A Mixed Finite Element Discretization of Dynamical Optimal Transport

Abstract

In this paper we introduce a new class of finite element discretizations of the quadratic optimal transport problem based on its dynamical formulation. These generalize to the finite element setting the finite difference scheme proposed by Papadakis et al. (SIAM J Imaging Sci, 7(1):212–238, 2014). We solve the discrete problem using a proximal splitting approach and we show how to modify this in the presence of regularization terms which are relevant for physical data interpolation.

Appendix A: Proof of Theorem 5.4

Applying Theorem 2.16 in [22], in order to prove Theorem 5.4 it is sufficient to check that the conditions listed in definition 2.9 of [22] are verified. Such conditions translated to our finite element settings are listed in Proposition A.2 below.

From now on, we assume $r=1$, and $X_h$ stands for $X_h^1$. We also denote by ${\mathcal {I}}$ is the standard nodal interpolant onto $X_h$, defined element by element.

First of all, we introduce some notation and list some technical results [13]. Denote by $P_{X_h}$ and $P_{V_h}$ the $L^2$ projections onto $X_h$ and $V_h$, respectively. Then,

$$\begin{aligned} \Vert P_{X_h} \varphi \Vert _{L^p} \le C \Vert \varphi \Vert _{L^p} \,, \quad \forall \varphi \in L^p \,,\, 1\le p\le \infty \,, \end{aligned}$$

and moreover $\forall \, T\in {\mathcal {T}}_h$

$$\begin{aligned} \Vert \varphi - P_{X_h} \varphi \Vert _{L^p(T)} \le C h_T \Vert \nabla \varphi \Vert _{L^p(T)} \,,\quad \forall \varphi \in W^{1,p}(T) \,,\, 1\le p\le \infty \,, \end{aligned}$$

where, with an abuse of notation, we have used $P_{X_h}$ to denote the $L^2$ projection onto $X_h(T)$. These imply the following lemma.

Lemma A.1

Given the regularity assumption in (4.1) on ${\mathcal {T}}_h$, we have

$$\begin{aligned} \left\| {\mathcal {I}} |P_{V_h} b|^2 \right\| _{L^\infty } \le C \Vert b\Vert ^2_{L^\infty }\,, \end{aligned}$$

for any $b\in L^\infty (D)$, and

$$\begin{aligned} \left\| {\mathcal {I}} |P_{V_h} b|^2 -|b|^2 \right\| _{L^\infty } \le C h | b|_{W^{1,\infty }}\Vert b\Vert _{L^{\infty }}\,, \end{aligned}$$

for any $b\in W^{1,\infty }(D)$.

Proof

For the first inequality, using standard inverse inequalities, we have

$$\begin{aligned} \left\| {\mathcal {I}} |P_{V_h} b| ^2 \right\| _{L^\infty }&\le \left\| |P_{V_h} b| ^2 \right\| _{L^\infty }\\&\le C h^{-d} \left\| |P_{V_h} b| ^2 \right\| _{L^1}\\&= C h^{-d} \left\| P_{V_h} b \right\| ^2_{L^2} \\&\le C h^{-d} \left\| P_{(X_h)^d} b \right\| ^2_{L^2} \\&\le C \left\| P_{(X_h)^d} b \right\| ^2_{L^\infty } \\&\le C \Vert b \Vert ^2_{L^\infty }\,. \end{aligned}$$

For the second inequality , we observe that

$$\begin{aligned} \left\| {\mathcal {I}} |P_{V_h} b|^2 -|b|^2 \right\| _{L^\infty } \le \left\| {\mathcal {I}} |P_{V_h} b|^2 - {\mathcal {I}} |b|^2 \right\| _{L^\infty } +\left\| {\mathcal {I}} |b|^2 -|b|^2 \right\| _{L^\infty } \,. \end{aligned}$$

