Numerical analysis of constrained total variation flows and its
application to the Kobayashi–Warren–Carter model∗
Koya Sakakibara†
Department of Applied Mathematics, Faculty of Science, Okayama University of Science;
RIKEN iTHEMS

1Introduction

A general constrained total variation (TV) flow for u : Ω × [0, T ) → M is given by

∂u∇u


= −πu −∇ ·in Ω × (0, T ),

|∇u|
 ∂t
∇u· νΩ = 0on ∂Ω × (0, T ),|∇u|

u|t=0 = u0in Ω,(1)where Ω ⊂ Rk (k ≥ 1) is a bounded region with Lipschitz boundary ∂Ω, M a manifold embedded into Rl (l ≥ 1),
u0 : Ω → M an initial datum, πp the orthogonal projection from the tangent space Tp Rl (= Rl ) to the tangent space
Tp M (⊂ Rl ) at p ∈ M , ν Ω the unit outward normal vector of ∂Ω, and T > 0. If πu is absent, (1) is the standard
vectorial total variation flow of the isotropic total variation of vector-valued maps:
ZTV(u) :=|∇u|Rk×l dxΩThe introduction of πu means that we restrict the gradient of total variation so that u always takes value in M .
Constrained TV flow (1) appears in diverse fields. The first application of this flow appears in [4], where theauthors considered the two-dimensional sphere S 2 as the target manifold to denoise color images while preserving
brightness. This system, where the target manifold is the space SO(3) of all three-dimensional rotations, appears
in the Kobayashi–Warren–Carter model [3, 2], a basic prototype of the continuum model of the time evolution ofgrain boundaries in a crystal.
Despite its broad applicability, there is still much room for research in the mathematical and numerical analysis
of constrained TV flows. This talk aims to develop a new numerical scheme for spatially discrete constrained total
variation flows based on the exponential map and the minimizing movement scheme.

2

Spatially discrete model and numerical scheme

2.1

Finite-dimensional Hilbert spaces and discrete constrained TV flows

Let ∆ be a finite set of indices and let Ω be a bounded region in Rk . A family Ω∆ = {Ωα }α∈∆ of subsets of Ω is arectangular partition of Ω if Ω∆ satisfies the following three conditions.
S(i) L k Ω \ α∈∆ Ωα = 0.
(ii) L k (Ωα ∩ Ωβ ) = 0 for α 6= β, (α, β) ∈ ∆ × ∆; here L k denotes the Lebesgue measure.
∗ This talk is based on joint work with Yoshikazu Giga (The University of Tokyo), Kazutoshi Taguchi (ARISE Inc.), Jun Okamoto(The University of Tokyo), and Masaaki Uesaka (Arithmer Inc./The University of Tokyo).
† ksakaki@xmath.ous.ac.jp

(iii) For each α ∈ ∆, there exists a rectangular region Rα ∈ Rk such that Ωα = Rα ∩ Ω holds. Here, by arectangular region, we mean for some aj < bj thatRα = {x = (x1 , . . . , xk ) | aj < xj < bj , j = 1, 2, . . . , k}.
We denote by e(∆) the set of edges associated with ∆ defined ase(∆) := {{α, β} ⊂ ∆ | Hk−1 (∂Ωα ∩ ∂Ωβ ) 6= 0, α 6= β},where Hk−1 denotes the (k − 1)-dimensional Hausdorff measure.
Subsequently, we fix the rectangular partition Ω∆ of Ω. Then, define the space of piecewise Rl -valued constantfunctions as()XH∆ := u =uα 1Ωα uα ∈ Rl , α ∈ ∆ .
α∈∆

We can regard that H∆ is a closed subspace of the Hilbert space L2 (Ω; Rl ) with an inner producthX, Y iH∆ := hX, Y iL2 (Ω;Rl ) .
The discrete total variation functional TV∆ : H∆ → R associated with Ω∆ is defined as follows:
XTV∆ (u) :=kuα − uβ kRl Hk−1 (∂Ωα ∩ ∂Ωβ ).
{α,β}∈e(∆)

Moreover, we define the space of piecewise M -valued constant functions as()XααM∆ := u =u 1Ωα u ∈ M, α ∈ ∆ .
α∈∆We finally introduce the orthogonal projection Pu : H∆ → M∆ at u ∈ M∆ byPu X(x) := πu(x) (X(x)) for a.e.

x ∈ Ω.

