{array}\right. } ,\quad e^{-\frac{2}{\alpha -1}}\epsilon \le h\le e^{-\frac{2}{\alpha }}\epsilon ,\\ \,{\left\{ \begin{array}{ll} \epsilon ^{\alpha }|\log \epsilon |^{\frac{1}{2}}, &{}\quad \text { if } \epsilon \ge e^{-\frac{1}{2(\alpha -2)}},\\ h^{\alpha -2}\epsilon ^{2}|\log \epsilon |^{\frac{1}{2}}, &{}\quad \text { if } \epsilon \le e^{-\frac{1}{2(\alpha -2)}},\ \end{array}\right. } \, \quad \quad h\ge e^{-\frac{2}{\alpha }}\epsilon .\ \end{array}\right. } \end{aligned}$$
  • \(\alpha > 3\):

    $$\begin{aligned} \inf _{\begin{array}{c} 0\le \beta \le 1 \\ 0<\eta +\beta \le \alpha \end{array} } \frac{h^{\beta } \epsilon ^{\alpha -\beta -\eta }}{(\alpha -\beta )^2\sqrt{\eta }} \lesssim h\epsilon ^2 . \end{aligned}$$

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Nobile, F., Tesei, F. A Multi Level Monte Carlo method with control variate for elliptic PDEs with log-normal coefficients. Stoch PDE: Anal Comp 3, 398–444 (2015). https://doi.org/10.1007/s40072-015-0055-9

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