The second term of the right-hand side is easy to control. For the first term, we have

$$\begin{aligned} \left\| {\mathcal {I}} |P_{V_h} b|^2 - {\mathcal {I}} |b|^2 \right\| _{L^\infty }&\le \left\| |P_{V_h} b|^2 - |b|^2 \right\| _{L^\infty } \\&\le \left\| |P_{V_h} b|^2 - |P_{(X_h)^d} b|^2 \right\| _{L^\infty } +\left\| |b|^2 - |P_{(X_h)^d} b|^2 \right\| _{L^\infty } \,. \end{aligned}$$

Again, the second term is easy to control. For the first tem, using the same reasoning as above,

$$\begin{aligned} \left\| |P_{V_h} b|^2 - |P_{(X_h)^d} b|^2 \right\| _{L^\infty }&\le C h^{-d} \left\| |P_{V_h} b|^2 - |P_{(X_h)^d} b|^2 \right\| _{L^1}\\&\le C h^{-d} \sum _{i=1}^d \left\| (P_{V_h} b)_i^2 -(P_{X_h} b_i)^2 \right\| _{L^1}\\&\le C h^{-d} \sum _{i=1}^d \left\| (P_{V_h} b)_i -P_{X_h} b_i \right\| _{L^1} \Vert b\Vert _{L^\infty }\\&\le C h^{-\frac{d}{2}} \left\| P_{V_h} b - P_{X_h} b \right\| _{L^2} \Vert b\Vert _{L^\infty } \\&\le C h \left\| \nabla P_{X_h} b \right\| _{L^\infty } \Vert b\Vert _{L^\infty } \le C h \left\| \nabla b \right\| _{L^\infty } \Vert b\Vert _{L^\infty } \,. \end{aligned}$$

$\square$

As mentioned in Sect. 4.1, there exist projection operators $\Pi _{Q_h}:L^2(D)\rightarrow V_h$ and $\Pi _{V_h}:{\mathcal {V}}_D \rightarrow Q_h$ commuting with the divergence operator, where ${\mathcal {V}}_D$ is a dense subset of $H(\mathrm {div};D)$. We pick these to be the canonical projections introduced in Section 5.2 of [2], and in particular $\Pi _{Q_h}$ as in Eq. (4.2).

Such operators verify the following approximation properties (see Theorem 5.3 in [2]): for any $\varphi \in H^1(D)$ and $\eta \in H^1(D)^d$

$$\begin{aligned} \Vert \Pi _{Q_h} \varphi - \varphi \Vert _{L^2(D)} \le C h \Vert \varphi \Vert _{H^1(D)} \,, \quad \Vert \Pi _{V_h} \eta -\eta \Vert _{L^2(D)^d} \le C h \Vert \eta \Vert _{H^1(D)^d}\,. \end{aligned}$$

(A.1)

Notice in particular that given the mesh regularity assumption (4.1), Eq. (A.1) is a standard property for $\Pi _{Q_h}$ as defined in Eq. (4.2).

Proposition A.2 below contains the properties needed for convergence: it can be seen as a specific instance of Definition 2.9 of [22]. Note that a few of the properties listed therein are omitted here because they are either unnecessary or true by construction in our setting. Note also that the sampling operators used in [22] are replaced here with the canonical projections $\Pi _{Q_h}$ and $\Pi _{V_h}$, where $\Pi _{Q_h}$ can be naturally extended to ${\mathcal {M}}(D)$ (see Eq. (4.2)) and $\Pi _{V_h}$ is considered to be defined on a dense subset of ${\mathcal {M}}(D)^{d}$. Moreover the reconstruction operators are simply the injection operators from $Q_h$ and $V_h$ to ${\mathcal {M}}(D)$ and ${\mathcal {M}}(D)^d$, respectively. Finally, we define for any $(\rho ,b)\in {\mathcal {M}}(D) \times C(D;{\mathbb {R}}^d)$

$$\begin{aligned} A^*(\rho ,b) :=\int _D \frac{|b|^2}{2} \rho \,, \end{aligned}$$

so that if $(\rho ,m) \in {\mathcal {M}}_+(D) \times {\mathcal {M}}(D)^d$ then

$$\begin{aligned} A(\rho ,m) = \sup _{b \in C(D;{\mathbb {R}}^d)} \langle m,b\rangle -A^*(\rho ,b); \end{aligned}$$

and for any $(\rho ,b) \in Q_h \times V_h$,

$$\begin{aligned} A^*_h(\rho ,b) :=\sup _{m\in V_h} \langle m,b \rangle -A_h(\rho ,m)\,. \end{aligned}$$