Definition 2.1. Let u0 ∈ M∆ and I := [0, T ). A map u ∈ W 1,2 (I; M∆ ) is said to be a solution to the discretemodel of (1) if u satisfies
 du ∈ −P ∂TV (u(t)) for a.e. t ∈ (0, T ),
∆u(t)
dt(2)
u|t=0 = u0 .

2.2

Numerical scheme(n−1)
Let τ be the time step and define tn := nτ . According to the minimizing movement scheme, from solution uτ
(n)at the current time, we want to define the solution uτ at the next time as the minimizer to the following problem:

1minimize Φτ (u; u(n−1)) := τ TV∆ (u) + ku − u(n−1)
k2H∆ subject to u ∈ M∆ .
ττ2
However, it is not easy to solve this problem since it is a non-smooth Riemannian constraint optimization problem.
To overcome this difficulty, we localize the energy by employing the exponential map. Thanks to the exponen(n−1)
tial map in M , each u ∈ M∆ can be written as a uτand X ∈ Tu(n−1) M∆ pair: u = Expu(n−1) (X), whereττExpu(n−1) : Tu(n−1) M∆ → M∆ is defined asττX(x) 7−→ expu(n−1) (x) (X(x))τfor a.e. x ∈ Ω.

Here, Tu M∆ denotes the tangent space of M∆ at u ∈ M∆ , and expx is the exponential map of the Riemannian(n−1)(n−1)
manifold M . Since Expu(n−1) (X) = uτ
+ X + o(X), we ignore the term o(X) and insert u = uτ
+ X intoτ(n−1)Φτ (u, uτ) to obtain the localized energy1Φτloc (X; u(n−1)) := τ TV∆ (u(n−1)+ X) + kXk2H∆ ,ττ
2As a result, we obtain the following modified minimizing movement scheme.

X ∈ H∆ .

(i) From solution uτ(n−1)

at the current time, we compute the unique minimizer Xτ(n−1)

minimize Φτloc (X; u(n−1)) subject toτ(ii) Set uτ(n)of the following problem:

X ∈ Tu(n−1) M∆ .
τ:= Expu(n−1) (Xτ(n−1) ).
τConcerning the above constructed numerical scheme, we obtain the following results.
Theorem 2.2 (Energy dissipation). Let M be a C 2 -compact manifold embedded into Rl , I := [0, T ), u0 ∈ M∆ be(n)an initial datum, τ > 0 be a step size, and {uτ }n be a sequence generated by the modified minimizing movement
(n+1)
(n)
scheme. If τ is sufficiently small, then TV∆ (uτ) ≤ TV∆ (uτ ) holds.
Namely, the proposed scheme inherits the property that the total variation dissipates along with the flow.
Theorem 2.3 (Error estimate). Let M be a path-connected and C 2 -compact manifold embedded into Rl , I := [0, T ),u0 ∈ M∆ be an initial datum, and τ > 0 be a step size. Denote by u the solution of the discrete model (2) and by(n){uτ }n the sequence obtained by the modified minimizing movement scheme. Then, there exist constants C0 , C1 ,and C2 independent of u and τ such that the following error estimate holds:
2n C0 tnkuτ (tn ) − u(n)C1 τ + C2 τ 2 , n = 0, 1, . . . .
τ kH∆ ≤ t e√
This theorem implies that the accuracy of the proposed scheme is the order of τ .
In the talk, we will present the details of the derivation of the above numerical scheme and the idea of the proof oftwo theorems. We will also show numerical results for constrained total variation flows, where the target manifolds
are S 2 and SO(3). Finally, we introduce the spatially discrete Kobayashi–Warren–Carter energy and numericallystudy the solution’s asymptotic behavior to the gradient flow of its energy compared with the Ambrosio–Tortorellienergy.
The contents of the constrained total variation flow are summarized in our paper [1].

References[1] Y. Giga, K. Sakakibara, K. Taguchi, and M. Uesaka. A new numerical scheme for discrete constrained total
variation flows and its convergence. Numer. Math., 146(1):181–217, 2020.
[2] R. Kobayashi and J.A. Warren. Modeling the formation and dynamics of polycrystals in 3D. Physica A,
356:127–132, 2005.
[3] R. Kobayashi, J.A. Warren, and W.C. Carter. A continuum model of grain boundaries. Physica D, 140:141–150,2000.
[4] B. Tang, G. Sapiro, and V. Caselles. Color image enhancement via chromaticity diffusion. IEEE Trans. Image
Process., 10, 2001.