An overlapping domain decomposition framework without dual formulation for variational imaging problems

Abstract

In this paper, we propose a novel overlapping domain decomposition method that can be applied to various problems in variational imaging such as total variation minimization. Most of recent domain decomposition methods for total variation minimization adopt the Fenchel–Rockafellar duality, whereas the proposed method is based on the primal formulation. Thus, the proposed method can be applied not only to total variation minimization but also to those with complex dual problems such as higher order models. In the proposed method, an equivalent formulation of the model problem with parallel structure is constructed using a custom overlapping domain decomposition scheme with the notion of essential domains. As a solver for the constructed formulation, we propose a decoupled augmented Lagrangian method for untying the coupling of adjacent subdomains. Convergence analysis of the decoupled augmented Lagrangian method is provided. We present implementation details and numerical examples for various model problems including total variation minimizations and higher order models.

Appendices

Appendix A: Convergence analysis of Algorithm 1

In this appendix, we analyze the convergence behavior of the decoupled augmented Lagrangian method. Throughout this section, we assume that $\widetilde {E} (\tilde {u})$ given in (2.3) is convex.

The proof of Theorem 3.4 is based on a Lyapunov functional argument, which is broadly used in the analysis of augmented Lagrangian methods [14, 16, 32]. That is, we show that there exists the Lyapunov functional that is bounded below and decreases in each iteration. The following lemma is a widely used property for convex optimization.

Lemma A.1

Let f:$\mathbb {R}^{n} \rightarrow \bar {\mathbb {R}}$ be a convex function, A:$\mathbb {R}^{n} \rightarrow \mathbb {R}^{m}$ a linear operator, and $b \in \mathbb {R}^{m}$. Then, a solution $x^{*} \in \mathbb {R}^{n}$ of the minimization problem:

$$ \min_{x \in \mathbb{R}^{n}} f(x) + \frac{\alpha}{2} \| Ax - b \|_{2}^{2} $$

is characterized by

$$ f(x) \geq f(x^{*} ) + \alpha \langle Ax^{*} - b , A(x^{*} - x) \rangle \forall x \in \mathbb{R}^{n} . $$

Proof

It is straightforward from the fact that − αA^∗(Ax^∗− b) ∈ ∂f(x^∗). □

We observe that if we choose an initial guess $\lambda ^{(0)} \in \widetilde {V}^{*}$ such that $J_{\widetilde {V}^{*}} \lambda ^{(0)} \in (\ker B)^{\bot }$, then we have $J_{\widetilde {V}^{*}} \lambda ^{(n)} \in (\ker B)^{\bot }$ for all n ≥ 0.

Proposition A.2

In Algorithm 1, we have $J_{\widetilde {V}^{*}} \lambda ^{(n)} \in (\ker B)^{\bot }$ for all n ≥ 1 if $J_{\widetilde {V}^{*}} \lambda ^{(0)} \in (\ker B)^{\bot }$.

Proof

Since $J_{\widetilde {V}^{*}} \left (\lambda ^{(n+1)} - \lambda ^{(n)} \right ) = (I -P_{B} ) \tilde {u}^{(n+1)} \in (\ker B)^{\bot }$ for all n ≥ 0, a simple induction argument yields the conclusion. □

With Lemma A.1 and Proposition A.2, we readily get the following characterization of $\tilde {u}^{(n+1)}$ in Algorithm 1.

Lemma A.3

In Algorithm 1, $\tilde {u}^{(n+1)} \in \widetilde {V}$ satisfies:

$$ \begin{array}{@{}rcl@{}} \widetilde{E} (\tilde{u}) \geq \widetilde{E} \left( \tilde{u}^{(n+1)}\right) + \langle J_{\widetilde{V}} (I - P_{B}) \left( \tilde{u}^{(n+1)} - \tilde{u}\right) , \lambda^{(n)} \rangle\\ + \eta \left\langle \tilde{u}^{(n+1)} - P_{B} \tilde{u}^{(n)} , \tilde{u}^{(n+1)} - \tilde{u} \right\rangle \quad \forall \tilde{u} \in \widetilde{V} \end{array} $$

for n ≥ 0.

