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1 Correction to: Stoch PDE: Anal Comp (2021) 9:437–471 https://doi.org/10.1007/s40072-020-00169-4
2 Introduction
Let \({\mathcal {O}}\subset {\mathbb {R}}^d\) be an open convex domain with piecewise smooth boundary. We consider numerical approximation of the stochastic total variation flow
which is constructed via the discretization of the regularized problem
Throughout the paper we employ the notation from [4]. The first error is corrected in Sect. 2 and the correction of the second error is provided in Sect. 3.
3 Definition of the SVI solution and the uniqueness proof
In the proof of [4, Theorem 3.1] the term IV in (29) is wrongly rewritten as
since \(X_1^{\varepsilon }\) is only in \(BV({\mathcal {O}})\) and possibly non-zero at the boundary. Hence, to show the uniqueness of the SVI solutions for \(\varepsilon >0\) requires a modification of the definition which takes into account the value at the solution at the boundary. This definition is consistent with the one from [2], which also shows uniqueness in case \(\varepsilon =0\).
We define the following functionals which include the corresponding boundary terms
and
where \(\gamma _0(u) \) is the trace of \(u\in BV({\mathcal {O}})\) on the boundary and \(\,\mathrm {d}{\mathcal {H}}^{n-1}\) is the Hausdorff measure on \(\partial {\mathcal {O}}\). \(\bar{{\mathcal {J}}}_{\varepsilon ,\lambda }\) and \(\bar{{\mathcal {J}}}_{\lambda }\) are both convex and lower semicontinuous on \(L^2({\mathcal {O}})\) and the lower semicontinuous hulls of \(\bar{{\mathcal {J}}}_{\varepsilon ,\lambda }\vert _{{\mathbb {H}}^1_0}\) or \(\bar{{\mathcal {J}}}_{\lambda }\vert _{{\mathbb {H}}^1_0}\) respectively, cf. [1, Proposition 11.3.2]. We define the SVI solution as follows.
Definition 2.1
Let \(0< T < \infty \), \(\varepsilon \in [0,1]\) and \(x_0 \in L^2(\Omega ,{\mathcal {F}}_0;{\mathbb {L}}^2)\) and \(g \in {\mathbb {L}}^2\). Then a \(({\mathcal {F}}_t)\)-adapted map \(X^{\varepsilon }\in L^2(\Omega ; C([0,T];{\mathbb {L}}^2))\cap L^1(\Omega ; L^1((0,T);BV({\mathcal {O}})))\) (denoted by \(X \in L^2(\Omega ; C([0,T];{\mathbb {L}}^2))\cap L^1(\Omega ; L^1((0,T);BV({\mathcal {O}})))\) for \(\varepsilon =0\)) is called an SVI solution of (2) (or (1) if \(\varepsilon =0\)) if \(X^{\varepsilon }(0)=x_0\) (\(X(0)=x_0\)), and for each \(({\mathcal {F}}_t)\)-progressively measurable process \(G\in L^2(\Omega \times (0,T),{\mathbb {L}}^2) \) and for each \(({\mathcal {F}}_t)\)-adapted \({\mathbb {L}}^2\)-valued process Z with \({\mathbb {P}}\)-a.s. continuous sample paths, s.t. \(Z \in L^2(\Omega \times (0,T);{\mathbb {H}}^1_0)\), which together satisfy the equation
it holds for \(\varepsilon \in (0,1]\) that
and analogously for \(\varepsilon =0\) it holds that
The existence of SVI solutions (3), (4) follows as in [4, Theorem 3.1] by the lower semicontinuity of \(\bar{ {\mathcal {J}}}_{\varepsilon , \lambda }\), \(\bar{ {\mathcal {J}}}_\lambda \), respectively. The uniqueness of SVI solution (4) follows from [2, Theorem 3.2].
