A duality based approach to the minimizing total variation flow in the space \(H^{s}\)
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Abstract
We consider a gradient flow of the total variation in a negative Sobolev space \(H^{s}\,(0\le s \le 1)\) under the periodic boundary condition. If \(s=0\), the flow is nothing but the classical total variation flow. If \(s=1\), this is the fourth order total variation flow. We consider a convex variational problem which gives an implicittime discrete scheme for the flow. By a duality based method, we give a simple numerical scheme to solve this minimizing problem numerically and discuss convergence of a forward–backward splitting scheme. Results of several numerical experiments are given.
Keywords
Total variation Rudin–Osher–Fatemi model Fractional Sobolev spaces Negative Sobolev spacesMathematics Subject Classification
Primary 94A08 Secondary 65M06 49N90 35K251 Introduction
This paper is organized as follows. We recall \(H^{s}\) space and the total variation in Sect. 2. We give a characterization of the subdifferential of the total variation in \(H^{s}\) in Sect. 3. In Sect. 4, we give a way of semidiscritization, which is a recursive minimization problem of convex but nondifferentiable functional. In Sect. 5, we formulate its dual problem. In Sect. 6, we discuss convergence of a forward–backward splitting scheme. In Sect. 7, we derive its explicit form. In Sect. 8, we discuss its ergodic convergence. In Sect. 9, we explain how to discretize the introduced scheme and provide an exact condition for a convergent solution in a finite dimensional space. In Sect. 10, we present results of numerical experiments to illustrate evolution of solutions to considered total variation flows, showing their characteristic features with respect to different values of index \(s\in [0,1]\).
2 Preliminaries
To give a rigours interpretation of the Eq. (1.5) with the periodic boundaryconditions, we first need to introduce preliminary definitions and notations.
3 A characterization of the subdifferential
Lemma 1
Let \(\Phi \), \(\Psi : H \rightarrow [0,\infty ]\). If \(\Phi \le \Psi \) on H, then \({\tilde{\Psi }}\le {\tilde{\Phi }}\) on \(H^*\).
Proof
See proof of [3, Lemma 1.5]. \(\square \)
Lemma 2
Suppose \(\Phi \) is convex, lower semicontinuous and positively homogeneous of degree one, then \(\tilde{{\tilde{\Phi }}}(u) =\Phi (u)\).
Proof
See proof of [3, Proposition 1.6]. \(\square \)
Definition 1
Lemma 3
Suppose \(\Phi \) is convex, lower semicontinuous, nonnegative, and positively homogeneous of degree one. Then, \(v\in \partial _{H} \Phi (u)\) if and only if \({\tilde{\Phi }}(v)\le 1\) and \(( u,v)_{H} =\Phi (u)\).
Proof
See proof of [3, Theorem 1.8]. \(\square \)
Lemma 4
Proof
Theorem 1
Proof
Theorem 2
 \(\mathrm {(1)}\)

for all \(t>0\) we have that \(u(t)\in D(\mathcal {A})\),
 \(\mathrm {(2)}\)

\(\frac{du}{dt}\in {L^\infty (0,\infty ; H_{\mathrm {av}}^{s}(\mathbb {T}^d))}\) and \(\left\ \frac{du}{dt} \right\ _{H_{\mathrm {av}}^{s}(\mathbb {T}^d)} \le \left\ \mathcal {A}^0 (u_0) \right\ _{H_{\mathrm {av}}^{s}(\mathbb {T}^d)}\),
 \(\mathrm {(3)}\)

\(\frac{du}{dt}\in \mathcal {A}(u)\) a.e. on \((0,\infty )\),
 \(\mathrm {(4)}\)

\(u(0)=u_0\),
Proof
The proof of this theorem can be found in Brezis [7]. \(\square \)
4 A semidiscretization
Theorem 3
The original paper by Crandall–Liggett contains also an error estimate, which is not optimal. The optimal one is of order \(O(\tau ^2)\) and it has been derived by Rulla [35, Theorem 4].