Proof

Take any $\tilde {u} \in \widetilde {V}$. By (3.6), Lemma A.1, and Proposition A.2, we obtain:

$$ \begin{array}{@{}rcl@{}} \widetilde{E} (\tilde{u}) &\geq& \widetilde{E} \left( \tilde{u}^{(n+1)}\right) + \left\langle \tilde{u}^{(n+1)} - \tilde{u} , J_{\widetilde{V}^{*}} \lambda^{(n)} \right\rangle + \eta \left\langle \tilde{u}^{(n+1)} - P_{B} \tilde{u}^{(n)} , \tilde{u}^{(n+1)} - \tilde{u} \right\rangle \\ &=& \widetilde{E} \left( \tilde{u}^{(n+1)}\right) + \left\langle \tilde{u}^{(n+1)} - \tilde{u} , (I-P_{B} ) J_{\widetilde{V}^{*}} \lambda^{(n)} \right\rangle\\ &&+ \eta \left\langle \tilde{u}^{(n+1)} - P_{B} \tilde{u}^{(n)} , \tilde{u}^{(n+1)} - \tilde{u} \right\rangle, \end{array} $$

which concludes the proof. □

Let $(\tilde {u}^{*} , \lambda ^{*}) \in \widetilde {V} \times \widetilde {V}^{*}$ be a critical point of (3.9). We define:

$$ \begin{array}{@{}rcl@{}} d_{n} &=& \eta \left\| P_{B} \left( \tilde{u}^{(n)} - \tilde{u}^{(n+1)}\right) \right\|_{2}^{2} + \frac{1}{\eta} \left\| \lambda^{(n)} - \lambda^{(n+1)} \right\|_{2}^{2} , \end{array} $$

(A.1a)

$$ \begin{array}{@{}rcl@{}} e_{n} &=& \eta \left\| P_{B} \left( \tilde{u}^{(n)} - \tilde{u}^{*}\right) \right\|_{2}^{2} + \frac{1}{\eta} \left\| \lambda^{(n)} - \lambda^{*} \right\|_{2}^{2} . \end{array} $$

(A.1b)

It is clear that the value d_n measures the difference between two consecutive iterates $(\tilde {u}^{(n)} , \lambda ^{(n)})$ and $(\tilde {u}^{(n+1)}, \lambda ^{(n+1)})$, while e_n measures the error of the n th iterate $(\tilde {u}^{(n)} , \lambda ^{(n)})$ with respect to a solution $(\tilde {u}^{*}, \lambda ^{*})$. The following lemma presents the Lyapunov functional argument. We note that the Lyapunov functional that we use in the proof is motivated from [32].

Lemma A.4

The value e_n defined in (A.1b) is decreasing in each iteration of Algorithm 1. More precisely, we have:

$$ e_{n} - e_{n+1} \geq d_{n} $$

(A.2)

for n ≥ 0, where d_n is given in (A.1a).

Proof

By the definition of $(\tilde {u}^{*} , \lambda ^{*})$, we clearly have $(I - P_{B}) \tilde {u}^{*} = 0$. Furthermore, since:

$$ \tilde{u}^{*} \in \underset{\tilde{u} \in \widetilde{V}}{\arg\min} \left\{ \widetilde{E}(\tilde{u}) + \langle J_{\widetilde{V}} (I-P_{B} )\tilde{u}, \lambda^{*} \rangle + \frac{\eta}{2} \| (I -P_{B} ) \tilde{u} \|_{2}^{2} \right\}, $$

by Lemma A.1, $\tilde {u}^{*}$ is characterized by:

$$ \widetilde{E} (\tilde{u}) \geq \widetilde{E} (\tilde{u}^{*}) - \langle J_{\widetilde{V}} (I-P_{B}) (\tilde{u} - \tilde{u}^{*}) , \lambda^{*} \rangle \forall \tilde{u} \in \widetilde{V}. $$

(A.3)

Taking $\tilde {u} = \tilde {u}^{(n+1)}$ in (A.3) yields:

$$ \widetilde{E} \left( \tilde{u}^{(n+1)}\right) \geq \widetilde{E} (\tilde{u}^{*}) - \left\langle J_{\widetilde{V}} (I-P_{B})\bar{u}^{(n+1)} , \lambda^{*} \right\rangle. $$

(A.4)