To show uniqueness of the SVI solution (3) we proceed as in [4, Theorem 3.1] with exception that the term IV in (29) takes a different form. In particular, to obtain uniqueness we have to show the following estimate:
We note that term IV is well defined since \(\mathrm {div}\frac{\nabla X^{\varepsilon ,\delta }_{2,n}}{ \sqrt{ \vert \nabla X^{\varepsilon ,\delta }_{2,n} \vert ^2 +\varepsilon ^2}} \in {\mathbb {L}}^2\) for a.a. \((\omega , t) \in \Omega \times (0,T)\). Indeed, from [4, Lemma 3.2] for \(\delta > 0\), \(n< \infty \) we deduce by parabolic regularity theory that \(X^{\varepsilon ,\delta }_{2,n}(\omega , t) \in {\mathbb {H}}^2\) for a.a. \((\omega , t) \in \Omega \times (0,T)\) and a direct calculation yields that
We show the inequality in (5) by the integration by parts formula using a density argument. We fix \((\omega , t)\in \Omega \times (0,T)\) and proceed below with \(X_1^{\varepsilon } \equiv X_1^{\varepsilon }(\omega ,t)\), \(X^{\varepsilon ,\delta }_{2,n}\equiv X^{\varepsilon ,\delta }_{2,n}(\omega ,t)\). We consider an approximating sequence \(x_k \in C^{\infty }({\mathcal {O}})\cap BV({\mathcal {O}})\), s.t. \(x_k \rightarrow X_1^{\varepsilon }\) strongly in \({\mathbb {L}}^1\) and
cf., [1, Theorems 10.1.2, 13.4.1 and Remark 10.2.1] or [6, Theorem 5.2].
Note that, since \(X_1^{\varepsilon }(\omega ,t)\in {\mathbb {L}}^2\) it is straightforward to modify the proof of [1, Theorems 10.1.2] (see for instance [1, Proposition 2.2.4] for the \({\mathbb {L}}^p\) properties of the mollifiers) such that the sequence \(x_k\) converges strongly in \({\mathbb {L}}^2\):
Using the integration by parts formula [1, Theorem 10.2.1] we obtain that
where \(\nu \) is the outer unit normal vector to \(\partial {\mathcal {O}}\) and \({\mathcal {H}}^{n-1}\) is the Hausdorff measure on \(\partial {\mathcal {O}}\).
Since \(X^{\varepsilon ,\delta }_{2,n}\in {\mathbb {H}}^1_0\) it holds that \(\gamma _0(X^{\varepsilon ,\delta }_{2,n})=0\) and the second boundary integral vanishes. The first boundary integral can be estimated as
where the last equality follows from the fact that the trace of \(x_k \in C^{\infty }({\mathcal {O}})\cap BV({\mathcal {O}})\) coincides with the trace of \(X_1^{\varepsilon }\), cf. [1, Remark 10.2.1].
By the convexity of \({\mathcal {J}}_{\varepsilon ,0}\) we deduce that
Hence, (5) follows after substituting (10), (9) into (8) and taking the limit for \(k\rightarrow \infty \) and noting (6), (7).
The rest of the proof follows analogously to the original proof of [4, Theorem 3.1].
4 Convergence of the full discretization
In the proof of [4, Lemma 4.4] it is concluded that
which is not justified. Lemma 4.4 is required to obtain the estimate (48) in [4, Lemma 4.5] (note that the continuos counterpart of the estimate in Lemma 3.2 is obtained using Proposition 2.1), which is in turn required to show [4, Theorem 4.1].
