Two-dimensional diffeomorphic model for multi-modality image registration

Abstract

In this paper, we propose a diffeomorphic multi-modality image registration model based on the Rényi’s statistical dependence measure by taking mesh folding elimination into consideration. The existence of solution for the proposed model is proved, and a multi-grid algorithm is presented for multi-modality image registration. And the convergence of minimizing sequence is proved. Moreover, numerical tests are performed to demonstrate that the proposed algorithm works for both mono-modality image registration and multi-modality image registration without mesh folding.

Appendices

Appendix A: The convergence of alternating direction method in (53)–(55)

To prove the convergence of sequences $({\overline{{\mathbf {x}}} ^k},{\overline{{\mathbf {y}}} ^k},{{{\mathbf {u}}}^k})$ obtained from subproblems (53)–(55), we assume the set of discontinuous points of $F( \cdot )$ and ${\nabla _F}$ at ${\mathbf {x}} \in \Omega$ are two zero measure sets, and F is piecewise two-times differentiable on $\Omega$. Then $\Delta \buildrel \Delta \over = \{ {{\mathbf {x}}}|\nabla F( \cdot )$ or $\nabla F( \cdot )$ is discontinuous at ${\mathbf {x}}\}$ is a zero measure set. Motivated by Chen and Ye (2010), we introduce two new functions ${{\varvec{\alpha }}} \in {{\mathbb {R}}^p},{{\varvec{\beta }}} \in {{\mathbb {R}}^l}$ to transform (53)–(55) into following problems:

$$\begin{aligned} ({{\varvec{\alpha }}},\overline{{\mathbf {x}}} ,{{\varvec{\beta }}},\overline{{\mathbf {y}}} ,{{\mathbf {u}}})\in \arg \mathop {\min }\limits _{{{\varvec{\alpha }}} \in {{\mathbb {R}}^p},\overline{{\mathbf {x}}} \in {{\mathbb {R}}^p},{{\varvec{\beta }}} \in {{\mathbb {R}}^l},\overline{{\mathbf {y}}} \in {{\mathbb {R}}^l},{{\mathbf {u}}} \in {{[H_0^\alpha (\Omega )]}^2}} {E_1}({{\varvec{\alpha }}},\overline{{\mathbf {x}}} ,{{\varvec{\beta }}},\overline{{\mathbf {y}}} ,{{\mathbf {u}}}), \end{aligned}$$

(A1)

where ${\theta _1} > 0$ is small enough, and ${E_1}({{\varvec{\alpha }}},{\mathbf {{\overline{x}}}},{{\varvec{\beta }}},{\mathbf {{\overline{y}}}},{{\mathbf {u}}}) = S({{\varvec{\alpha }}},{{\varvec{\beta }}},{{\mathbf {u}}}) + \delta {R_0}({{\mathbf {u}}}) + \Theta R({{\mathbf {u}}}) + \frac{1}{{2\theta _1 }}{({\mathbf {{\overline{x}}}} - {{\varvec{\alpha }}})^2} + \frac{1}{{2\theta _1 }}{({\mathbf {{\overline{y}}}} - {{\varvec{\beta }}})^2}$.

Based on these notations, we transform (A1) into the following five subproblems:

$$\begin{aligned}&{{{\varvec{\alpha }}}^{k + 1}} \in \arg \mathop {\min }\limits _{{{\varvec{\alpha }}} \in {{\mathbb {R}}^p}} {E_1}({{\varvec{\alpha }}},{\overline{{\mathbf {x}}} ^k},{{{\varvec{\beta }}}^k},{\overline{{\mathbf {y}}} ^k},{{{\mathbf {u}}}^k}), \end{aligned}$$

(A2)

$$\begin{aligned}&{\overline{{\mathbf {x}}} ^{k + 1}} \in \arg \mathop {\min }\limits _{\overline{{\mathbf {x}}} \in {{\mathbb {R}}^p}} {E_1}({{{\varvec{\alpha }}}^{k + 1}},\overline{{\mathbf {x}}} ,{{{\varvec{\beta }}}^k},{\overline{{\mathbf {y}}} ^k},{{{\mathbf {u}}}^k}), \end{aligned}$$

(A3)

$$\begin{aligned}&{{{\varvec{\beta }}}^{k + 1}} \in \arg \mathop {\min }\limits _{{{\varvec{\beta }}} \in {{\mathbb {R}}^l}} {E_1}({{{\varvec{\alpha }}}^{k + 1}},{\overline{{\mathbf {x}}} ^{k + 1}},{{\varvec{\beta }}},{\overline{{\mathbf {y}}} ^k},{{{\mathbf {u}}}^k}), \end{aligned}$$

(A4)

$$\begin{aligned}&{\overline{{\mathbf {y}}} ^{k + 1}} \in \arg \mathop {\min }\limits _{\overline{{\mathbf {y}}} \in {{\mathbb {R}}^l}} {E_1}({{{\varvec{\alpha }}}^{k + 1}},{\overline{{\mathbf {x}}} ^{k + 1}},{{{\varvec{\beta }}}^{k + 1}},\overline{{\mathbf {y}}} ,{{{\mathbf {u}}}^k}), \end{aligned}$$

(A5)

$$\begin{aligned}&{{{\mathbf {u}}}^{k + 1}} \in \arg \mathop {\min }\limits _{{{\mathbf {u}}} \in {{[H_0^\alpha (\Omega )]}^2}} {E_1}({{{\varvec{\alpha }}}^{k + 1}},{\overline{{\mathbf {x}}} ^{k + 1}},{{{\varvec{\beta }}}^{k + 1}},{\overline{{\mathbf {y}}} ^{k + 1}},{{\mathbf {u}}}). \end{aligned}$$

(A6)

Concerning the convergence of (A2)–(A6), we have the following theorem.