Let $\bar {u}^{(n)} = \tilde {u}^{(n)} - \tilde {u}^{*}$ and $\bar {\lambda }^{(n)} = \lambda ^{(n)} - \lambda ^{*}$. Taking $\tilde {u} = \tilde {u}^{*}$ in Lemma A.3 yields:

$$ \widetilde{E} (\tilde{u}^{*}) \geq \widetilde{E} \left( \tilde{u}^{(n+1)} \right) + \left\langle J_{\widetilde{V}} (I - P_{B}) \bar{u}^{(n+1)} , \lambda^{(n)} \right\rangle + \eta \left\langle \tilde{u}^{(n+1)} - P_{B} \tilde{u}^{(n)} , \bar{u}^{(n+1)}\right\rangle. $$

(A.5)

Then, by adding (A.4) and (A.5) and using $P_{B} \tilde {u}^{(n)} = P_{B} \bar {u}^{(n)}$, we have:

$$ \begin{array}{@{}rcl@{}} 0 &\geq& \left\langle J_{\widetilde{V}} (I-P_{B}) \bar{u}^{(n+1)} , \bar{\lambda}^{(n)} \right\rangle + \eta \left\langle \bar{u}^{(n+1)} - P_{B} \bar{u}^{(n)} , \bar{u}^{(n+1)}\right\rangle \\ &=& \left\langle J_{\widetilde{V}} (I-P_{B}) \bar{u}^{(n+1)} , \bar{\lambda}^{(n)} \right\rangle + \eta \left\| (I-P_{B}) \bar{u}^{(n+1)} \right\|_{2}^{2}\\ &&- \eta \left\langle P_{B}\left( \bar{u}^{(n)} - \bar{u}^{(n+1)}\right), \bar{u}^{(n+1)}\right\rangle. \end{array} $$

That is, we obtain:

$$ \begin{array}{@{}rcl@{}} S&:=& -\left\langle J_{\widetilde{V}} (I-P_{B}) \bar{u}^{(n+1)} , \bar{\lambda}^{(n)} \right\rangle - \eta \left\| (I-P_{B}) \bar{u}^{(n+1)} \right\|_{2}^{2}\\ &&+ \eta \left\langle P_{B}\left( \bar{u}^{(n)} - \bar{u}^{(n+1)}\right), \bar{u}^{(n+1)}\right\rangle \geq 0. \end{array} $$

(A.6)

Using $\bar {\lambda }^{(n+1)} = \bar {\lambda }^{(n)} + \eta J_{\widetilde {V}} (I-P_{B} ) \bar {u}^{(n+1)}$, we get:

$$ e_{n} - e_{n+1} = 2S + \eta \left\| P_{B}\left( \bar{u}^{(n)} - \bar{u}^{(n+1)}\right) \right\|_{2}^{2} + \eta \left\| (I-P_{B}) \bar{u}^{(n+1)} \right\|_{2}^{2} \geq d_{n}, $$

which yields (A.2). The last inequality is due to (A.6). □

Now, we present the proof of Theorem 3.4.

Proof Proof of Theorem 3.4

As $\widetilde {E}$ is convex, Lemma A.4 ensures that (A.2) holds. Since $\left \{e_{n} \right \}$ is bounded, we conclude that $\left \{ P_{B} {\tilde {u}}^{(n)} \right \}$ and $\left \{ \lambda ^{(n)} \right \}$ are bounded. We sum (A.2) from n = 0 to N − 1 and let $N \rightarrow \infty$ to obtain:

$$ e_{0} - \lim\limits_{n \rightarrow \infty} e_{n} \geq \sum\limits_{n=0}^{N} d_{n} = \eta \sum\limits_{n=0}^{\infty} \left\| P_{B} \left( \tilde{u}^{(n)} - \tilde{u}^{(n+1)}\right)\right\|_{2}^{2} + \eta \sum\limits_{n=0}^{\infty} \left\| (I-P_{B}) \tilde{u}^{(n+1)} \right\|_{2}^{2} , $$

which implies that $P_{B}\left (\tilde {u}^{(n)} - \tilde {u}^{(n+1)}\right ) \rightarrow 0$ and $(I-P_{B}) \tilde {u}^{(n+1)} \rightarrow 0$. Therefore, $\left \{ \tilde {u}^{(n)} \right \}$ is bounded and we have:

$$ \tilde{u}^{(n)} - \tilde{u}^{(n+1)} = P_{B}\left( \tilde{u}^{(n)} - \tilde{u}^{(n+1)}\right) + (I-P_{B}) \tilde{u}^{(n)} - (I-P_{B}) \tilde{u}^{(n+1)} \rightarrow 0 $$

