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A descent method for the free energy of multicomponent systems

Collaborator: H. Gajewski (FG 1), J.A. Griepentrog (FG 1), K. Gärtner (FG 3)

Cooperation with: J. Beuthan, O. Minet (Charité, Institut für Medizinische Physik und Lasermedizin, Berlin)

Supported by: BMBF-Verbundprojekt: ,,Anwendung eines nichtlokalen Phasenseparationsmodells zur Bildbewertung in der Rheumadiagnostik'' (Joint Project: Application of a nonlocal phase separation model to optical diagnosis of rheumatic diseases)

Description:

To describe phase-separation processes, we consider a closed system with interacting particles of type i $\in$ {0, 1,..., m} occupying a spatial Lipschitz domain $\Omega$ $\subset$ $\mathbb {R}$ⁿ. We assume that the particles jump around on a given microscopically scaled lattice following a stochastic exchange process. Exactly one particle sits on each lattice site (exclusion principle). Two particles of type i and $\ell$ change their sites x and y with a certain probability p_i$\ell$(x, y) due to diffusion and interaction. The hydrodynamical limit (see [2]) leads to a system of m + 1 conservation laws:

$$\nabla \Omega \nu \partial \Omega \Omega$$ (1)

with mass densities u₀,..., u_m, their initial values u₀₀,..., u_0m, and current densities j₀,..., j_m. Due to the exclusion principle, we can assume $\sum_{{i=0}}^{m}$u_i = 1, $\sum_{{i=0}}^{m}$u_0i = 1, and $\sum_{{i=0}}^{m}$j_i = 0. Hence, we can describe the state of the system by u = (u₁,..., u_m) and u₀ = 1 - $\sum_{{i=1}}^{m}$u_i.

Equilibrium distributions u^* = (u₁^*,..., u_m^*) : $\Omega$$\to$$\mathbb {R}$^m of the multicomponent system, and more generally, steady states of the evolution system can be supposed to be local minimizers of the free energy functional F under the constraint of mass conservation:

F(u^*) = min$\big\{$F(u) : $\int_{\Omega}^{}$(u_i - u_0i) dx = 0 for i $\in$ {1,..., m}$\big\}$,

or solutions (u^*,$\mu^{*}_{}$) of the Euler-Lagrange equations including Lagrange multipliers $\mu^{*}_{}$ $\in$ $\mathbb {R}$^m:

$$\sum_{{i=1}}^{m} \mu^{*}_{i} \big\langle \big\rangle \int_{\Omega}^{} \big\langle \big\rangle \in$$ (2)

In many applications, one is originally interested in u^*. However, F is in general not convex, so it seems to be difficult to solve (2) directly. One can try to construct u^* as steady state of the evolution system (1): In view of the fact that the Lagrange multipliers $\mu^{*}_{i}$ should be constant, one assumes their antigradients to be driving forces towards equilibrium. This leads to the evolution system (1) with current densities j_i = - $\sum_{{\ell=1}}^{m}$a_i$\ell$(u)$\nabla$$\mu_{\ell}^{}$ and positively semidefinite mobility matrix (a_i$\ell$) (see [2], [3]). Evidently, F is a Lyapunov function of (1), and the authors have developed a dissipative discretization scheme with respect to space and time. However, from the practical point of view that approach becomes questionable if meta-stable states occur. Hence, we want to establish a descent method to solve (2) directly (see [1]):

To formulate our problem, we use standard spaces H = L²($\Omega$;$\mathbb {R}$^m) and V = L^$\infty$($\Omega$;$\mathbb {R}$^m). We consider the decomposition of H = H₀ + H₁ into the closed subspace H₀ = {u $\in$ H : $\int_{\Omega}^{}$u dx = 0} and the m-dimensional subspace H₁ $\subset$ V of constants. Let J $\in$ $\mathcal {L}$(H;H^*) be the duality map between H and H^*. Then the annihilator H₀⁰ = J[H₁] is the m-dimensional subspace of elements f = $\sum_{{i=1}}^{m}$$\mu_{i}^{}$g_i where $\mu$ $\in$ $\mathbb {R}$^m and g₁,..., g_m $\in$ J[V] are given by $\big\langle$g_i, u$\big\rangle$ = $\int_{\Omega}^{}$u_i dx, u $\in$ H.