5 A dual problem
Lemma 5
 (i)The setis closed under the strong \(H_{\mathrm {av}}^{s}(\mathbb {T}^d)\)topology.$$\begin{aligned} K := \{v\in \mathcal {D}'(\mathbb {T}^d) \ : \ v=(\Delta _{\mathrm {av}})^s\mathrm {div\,}z, \ \mathrm {div\,}z \in H_{\mathrm {av}}^s(\mathbb {T}^d),\ \left\ z \right\ _{\infty }\le 1\} \end{aligned}$$(5.2)
 (ii)
If we consider \(\Phi \) given by (3.2) as a functional defined on \(H_{\mathrm {av}}^{s}(\mathbb {T}^d)\), then the convex conjugate of \(\Phi \) is given by \(\Phi ^*(v)=\chi _{K}(v)\).
Proof
 (i)
Let \(\{v_j\}\in K\) be a sequence converging to v in \(H_{\mathrm {av}}^{s}\). This means that \(\mathrm {div\,}z_j\) converges to some w in \(H_{\mathrm {av}}^s\), where \(v_j = (\Delta _{\mathrm {av}})^s\mathrm {div\,}z_j\). Since \(z_j\) is bounded in \(L^\infty \), it converges to some z in the sense of weak\(^*\) topology of \(L^\infty \) by taking a subsequence. This implies \(w = \mathrm {div\,}z\) so we get the desired result.
 (ii)Let \(u\in BV(\mathbb {T}^d)\cap H_{\mathrm {av}}^{s}(\mathbb {T}^d)\), then the convex conjugate of function \(\chi _{K}\) is given bySince the set K is convex, the function \(\chi _{K}\) is convex. Moreover, since \(\chi _{K}\) is lower semicontinuous, then it follows from [13, Proposition I.4.1] that \(\Phi ^*(v)=\chi _{K}^{**}(v)=\chi _{K}(v)\).$$\begin{aligned} \begin{aligned} \chi _{K}^*(u)&=\sup _{v}\{(v,u)_{H_{\mathrm {av}}^{s}} \ : \ v\in K\}\\&=\sup _{z}\{\langle u,\mathrm {div\,}z \rangle \ : \ z\in \mathcal {D}(\mathbb {T}^d),\ \left\ z \right\ _{\infty }\le 1\}=\Phi (u). \end{aligned} \end{aligned}$$(5.3)
Proposition 1
Proof
Corollary 1
The solution of problem (5.4) satisfies \(u = f\tau P_K^{H_{\mathrm {av}}^{s}}(f/\tau )\), where \(P_K^{H_{\mathrm {av}}^{s}}\) denotes the orthogonal projection on the set K with respect to the inner product in \(H_{\mathrm {av}}^{s}\).
Remark 1
6 Convergence of a forward–backward splitting scheme
In this section, using the forwardbackward splitting approach, we construct theminimizing dual problem (5.5) sequence \(\{v_k\}\) and prove its convergence in \(H_{\mathrm {av}}^{s}\).
Remark 2
Remark 3
In the proof of convergence of the sequence \(\{v^k\}\) we will need the lemma below:
Lemma 6
For \(v\in K\) we have that if \(u\in \partial _{H_{\mathrm {av}}^{s}} \Phi ^*(v)\), then \((u,vw)_{H_{\mathrm {av}}^{s}}\ge 0\) for all \(w\in K\).
Proof
By the definition of the subdifferential \(\partial _{H_{\mathrm {av}}^{s}} \Phi ^*(v)\), we have that \(u\in \partial _{H_{\mathrm {av}}^{s}} \Phi ^*(v)\) if and only if \(\Phi ^*(w) \ge \Phi ^*(v) + (u,wv)_{H_{\mathrm {av}}^{s}}\) for all \(w\in H_{\mathrm {av}}^{s}(\mathbb {T}^d)\). Since \(\Phi ^*\) is the indicator function of the set K, we have \((u,vw)_{H_{\mathrm {av}}^{s}}\ge 0\) for all \(w\in K\). \(\square \)
Proposition 2
Assume that \(0<\lambda \tau <2\), then \(v^k\rightharpoonup v^*\) and \(u^k\rightharpoonup u^*\) in \(H_{\mathrm {av}}^{s}\), where \(v^*\in K\) is such that \(v^*\in \partial _{H_{\mathrm {av}}^{s}} \Phi (u^*)\) and \(u^*= f\tau v^*\).
Proof
7 An explicit form of a scheme
Since the nonlinear projection, given in Corollary 1, is difficult to compute in practice, we use the characterization of subdifferential \(\partial _{H_{\mathrm {av}}^{s}} \Phi \), provided in Theorem 1, toconstruct an explicit scheme for minimizing sequence of the dual problem (5.10). More precisely, we apply the same forward–backward scheme as in the previous section to generate the sequence \(\{z^k\}\) minimizing (5.10). This sequence is related to \(\{v^k\}\) by formula \(v^k = \tau (\Delta _{\mathrm {av}})^s\mathrm {div\,}z^k\).