(A.7a)

and

$$ \lambda^{(n)} - \lambda^{(n+1)} = -\eta J_{\widetilde{V}} (I-P_{B}) \tilde{u}^{(n+1)} \rightarrow 0. $$

(A.7b)

By the Bolzano–Weierstrass theorem, there exists a limit point $(\tilde {u}^{(\infty )}, \lambda ^{(\infty )})$ of the sequence $\left \{ (\tilde {u}^{(n)} , \lambda ^{(n)}) \right \}$. We choose a subsequence $\{ (\tilde {u}^{(n_{j})} , \lambda ^{(n_{j})}) \}$ of $\left \{ (\tilde {u}^{(n)} , \lambda ^{(n)}) \right \}$ such that:

$$ \left( \tilde{u}^{(n_{j})}, \lambda^{(n_{j})}\right) \rightarrow \left( \tilde{u}^{(\infty)} , \lambda^{(\infty)}\right) \textrm{ as } j \rightarrow \infty. $$

(A.8)

By (A.7a), we have:

$$ \left( \tilde{u}^{(n_{j} - 1)}, \lambda^{(n_{j} - 1)}\right) \rightarrow \left( \tilde{u}^{(\infty)} , \lambda^{(\infty)}\right) \textrm{ as } j \rightarrow \infty. $$

In the λ-update step with n = n_j − 1:

$$ \lambda^{(n_{j})} = \lambda^{(n_{j} -1)} + \eta J_{\widetilde{V}} (I - P_{B}) \tilde{u}^{(n_{j}) }, $$

we readily obtain $(I -P_{B}) \tilde {u}^{(\infty )} = 0$ as j tends to $\infty$. On the other hand, (3.6) with n = n_j − 1 is equivalent to:

$$ \partial \widetilde{E} \left( \tilde{u}^{(n_{j})}\right) + J_{\widetilde{V}^{*}} \lambda^{(n_{j} - 1)} + \eta \left( \tilde{u}^{(n_{j} )} - P_{B} \tilde{u}^{(n_{j} - 1)}\right) \ni 0. $$

By the graph-closedness of $\partial \widetilde {E}$ (see Theorem 24.4 in [27]), we get:

$$ \partial \widetilde{E} \left( \tilde{u}^{(\infty)} \right) + J_{\widetilde{V}^{*}} \lambda^{(\infty)} + \eta (I-P_{B} ) \tilde{u}^{(\infty)} \ni 0 $$

as $j \rightarrow \infty$. By Proposition A.2, we conclude that:

$$ \partial \widetilde{E} \left( \tilde{u}^{(\infty)} \right) + (I-P_{B} ) J_{\widetilde{V}^{*}} \lambda^{(\infty)} \ni 0. $$

Therefore, $\left (\tilde {u}^{(\infty )}, \lambda ^{(\infty )}\right )$ is a critical point of (3.9).

Finally, it remains to prove that the whole sequence $\left \{ \left (\tilde {u}^{(n)}, \lambda ^{(n)} \right ) \right \}$ converges to the critical point $\left (\tilde {u}^{(\infty )}, \lambda ^{(\infty )}\right )$. Since the critical point $(\tilde {u}^{*}, \lambda ^{*})$ was arbitrarily chosen, Lemma A.4 is still valid if we set $(\tilde {u}^{*}, \lambda ^{*}) = \left (\tilde {u}^{(\infty )}, \lambda ^{(\infty )}\right )$ in (A.1b). That is, the sequence:

$$ e_{n} = \eta \left\| P_{B} \left( \tilde{u}^{(n)} - \tilde{u}^{(\infty)}\right) \right\|_{2}^{2} + \frac{1}{\eta} \left\| \lambda^{(n)} - \lambda^{(\infty)} \right\|_{2}^{2} $$

is decreasing. On the other hand, by (A.8), the subsequence $\{ e_{n_{j}} \}$ tends to 0 as j goes to $\infty$. Therefore, the whole sequence {e_n} tends to 0 and we deduce that $\left \{ \left (\tilde {u}^{(n)}, \lambda ^{(n)} \right ) \right \}$ converges to $\left (\tilde {u}^{(\infty )}, \lambda ^{(\infty )}\right )$. □