Proposition A.2

The following properties hold:

• (1) For any $\rho \in {\mathcal {M}}_+(D)$, $\Pi _{Q_h} \rho \rightarrow \rho$ as $h\rightarrow 0$ weakly in ${\mathcal {M}}(D)$.

• (2) Let $B\subset (C^1(D))^d$ a bounded subset. Then there exists a constant $\varepsilon _h$ tending to 0 as $h\rightarrow 0$ such that for any $b\in B$ and $\rho \in Q_h$

  $$\begin{aligned} A^*_h(\rho ,P_{V_h} b) \le A^*(\rho ,b) +\epsilon _h \Vert \rho \Vert \,, \end{aligned}$$

  where $P_{V_h}$ denotes the $L^2$ projection onto $V_h$. Moreover there exists a constant $C\ge 1$ such that for any $b\in C(D)^d$, there holds

  $$\begin{aligned} A^*_h(\rho ,P_{V_h} b) \le \frac{C}{2} \Vert \rho \Vert \Vert b \Vert ^2_{L^\infty }\,. \end{aligned}$$

• (3) Let $B\subset C^0(D)\cap H^1(D)$ a bounded subset such that for all $\rho \in B$ there holds $\rho>C>0$, and let $B' \subset (C^0(D) \cap H^1(D))^d$ a bounded subset. There exists a constant $\varepsilon _h$ tending to 0 as $h\rightarrow 0$ such that, given $(\rho ,m)\in {\mathcal {M}}(D)^{d+1}$ such that $\rho$ has density in B and m in $B'$, then

  $$\begin{aligned} A_{h}(\Pi _{Q_h} \rho , \Pi _{V_h} m) \le A(\rho ,m) + \varepsilon _h. \end{aligned}$$

• (4) There exists $\varepsilon _h$ tending to 0 as $h\rightarrow 0$ and a continuous function $\omega$ satisfying $\omega (0)=0$ such that: for any $x,y\in D$ and $h>0$ there exists $\rho \in Q_h^+$ and $m_1,m_2 \in V_h$ satisfying

  $$\begin{aligned} \left\{ \begin{array}{l} \mathrm{div}\, m_1 = \rho - \Pi _{Q_h} (\delta _x) \\ \mathrm{div}\, m_2 = \rho - \Pi _{Q_h} (\delta _y) \end{array} \right. \quad \text {and} \quad A_h(\rho ,m_i) \le \omega (|x-y|) +\varepsilon _h\,,\forall i\in \{1,2\}. \end{aligned}$$

  (A.2)

Remark A.3

In [22] point (3) of Proposition A.2 is stated with B and $B'$ bounded subsets of $C^1(D)$ and $C^1(D)^d$, respectively. The condition we require here is stronger, but it is needed since we considered a convex polytope domain rather than a domain with a smooth boundary as in [22]. As a matter of fact, in [22] one applies the condition (3) on a regularized measure $(\tilde{\rho },\tilde{m})\in {\mathcal {M}}(D)^{d+1}$ obtained by convolution with the heat kernel and by solving an appropriate elliptic problem (see proposition 3.2 in [22]). For a convex polytope domain this procedure yields a couple $(\tilde{\rho },\tilde{m})$ with densities which are not $C^\infty$ given the singularities of the boundary. By classical elliptic regularity estimates on non-smooth domains (e.g., [31] and [25]), the regularity we require in condition (3) is however sufficient for the proof in [22] to apply without changes.