For practical implementation of this scheme, it remains to find its closedform expression.
Lemma 7
Assume that \(z\in L^2(\mathbb {T}^d,\mathbb {R}^d)\) is such that \(F(z)<\infty \). Then, we have \(D(\partial _{L^2} F) = \{z \in L^2(\mathbb {T}^d,\mathbb {R}^d)\, :\, F(z)<\infty ,\ \nabla (\tau (\Delta _{\mathrm {av}})^s\mathrm {div\,}z + f)\in L^2(\mathbb {T}^d,\mathbb {R}^d)\}\) and \(\partial _{L^2} F(z) = \{\nabla w\}\), where \(w = \tau (\Delta _{\mathrm {av}})^s\mathrm {div\,}z + f\).
Proof
Lemma 8
Proof
Remark 4
8 Ergodic convergence
Proposition 3
Let \(\{v^k\}\) be a weakly convergent sequence in K generated by the scheme (6.1) and let \(\{\beta _k\}\) be a sequence of positive real numbers such that \(\{\beta _k\}\in l^2{\setminus } l^1\). Then, for \(\alpha _k = \beta _k/\sum _{j=1}^n \beta _j\) the sequence \(\{{\bar{v}}^n\}\) given by (8.1) converges strongly in \(H_{\mathrm {av}}^{s}(\mathbb {T}^d)\) to \(v^*\in K\) as \(n\rightarrow \infty \). Moreover, the sequence \(\{{\bar{z}}^n\}\) given by (8.2), where \(\{z^k\}\) is generated by the scheme (7.7), converges weakly in \(QL^2(\mathbb {T}^d)\) to \(z^*\in Z\) as \(n\rightarrow \infty \).
Proof
9 Implementation
In this section, we explain how to discretize the system (7.7) and to solvenumerically one iteration of the semiimplicit scheme (4.1). Recursive application of this procedure leads to a numerical solution of the nonlinear evolution equation (3.8). For simplicity of presentation, we construct a method for the onedimensional problem, but it can be easily extended to higher dimensions. We also present results concerning the convergence of the introduced scheme in a finite dimensional space.
Lemma 9
Proof
Proposition 4
Assume that \(C \lambda \tau <2\), where the constant \(C>0\) is as in Lemma 9, then \(z^k\rightarrow z^*\) in X, where \(z^*\in Z\) is such that \(0\in \partial _X F(z^*) + \partial _X G(z^*)\).
Proof
10 Numerical results
In this section, we present results of numerical experiments, obtained by application of the introduced earlier schemes, to solve the evolution equation (3.8). Theseexperiments have been performed on onedimensional data for more accuratepresentation of differences in solutions with respect to different values of the index s and their characteristic features.
In all calculations, we have fixed values of parameters \(h = 0.1\) and \(\tau = 0.1\). In each case, a value of \(\lambda \) was selected so that the criterion for the convergence given in Proposition 4 would be satisfied, i.e., we set \(\lambda = 4\cdot 10^{2}\), \(\lambda = 2\cdot 10^{3}\) and \(\lambda = 10^{4}\) for \(s = 0\), \(s=0.5\) and \(s=1\), respectively. In Fig. 2, we present evolution of solutions to the \(H^{s}\) total variation flow for initial data f and g, and for different values of the index s (\(s=0, 0.5, 1\)).
From our computation, we conjecture that a solution may be instantaneously discontinuous for \(s \in (0,1]\) for Lipschitz initial data. This is rigorously proved for \(s=1\) in [16]. Also, we see from this computation, that the motion becomes slower as s becomes larger.
Notes
Acknowledgements
This work was partially supported by the EU IRSES program “FLUX” and the Polish Ministry of the Science and Higher Education Grant number 2853/7.PR/2013/2. The work of the Yoshikazu Giga was partially supported by Japan Society for the Promotion of Science through Grant Nos. 26220702 (Kiban S) and 16H03948 (Kiban B). A part of the research for this paper was performed, when Monika Muszkieta and Piotr Rybka visited the University of Tokyo. Its hospitality is gratefully acknowledged. The authors also thank the anonymous reviewer for the careful reading of our manuscript and for insightful comments.
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