Remark A.5

In practice, local problems (3.8) are solved by iterative algorithms and an inexact solution $\tilde {u}^{(n+1)}$ to (3.6) is obtained in each iteration of Algorithm 1. That is, for n ≥ 0, we have:

$$ 0 \in \partial_{\epsilon_{n}} J_{n} \left( \tilde{u}^{(n)}\right) $$

for some 𝜖_n > 0, where:

$$ J_{n} (\tilde{u}) = \widetilde{E} (\tilde{u}) + \left\langle J_{\widetilde{V}} \tilde{u}, \lambda^{(n)} \right\rangle_{\widetilde{V}^{*}} + \frac{\eta}{2} \left\| \tilde{u} - P_{B} \tilde{u}^{(n)} \right\|_{2, \widetilde{V}}^{2}. $$

One may refer to e.g. [27] for the definition of the 𝜖-subgradient ∂_𝜖. In this case, the conclusion of Lemma A.3 is replaced by:

$$ \begin{array}{@{}rcl@{}} \widetilde{E} (\tilde{u}) &\geq& \widetilde{E} \left( \tilde{u}^{(n+1)}\right) + \left\langle J_{\widetilde{V}} (I - P_{B}) \left( \tilde{u}^{(n+1)} - \tilde{u}\right) , \lambda^{(n)} \right\rangle\\ &&+ \eta \left\langle \tilde{u}^{(n+1)} - P_{B} \tilde{u}^{(n)} , \tilde{u}^{(n+1)} - \tilde{u} \right\rangle - \epsilon_{n} \quad \forall \tilde{u} \in \widetilde{V} \end{array} $$

(A.9)

for all n ≥ 0. By slightly modifying the above proofs using (A.9), one can prove without major difficulty that the conclusion of Theorem 3.4 holds under an assumption:

$$ \sum\limits_{n=0}^{\infty} \epsilon_{n} < \infty. $$

The above summability condition of errors is popular in the field of mathematical optimization; see, e.g., [26].

To prove Theorem 3.5, we first show that d_n is decreasing.

Lemma A.6

The value d_n defined in (A.1a) is decreasing in each iteration of Algorithm 1.

Proof

Let n ≥ 1. Taking $\tilde {u} = \tilde {u}^{(n)}$ in Lemma A.3 yields:

$$ \begin{aligned} \widetilde{E}\left( \tilde{u}^{(n)}\right) &\geq \widetilde{E} \left( \tilde{u}^{(n+1)}\right) + \left\langle J_{\widetilde{V}} (I-P_{B})\left( \tilde{u}^{(n+1)} - \tilde{u}^{(n)}\right), \lambda^{(n)} \right\rangle\\ &\quad+ \eta \left\langle\tilde{u}^{(n+1)} - P_{B} \tilde{u}^{(n)} , \tilde{u}^{(n+1)} - \tilde{u}^{(n)} \right\rangle. \end{aligned} $$

(A.10)

Also, substituting n by n − 1 and taking $\tilde {u} = \tilde {u}^{(n+1)}$ in Lemma A.3, we have:

$$ \begin{aligned} \widetilde{E}\left( \tilde{u}^{(n+1)}\right) &\geq \widetilde{E} \left( \tilde{u}^{(n)}\right) + \left\langle J_{\widetilde{V}} (I-P_{B})\left( \tilde{u}^{(n)} - \tilde{u}^{(n+1)}\right), \lambda^{(n-1)} \right\rangle\\ &\quad+ \eta \left\langle\tilde{u}^{(n)} - P_{B} \tilde{u}^{(n-1)} , \tilde{u}^{(n)} - \tilde{u}^{(n+1)} \right\rangle. \end{aligned} $$

(A.11)

Summation of (A.10) and (A.11) yields:

$$ \begin{array}{@{}rcl@{}} 0 &\geq& \left\langle J_{\widetilde{V}} (I-P_{B} ) \left( \tilde{u}^{(n)} - \tilde{u}^{(n+1)}\right) , \lambda^{(n-1)}-\lambda^{(n)} \right\rangle \\ &&+ \eta \left\langle \tilde{u}^{(n)} - \tilde{u}^{(n+1)}, \tilde{u}^{(n)} - P_{B} \tilde{u}^{(n-1)} - \tilde{u}^{(n+1)} + P_{B} \tilde{u}^{(n)} \right\rangle \\ &=& \eta \left\langle \tilde{u}^{(n)} - \tilde{u}^{(n+1)}, -P_{B} \tilde{u}^{(n-1)} + 2P_{B} \tilde{u}^{(n)} - \tilde{u}^{(n+1)} \right\rangle, \end{array} $$