In this section we show an analogue of the estimate in [4, Lemma 4.4] for a slightly modified numerical scheme in dimension \(d=1\). Given \(J \in {\mathbb {N}}\) and a mesh size \(h=1/J\) we consider a uniform partiton \({\mathcal {T}}_h=\cup _{j=1}^J T_j\) of the spatial domain \({\mathcal {O}}=(0,1)\) into subintervals \(T_j=(x_{i-1},x_i)\) with nodes \(x_j=jh\), \(j=0,\dots , J\). As in [4] we consider a finite element space \({\mathbb {V}}_h\subset {\mathbb {H}}^1_0\) of piecewise linear globally continuous functions on subordinated to \({\mathcal {T}}_h\). The standard nodal interpolation operator \({\mathcal {I}}_h:C({\bar{{\mathcal {O}}}})\rightarrow {\mathbb {V}}_h\) is defined as
We define the discrete (mass-lumped) \({\mathbb {L}}^2\)-inner product \(\left( \cdot ,\cdot \right) _h\) on \({\mathbb {V}}_h\) as
with the corresponding discrete norm \(\Vert \psi \Vert ^2_h=\left( \psi ,\psi \right) _h\).
It is well known that the above discrete inner product and the norm satisfy (cf. [5]):
We define the mass-lumped Discrete Laplace operator \(\Delta _h: {\mathbb {V}}_h \rightarrow {\mathbb {V}}_h\) through the identity
The next lemma is the counterpart of [4, Lemma 4.4] for the 1d discrete Laplace operator (14). Numerical experiments (not stated in this paper) indicate that the result also holds for \(d>1\) (possibly under some additional assumptions on the shape of the mesh). Nevertheless, the proof of the result for \(d>1\) remains open, so far.
Lemma 3.1
Let \(\Delta _h\) be the discrete Laplacian defined by (14). Then for any \(v_h \in {\mathbb {V}}_h\), \(\varepsilon , h > 0\) the following inequality holds:
Proof
Since \({\mathbb {V}}_h\) is the space of piecewise linear functions over \({\mathcal {T}}_h\), it holds for \(v_h \in {\mathbb {V}}_h\) that
By definition (11) and (14) we deduce that
and \(\Delta _h v_h^0 = \Delta _h v_h^J = 0\).
By the above properties we deduce that
where we used the convexity of \(\sqrt{|\cdot |^2+\varepsilon ^2}\) to deduce the last inequality. \(\square \)
Using the above lemma one can show the convergence for a slight modification of the fully discrete numerical scheme of [4] where the standard \({\mathbb {L}}^2\)-inner product is replaced by the discrete inner product (11) as follows: given \(x_0,\, g \in {\mathbb {L}}^2\) we set \(X^{0}_{\varepsilon ,h}={\mathcal {P}}_h x_0\), \(g^h:={\mathcal {P}}_h g\) and obtain \(X^{i}_{\varepsilon ,h}\) for \(i=1,\dots , N\) as the solution of the following system:
By the equivalence of the norms \(\Vert \cdot \Vert \) and \(\Vert \cdot \Vert _h\) (cf. (12)) the convergence of the above numerical approximation for \(d=1\) follows as in [4] with [4, Lemma 4.4] replaced by Lemma 3.1.
We note that the convergence proof remains valid for \(d\ge 1\) in the case of the time-semi discrete variant of the original numerical scheme from [4]:
In the semi-discrete setting one employs the continuous counterpart of Lemma 3.1 and proceeds as in the proof of [4, Lemma 3.2] to obtain the space-continuous version of the stronger estimate (48) in Lemma 4.5 from [4]. Then the convergence proof of the above semi-discrete numerical scheme follows analogically as in the case of the fully discrete numerical approximation; we skip the detailed exposition for brevity and instead refer to [3, Section 4], from where the necessary components of the proof can be deduced.
Finally, we conclude that a convergence proof of the fully discrete numerical approximation for \(d \ge 1\), which avoids the use of [4, Lemma 4.4], is provided in the upcoming paper [3].
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Acknowledgements
The authors would like to thank Martin Ondreját for pointing the two mistakes in the original paper to us.
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Baňas, Ĺ., Röckner, M. & Wilke, A. Correction to: Convergent numerical approximation of the stochastic total variation flow. Stoch PDE: Anal Comp 11, 1732–1739 (2023). https://doi.org/10.1007/s40072-022-00267-5
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DOI: https://doi.org/10.1007/s40072-022-00267-5