Proof

The first point is immediate from the definition of $\Pi _{Q_h}$ in Eq. (4.2). For (2), we observe that

$$\begin{aligned} A_h(\rho ,m) = \sup _{b\in X_h} \langle m,b\rangle -\frac{1}{2}\langle \rho , {\mathcal {I}} |b|^2 \rangle \,, \end{aligned}$$

where we recall that ${\mathcal {I}}$ is the standard element-wise nodal interpolant onto $X_h$. In fact, for any $b \in (X_h)^d$, we have $b^2 \le {\mathcal {I}}|b|^2$, and therefore when $\rho \ge 0$ we can “saturate" the constraint setting $a = -{\mathcal {I}}|b|^2/2$. On the other hand if $\rho <0$ on some element both sides of the equality are $+\infty$. For $(\rho ,b,m) \in Q_h \times V_h\times V_h$ define

$$\begin{aligned} {A}^{*}_{{\mathcal {I}},h} (\rho ,b) :=\frac{1}{2} \langle \rho , {\mathcal {I}} |b|^2 \rangle , \quad \bar{A}_{{\mathcal {I}},h}(\rho ,m) :=\sup _{b\in V_h} \langle m,b\rangle -{A}^{*}_{{\mathcal {I}},h} (\rho ,b). \end{aligned}$$

Then, since when $\rho <0$ on some element $A^*_h(\rho ,b)= -\infty$,

$$\begin{aligned} A_h(\rho ,m) \ge \bar{A}_{{\mathcal {I}},h}(\rho ,m) , \quad A^*_h(\rho ,b) \le {\bar{A}}^{*}_{{\mathcal {I}},h} (\rho ,b) \le {A}^{*}_{{\mathcal {I}},h}(\rho ,b), \end{aligned}$$

and we can prove (2) for $A^*_{{\mathcal {I}},h}$. In particular, we have

$$\begin{aligned} A^*_{{\mathcal {I}},h}(\rho ,P_{V_h}b) \le A^*(\rho ,b) +\frac{1}{2} \left\| {\mathcal {I}} |P_{V_h} b|^2 -|b|^2 \right\| _{L^\infty } \Vert \rho \Vert , \end{aligned}$$

and we obtain the result applying Lemma A.1. Using again Lemma A.1, we easily obtain the second bound as well.

For point (3), observe first that $A_h(\Pi _{Q_h}\rho , \Pi _{V_h} m)\le A(\Pi _{Q_h}\rho , \Pi _{V_h} m)$ by definition. Then given the assumption on $\rho$ and m we can simply write

$$\begin{aligned} A_h(\Pi _{Q_h}\rho , \Pi _{V_h} m) - A(\rho ,m)&\le \int _D \frac{|\Pi _{V_h} m|^2}{2 \Pi _{Q_h}\rho } -\frac{|m|^2}{2\rho } \\&\le \frac{1}{2} \int _D \left| \frac{|\Pi _{V_h} m|^2 -|m|^2}{\Pi _{Q_h}\rho }\right| + \left| \frac{| m|^2}{ \Pi _{Q_h}\rho } -\frac{|m|^2}{\rho }\right| \\&\le C \left( \Vert \Pi _{Q_h} \rho - \rho \Vert _{L^2} +\left\| |\Pi _{V_h} m|^2-|m|^2 \right\| _{L^1}\right) \,, \end{aligned}$$

where the constant C depends on the uniform lower bound on $\rho$ and on the $L^\infty$ norm of |m|. We conclude using Cauchy–Schwarz inequality on the second term and then Eq. (A.1).

For the last point, we will establish a connection between our scheme and the one proposed by Gladbach, Kopfer and Maas [16] and then use propoperty (A.2) for this scheme which was proved in [22]. We will consider only the case of a simplicial mesh and $V_h = {\mathcal {R}}{\mathcal {T}}_0$ (which covers also the case of $V_h = {\mathcal {B}}{\mathcal {D}} {\mathcal {M}}_1$, since ${\mathcal {R}}{\mathcal {T}}_0\subset {\mathcal {B}}{\mathcal {D}}{\mathcal {M}}_1$). The quadrilateral case with $V_h = {\mathcal {R}}{\mathcal {T}}_{[0]}$ can be dealt with in a completely analogous way.