where we used $\lambda ^{(n)} = \lambda ^{(n-1)} + \eta J_{\widetilde {V}} (I-P_{B}) \tilde {u}^{(n)}$ in the equality. Therefore, we get:

$$ T:= \left\langle \tilde{u}^{(n)} - \tilde{u}^{(n+1)}, P_{B} \tilde{u}^{(n-1)} - 2P_{B} \tilde{u}^{(n)} + \tilde{u}^{(n+1)} \right\rangle \geq 0. $$

(A.12)

On the other hand, direct computation yields:

$$ \frac{1}{\eta} (d_{n-1} - d_{n} ) = 2T + \left\| \tilde{u}^{(n)} - \tilde{u}^{(n+1)} - P_{B} \left( \tilde{u}^{(n-1)} - \tilde{u}^{(n)}\right) \right\|_{2}^{2} \geq 0, $$

which concludes the proof. The last inequality is due to (A.12). □

Combining Lemmas A.4 and A.6, we get the proof Theorem 3.5, which closely follows [14].

Proof Proof of Theorem 3.5

Invoking Lemmas A.6 and (A.2) yields:

$$ (n+1)d_{n} \leq \sum\limits_{k=0}^{n} d_{k} \leq e_{0} - e_{n+1} \leq e_{0}. $$

This completes the proof. □

Appendix B: A remark on the continuous setting

As we noticed in Section 4, the proposed domain decomposition framework reduces to the one proposed in [11] when it is applied to the convex Chan–Vese model [5]. However, while the authors of [11] introduced their method as a nonoverlapping DDM, we classified it as an overlapping one. In this section, we claim that the proposed method belongs to a class of overlapping DDMs in the continuous setting.

For simplicity, we consider the case $\mathcal {N} = 2$ only. Let $\left \{ {\varOmega }_{s} \right \}_{s=1}^{2}$ be a nonoverlapping domain decomposition of Ω with the interface Γ = ∂Ω₁ ∩ ∂Ω₂. Recall the convex Chan–Vese model (4.2):

$$ \min_{u \in BV(\varOmega)} \left\{ \alpha {\int}_{\varOmega} ug dx + \chi_{\left\{ 0 \leq \cdot \leq 1\right\}} (u) + TV_{\varOmega}(u) \right\}, $$

(B.1)

where g = (f − c₁)² − (f − c₂)² and TV_Ω(u) is defined as:

$$ TV_{\varOmega} (u) = \sup \left\{{\int}_{\varOmega} u \text{div} \mathbf{p} dx : \mathbf{p} \in {C_{0}^{1}}\left( \varOmega, \mathbb{R}^{2}\right), |\mathbf{p}| \leq 1 \right\}. $$

In Section 3.1 of [11], it was claimed that a solution of (B.1) can be constructed by u = u₁ ⊕ u₂, where (u₁,u₂) is a solution of the constrained minimization problem:

$$ \min_{\substack{u_{s} \in BV({\varOmega}_{s}) \\ s=1,2}} \sum\limits_{s=1}^{2} \left( \alpha {\int}_{{\varOmega}_s} u_{s} g dx + \chi_{\left\{ 0 \leq \cdot \leq 1 \right\} } (u_{s}) + TV_{{\varOmega}_{s}}(u_{s}) \right) \textrm{subject to } u_{1} = u_{2} \textrm{ on } {\Gamma}. $$

(B.2)

Here, the condition u₁ = u₂ on Γ is of the trace sense [12]. Unfortunately, this argument is not valid since the solution space BV (Ω) of (B.1) allows discontinuities on Γ. We provide a simple counterexample inspired from [5].