We split the free energy functional F : H $\longrightarrow$ $\mathbb {R}$ $\cup$ { + $\infty$} into a sum F = $\Phi$ + $\Psi$ of a chemical part $\Phi$ : H$\to$$\mathbb {R}$ $\cup$ { + $\infty$} and a nonlocal interaction part $\Psi$ : H $\longrightarrow$ $\mathbb {R}$ as follows:

We introduce a lower semicontinuous and strongly convex functional $\Phi$ : H $\longrightarrow$ $\mathbb {R}$ $\cup$ { + $\infty$} by

$$\Phi \begin{cases} \int_\Omega \varphi(u) dx & \text{if $u \in \mathrm{dom}(\Phi)$}, \\ + \infty & \text{otherwise}, \end{cases} \varphi \begin{cases} \sum_{i=0}^m z_i \log(z_i) & \text{if $z \in \mathrm{dom}(\varphi)$}, \\ + \infty & \text{otherwise}, \end{cases}$$ (3)

where dom($\Phi$) = {u $\in$ H : 0$\le$u₀, u₁,..., u_m$\le$1} and dom($\varphi$) = {z $\in$ $\mathbb {R}$^m : 0$\le$z₀, z₁,..., z_m$\le$1}. Additionally, we fix l $\in$ J[V] and d $\in$ $\mathbb {R}$ and define a quadratic functional $\Psi$ : H $\longrightarrow$ $\mathbb {R}$ by

$$\Psi {\frac{{1}}{{2}}} \langle \rangle \langle \rangle \in$$ (4)

where T $\in$ $\mathcal {L}$(H;H^*) is a self-adjoint and completely continuous operator such that T| V is a completely continuous operator in $\mathcal {L}$(V;J[V]). Then we can find constants $\alpha$, $\beta$ > 0 such that

$$\big\langle$$$$\partial$$$$\Phi$$(u) - $$\partial$$$$\Phi$$(v), u - v$$\big\rangle$$$$\ge$$$$\alpha$$ | u - v|²_H,| D$$\Psi$$(u) - D$$\Psi$$(v)|_H^*$$\le$$$$\beta$$ | u - v|_H, u, v $$\in$$ dom($$\Phi$$).

If we specify $\overline{{u}}$ $\in$ int  dom($\varphi$) and the set K = $\big\{$u $\in$ dom($\Phi$) : $\int_{\Omega}^{}$(u - $\overline{{u}}$) dx = 0$\big\}$ of mass constraints, then the functional F : H$\to$$\mathbb {R}$ $\cup$ { + $\infty$} is bounded from below, and there exists a solution u^* $\in$ K of the constrained minimum problem F(u^*) = min$\big\{$F(u) : u $\in$ K$\big\}$.

In view of the above assumptions on monotonicity, continuity, and compactness, we can find convex subsets C $\subset$ K and M $\subset$ H₀⁰ which are closed in V and J[V], respectively, such that for all initial values u₀ $\in$ K, we can define iteration sequences (u_k) $\subset$ K and (v_k, f_k) $\in$ C x M by

$$\tau \tau \in \partial \Phi \Psi \in \mathbb {N}$$ (5)

Here, the relaxation parameter $\tau$ $\in$ (0, 1] satisfies $\alpha$ > $\beta$$\tau$ which ensures the decay and the convergence of (F(u_k)) due to the descent property

| u_k - u_k+1|_H²$\le$${\frac{{2\tau}}{{\alpha-\beta\tau}}}$$\big($F(u_k) - F(u_k+1)$\big)$, k $\in$ $\mathbb {N}$.