First, we introduce some notation. For each $T\in {\mathcal {T}}_h$, let ${\mathcal {T}}_{h,T}$ be the set of neighbouring elements $L\in {\mathcal {T}}_h$ such that $f_{T,L}:=\overline{T}\cap \overline{L} \ne \emptyset$, which we assume to be oriented. Define by ${\mathcal {F}}_h$ the set of $(d-1)$-dimensional facets in the triangulation. Let $T,L \in {\mathcal {T}}_h$ be neighbouring elements, we denote by $\varphi _{T,L}\in {\mathcal {R}} {\mathcal {T}}_0$ the canonical basis function associated with the oriented facet $f_{T,L}$. Then, any $m\in {\mathcal {R}} {\mathcal {T}}_0$ can be written as

$$\begin{aligned} m = \sum _{f_{T,L} \in {\mathcal {F}}_h} m_{T,L} \varphi _{T,L}\,, \end{aligned}$$

where $m_{T,L}$ is the flux of m on the oriented facet $f_{T,L}$. In other words we can identify functions in $(\rho ,m) \in Q_h \times {\mathcal {R}}{\mathcal {T}}_0$ with their finite volume representation $\{\rho _T, m_{T,L}\}_{T,L}$. Then, we can interpret the action for the finite volume scheme [16], which we denote by ${A}_h^{FV}(\rho ,m)$, as a function on $Q_h\times {\mathcal {R}}{\mathcal {T}}_0$. This is given by the following expression

$$\begin{aligned} {A}_h^{FV}(\rho ,m) :=\sum _{f_{T,L}\in {\mathcal {F}}_h} \frac{m_{T,L}^2}{2\theta (\rho _T,\rho _{L})} |f_{T,L}||x_T-x_L|\,, \end{aligned}$$

where $\theta :{\mathbb {R}}^+\times {\mathbb {R}}^+ \rightarrow {\mathbb {R}}^+$ is an appropriate function (see [16]) which we take to be the harmonic mean.

Now, in order to construct $\rho \in Q_h^+$ and $m_1,m_2 \in {\mathcal {R}}{\mathcal {T}}_0$ satisfying (A.2), we use the same construction as in [22] for the finite volume scheme, and interpolate these to the spaces ${\mathcal {R}}{\mathcal {T}}_0$ and $Q_h^+$ to obtain $\rho$, $m_1$ and $m_2$ satisfying

$$\begin{aligned} \left\{ \begin{array}{l} \mathrm{div}\, m_1 = \rho - \Pi _{Q_h} (\delta _x) \,,\\ \mathrm{div}\, m_2 = \rho - \Pi _{Q_h} (\delta _y)\,. \end{array} \right. \end{aligned}$$

In particular the support of $\rho$, $m_1$ and $m_2$ is a chain of neighbouring elements $T_1,\ldots , T_N$. To prove the bound on the action, we observe that $A_h(\rho ,m_i)\le A(\rho ,m_i)$. Then, we only need to bound $A(\rho ,m_i)$ by the action of the finite-volume scheme ${A}_h^{FV}(\rho ,m_i)$, since $A_h^{FV}$ satisfies the desired inequality thanks to the regularity assumption (4.1) on the mesh [22]. By the regularity assumption on the triangulation, we can assume

$$\begin{aligned} \int _{T\cup L} |\varphi _{T,L}|^2 \, \mathrm{d}x \le C |f_{T,L}||x_T-x_L| \end{aligned}$$

uniformly. Then, by explicit calculations we obtain $A(\rho ,m_i) \le C{A}_h^{FV}(\rho ,m_i)$ and we are done. $\square$