Example 1

Let $\varOmega = (-1, 1) \subset \mathbb {R}$, Ω₁ = (− 1, 0), and Ω₂ = (0, 1). We set

$$ g(x) = \begin{cases} -1 & \textrm{ if } x \in {\varOmega}_{1} , \\ 1 & \textrm{ if } x \in {\varOmega}_{2} . \end{cases} $$

We will show that:

$$ u^{*}(x) = \begin{cases} 1 & \textrm{ if } x \in {\varOmega}_{1} , \\ 0 & \textrm{ if } x \in {\varOmega}_{2} \end{cases} $$

is a unique solution of (B.1) for sufficiently large α, while it cannot be a solution of (B.2) since it is not continuous on Γ. We clearly have TV_Ω(u^∗) = 1. There exists $p^{*} \in {C_{0}^{1}} (\varOmega )$ with |p^∗|≤ 1 which attains the supremum in the definition of total variation for u^∗. Indeed, with p^∗(x) = 1 − x², we have:

$$ 1 = TV_{\varOmega}(u^{*} ) = \sup\left\{ {\int}_{\varOmega} u^{*} p^{\prime} dx : p \in {C_{0}^{1}} (\varOmega), |p| \leq 1 \right\} \\ \geq {\int}_{\varOmega} u^{*} (p^{*})^{\prime} dx = 1. $$

Choose $\alpha > 2 = \max \limits _{x \in \varOmega } |(p^{*})^{\prime }(x)|$. For any u ∈ BV (Ω) with 0 ≤ u ≤ 1, we have:

$$ \begin{array}{@{}rcl@{}} \alpha {\int}_{\varOmega} ug dx + TV_{\varOmega}(u) &\geq& \alpha {\int}_{\varOmega} ug dx + {\int}_{\varOmega} u(p^{*})^{\prime} dx \\ &=& \alpha {\int}_{\varOmega} u^{*} g dx + TV_{\varOmega}(u^{*}) + {\int}_{\varOmega} (u-u^{*}) (\alpha g + (p^{*})^{\prime}) dx. \\ \end{array} $$

In addition, we have:

$$ {\int}_{\varOmega} (u-u^{*}) (\alpha g + (p^{*})^{\prime}) dx = {\int}_{-1}^{0} (1-u) (\alpha - (p^{*})^{\prime}) dx + {{\int}_{0}^{1}} u (\alpha + (p^{*})^{\prime}) dx \geq 0. $$

Since $\alpha \pm (p^{*})^{\prime }$ is strictly positive, the equality holds if and only if u = u^∗ a.e.. Therefore, u^∗ is a unique solution of (B.1).

On the other hand, it is possible to construct an equivalent constrained minimization problem with an overlapping domain decomposition. Let S be a neighborhood of Γ with positive measure. Note that traces γ₁u and γ₂u of $u \in BV(\mathcal {O})$ along Γ with respect to $\mathcal {O} \cap {\varOmega }_{1}$ and $\mathcal {O} \cap {\varOmega }_{2}$, respectively, are well-defined for any open subset $\mathcal {O}$ of Ω such that $S \subset \mathcal {O}$. Also, they satisfy the formula:

$$ TV_{\varOmega} (u) = TV_{{\varOmega}_{1}} (u) + TV_{{\varOmega}_{2}} (u) + {\int}_{\Gamma} |\gamma_{1} u - \gamma_{2} u | ds . $$

Set $\widetilde {\varOmega }_{1} = {\varOmega }_{1} \cup S$ and $\widetilde {\varOmega }_{2} = {\varOmega }_{2}$. Then, $\{ \widetilde {\varOmega }_{s} \}_{s=1}^{2}$ forms an overlapping domain decomposition of Ω; i.e., $\widetilde {\Gamma } = \widetilde {\varOmega }_{1} \cap \widetilde {\varOmega }_{2}$ has positive measure. We define local energy functionals E_s: $\widetilde {\varOmega }_{s} \rightarrow \mathbb {R}$ as follows:

$$ \begin{array}{@{}rcl@{}} E_{1} (\tilde{u}_{1}) &=& \alpha {\int}_{{\varOmega}_{1}} \tilde{u}_{1} g dx + \chi_{\left\{ 0 \leq \cdot \leq 1 \right\}} (\tilde{u}_{1}) + TV_{{\varOmega}_{1}} (\tilde{u}_{1}) + {\int}_{\Gamma} |\gamma_{1} \tilde{u}_{1} - \gamma_{2} \tilde{u}_{1} | ds , \\ E_{2} (\tilde{u}_{2}) &=& \alpha {\int}_{{\varOmega}_{2}} \tilde{u}_{2} g dx + \chi_{\left\{ 0 \leq \cdot \leq 1 \right\}} (\tilde{u}_{2}) + TV_{{\varOmega}_{2}} (\tilde{u}_{2}). \end{array} $$

Consider the following constrained minimization problem:

$$ \min_{\substack{\tilde{u}_{s} \in BV(\widetilde{\varOmega}_{s})\\ s=1,2}} \sum\limits_{s=1}^{2} E_{s} (\tilde{u}_{s}) \textrm{subject to } \tilde{u}_{1} = \tilde{u}_{2} \textrm{ on } \widetilde{\Gamma} . $$

(B.3)

Then, we have the following equivalence theorem.