Moreover, using analyticity properties of $\varphi$, we can apply a ojasiewicz-Simon-type inequality to prove that the sequence (u_k, f_k) $\subset$ K x M defined by (5) converges to a solution (u^*, f^*) $\in$ C x M of the Euler-Lagrange equation (see (2))

f^* = $\sum_{{i=1}}^{m}$$\mu^{*}_{i}$g_i $\in$ $\partial$$\Phi$(u^*) + D$\Psi$(u^*),

in the sense of $\lim_{{k \to \infty}}^{}$F(u_k) = F(u^*), $\lim_{{k \to \infty}}^{}$| u_k - u^*|_V = 0, and $\lim_{{k \to \infty}}^{}$| f_k - f^*|_J[V] = 0.

To apply our descent method to image processing, we describe the nonlocal interaction by means of inverse operators corresponding to second-order elliptic operators with appropriate regularity properties. To do so, for r > 0 we consider the family of elliptic operators E_r $\in$ $\mathcal {L}$$\big($H¹($\Omega$);H¹($\Omega$)^*$\big)$ (including Neumann boundary conditions) given by

$$\big\langle$$E_rv, h$$\big\rangle$$ = $\int_{\Omega}^{}$$\big($r²$\nabla$v ^. $\nabla$h + vh$\big)$ dx, v, h $\in$ H¹($\Omega$).

We want to emphasize that the inverse E_r^-1 $\in$ $\mathcal {L}$$\big($H¹($\Omega$)^*;H¹($\Omega$)$\big)$ is completely continuous from L²($\Omega$) into L²($\Omega$) as well as from L^$\infty$($\Omega$) into L^$\infty$($\Omega$).

To control the qualitative behavior of nonlocal interaction, we prescribe effective ranges $\varrho$, r > 0 and intensities $\sigma_{{i\ell}}^{}$, s_i$\ell$ $\in$ $\mathbb {R}$ of interaction forces between particles of type i and $\ell$ $\in$ {0, 1,..., m}, respectively. Clearly, both matrices ($\sigma_{{i\ell}}^{}$) and (s_i$\ell$) are assumed to be symmetric. The cases $\sigma_{{i\ell}}^{}$ > 0 and $\sigma_{{i\ell}}^{}$ < 0 represent repulsive and attractive interaction, respectively. The nontrivial choice of (s_i$\ell$) enables us to get final states close to the corresponding initial values u₀ $\in$ K if we define the quadratic functional $\Psi$ : H $\longrightarrow$ $\mathbb {R}$ according to (4):

$$\Psi$$(u) = ${\frac{{1}}{{2}}}$$\sum_{{i=0}}^{m}$$\sum_{{\ell=0}}^{m}$$\int_{\Omega}^{}$$\big($u_i$\sigma_{{i\ell}}^{}$E_$\varrho$^-1u_$\ell$ + (u_i - u_0i)s_i$\ell$E_r^-1(u_$\ell$ - u_0$\ell$)$\big)$ dx, u $\in$ H.

In our joint BMBF project, we use the above nonlocal image segmentation method to analyze medical images regarding the scattering light distribution of the near-infrared spectral range on rheumatoid finger joints. Rheumatoid arthritis is the most common inflammatory arthropathy; it often affects the small joints, especially the finger joints. Inflammation of joints caused by rheumatic diseases starts with an inflammatory process of capsule synovial structures. Later, granulation tissue develops in the synovialis and destroys the cartilage and even the bone structure. Figures 1 and 2 show two examples of healthy and rheumatoid finger joints and the corresponding results of image segmentation with respect to three components (bone, cartilage, and synovial fluid):

Fig. 1: Healthy finger joint

Fig. 2: Rheumatoid finger joint

References:

1. H. GAJEWSKI, J.A. GRIEPENTROG, A descent method for the free energy of multicomponent systems, WIAS Preprint no. 980, 2004 .
2. G. GIACOMIN, J.L. LEBOWITZ, R. MARRA, Macroscopic evolution of particle systems with short- and long-range interactions, Nonlinearity, 13 (2000), pp. 2143-2162.
3. J.A. GRIEPENTROG, On the unique solvability of a phase separation problem for multicomponent systems, Banach Center Publ., 66 (2004), pp. 153-164.

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