Theorem B.1

Let $(\tilde {u}_{1}^{*}, \tilde {u}_{2}^{*} ) \in BV(\widetilde {\varOmega }_{1}) \times BV(\widetilde {\varOmega }_{2})$ be a solution of (B.3). Then, u^∗∈ BV (Ω) defined by:

$$ u^{*}(x) = \begin{cases} \tilde{u}_{1}^{*} (x) & \textrm{ if } x \in \widetilde{\varOmega}_{1} , \\ \tilde{u}_{2}^{*} (x) & \textrm{ if } x \in {\varOmega} \setminus \widetilde{\varOmega}_{1} \end{cases} $$

is a solution of (B.1). Conversely, if u^∗∈ BV (Ω) is a solution of (B.1), then $(\tilde {u}_{1}^{*} , \tilde {u}_{2}^{*}) = (u^{*}|_{\widetilde {\varOmega }_{1}} , u^{*}|_{{\widetilde {\varOmega }}_{2}}) \in BV({\widetilde {\varOmega }}_{1}) \times BV({\widetilde {\varOmega }}_{2})$ is a solution of (B.3).

Proof

First, suppose that $\left (\tilde {u}_{1}^{*}, \tilde {u}_{2}^{*} \right )$ is a solution of (B.3). For any u ∈ BV (Ω) with 0 ≤ u ≤ 1, we have:

$$ \begin{array}{ll} \alpha {\int}_{\varOmega} ug dx + TV_{\varOmega}(u) &= E_{1} \left( u|_{\widetilde{\varOmega}_{1}}\right) + E_{2} \left( u|_{\widetilde{\varOmega}_{2}}\right) \\ &\geq E_{1} \left( \tilde{u}_{1}^{*}\right) + E_{2} \left( \tilde{u}_{2}^{*}\right) = \alpha {\int}_{\varOmega} u^{*} g dx + TV_{\varOmega}(u^{*}). \end{array} $$

Hence, u^∗ minimizes (B.1).

Conversely, we assume that u^∗∈ BV (Ω) is a solution of (B.1) and set $\left (\tilde {u}_{1}^{*} , \tilde {u}_{2}^{*}\right ) = \left (u^{*}|_{\widetilde {\varOmega }_{1}} , u^{*}|_{\widetilde {\varOmega }_{2}}\right )$. Take any $(\tilde {u}_{1} , \tilde {u}_{2}) \in BV(\widetilde {\varOmega }_{1}) \times BV(\widetilde {\varOmega }_{2})$ such that $0 \leq \tilde {u}_{1} \leq 1$, $0 \leq \tilde {u}_{2} \leq 1$, and $\tilde {u}_{1} = \tilde {u}_{2}$ on $\widetilde {\Gamma }$. Let:

$$ u(x) = \begin{cases} \tilde{u}_{1} (x) & \textrm{ if } x \in \widetilde{\varOmega}_{1} , \\ \tilde{u}_{2} (x) & \textrm{ if } x \in \varOmega \setminus \widetilde{\varOmega}_{1} . \end{cases} $$

Then, we have:

$$ \begin{array}{ll} E_{1} (\tilde{u}_{1}) + E_{2} (\tilde{u}_{2}) &= \alpha {\int}_{\varOmega} ug dx + TV_{\varOmega} (u) \\ &\geq \alpha {\int}_{\varOmega} u^{*} g dx + TV_{\varOmega} (u^{*}) = E_{1} \left( \tilde{u}_{1}^{*}\right) + E_{2} \left( \tilde{u}_{2}^{*}\right). \end{array} $$

Therefore, $\left (\tilde {u}_{1}^{*} , \tilde {u}_{2}^{*}\right )$ is a solution of (B.3). □

In conclusion, it is more appropriate to classify the proposed DDM in [11] as an overlapping one instead of a nonoverlapping one.