Theorem 3

Suppose $\mathrm {ess}$ ${\sup _{{\mathbf {x}} \in \Omega }}|{F({\mathbf {x}} )} |\le {\overline{M}}$ and $\mathrm {ess}$ $\sup _{{\mathbf {x}} \in \Omega } |T({\mathbf {x}})|\le {\overline{M}}$ for some ${\overline{M}}<+\infty$, $F({\mathbf {x}})\not \equiv c$, $T({\mathbf {x}})\not \equiv c$ (c is a constant) for any ${\mathbf {x}} \in \Omega$, $0< {\theta _1} < {\frac{1}{{4\lambda |\Omega |{{\left\| {\nabla ^2 \mathrm{CC}(P( \cdot ),Q( \cdot ))} \right\| }_\infty }}}}$, $\delta > 2b\lambda |\Omega |{\left\| {{\nabla ^2}_{{\mathbf {u}}}\mathrm{CC}(P( \cdot ),Q( \cdot ))} \right\| _\infty }$; then the sequence $({{{\varvec{\alpha }}}^k},{\overline{{\mathbf {x}}} ^k},{{{\varvec{\beta }}}^k},{\overline{{\mathbf {y}}} ^k},{{{\mathbf {u}}}^k})$ converges to the solution of (A1) as $k \rightarrow \infty$.

Proof

Step 1: we claim that there exists $({{\varvec{\alpha }}},\overline{{\mathbf {x}}} ,{{\varvec{\beta }}},\overline{{\mathbf {y}}} ,{{\mathbf {u}}}) \in {{\mathbb {R}}^p} \times {{\mathbb {R}}^p} \times {{\mathbb {R}}^l} \times {{\mathbb {R}}^l} \times {[H_0^\alpha (\Omega )]^2}$ such that

$$\begin{aligned} ({{{\varvec{\alpha }}}^k},{\overline{{\mathbf {x}}} ^k},{{{\varvec{\beta }}}^k},{\overline{{\mathbf {y}}} ^k},{{{\mathbf {u}}}^k}) {\mathop {\longrightarrow }\limits ^{k}} ({{\varvec{\alpha }}},\overline{{\mathbf {x}}} ,{{\varvec{\beta }}},\overline{{\mathbf {y}}} ,{{\mathbf {u}}}). \end{aligned}$$

(A7)

The Euler–Lagrange equation of (A2) is

$$\begin{aligned}&- 2\lambda |\Omega |\left( 1 - CC\left( P\left( {{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^k}\right) ,Q\left( {{{\varvec{\beta }}}^k}\right) \right) \right) \cdot \mathrm{Var}{^{ - \frac{3}{2}}}\left( P\left( {{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^k}\right) \right) \mathrm{Var}{^{ - \frac{1}{2}}}\left( Q\left( {{{\varvec{\beta }}}^k}\right) \right) \nonumber \\&\quad \cdot \left[ \mathrm{Cov}\left( {S_s}\left( {{\mathbf {u}}^k}\right) ,Q\left( {{{\varvec{\beta }}}^k}\right) \right) Var\left( P\left( {{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^k}\right) \right) - \mathrm{Cov}\left( P\left( {{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^k}\right) ,{S_s}\left( {{\mathbf {u}}^k}\right) \right) \right. \nonumber \\&\qquad \cdot \left. \mathrm{Cov}\left( P\left( {{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^k}\right) ,Q\left( {{{\varvec{\beta }}}^k}\right) \right) \right] -\frac{1}{\theta }\left( {\overline{{\mathbf {x}}} ^k} - {{\varvec{\alpha }} ^{k + 1}}\right) = 0. \end{aligned}$$

(A8)

On the other hand,

$$\begin{aligned}&{E_1}\left( {{{\varvec{\alpha }}}^k},{\overline{{\mathbf {x}}} ^k},{{{\varvec{\beta }}}^k},{\overline{{\mathbf {y}}} ^k},{{{\mathbf {u}}}^k}\right) - {E_1}\left( {{{\varvec{\alpha }}}^{k + 1}},\overline{{\mathbf {x}}}^k ,{{{\varvec{\beta }}}^k},{\overline{{\mathbf {y}}} ^k},{{{\mathbf {u}}}^k}\right) \nonumber \\&\quad = S\left( {{{\varvec{\alpha }}}^k},{{{\varvec{\beta }}}^k},{{{\mathbf {u}}}^k}\right) - S\left( {{{\varvec{\alpha }}}^{k + 1}},{{{\varvec{\beta }}}^k},{{{\mathbf {u}}}^k}\right) + \frac{1}{{2{\theta _1}}}{\left( {\overline{{\mathbf {x}}} ^k} - {{{\varvec{\alpha }}}^k}\right) ^2} - \frac{1}{{2{\theta _1}}}{\left( {\overline{{\mathbf {x}}} ^k} - {{{\varvec{\alpha }}}^{k + 1}}\right) ^2}\nonumber \\&\quad = \lambda |\Omega |{\left[ 1 - CC\left( P\left( {{{\varvec{\alpha }}}^k},{{{\mathbf {u}}}^k}\right) ,Q\left( {{{\varvec{\beta }}}^k}\right) \right) \right] ^2} - \lambda |\Omega |{\left[ 1 - CC\left( P\left( {{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^k}\right) ,Q\left( {{{\varvec{\beta }}}^k}\right) \right) \right] ^2}\nonumber \\&\qquad + \frac{1}{{2{\theta _1}}}{\left( {\overline{{\mathbf {x}}} ^k} - {{{\varvec{\alpha }}}^k}\right) ^2} - \frac{1}{{2{\theta _1}}}{\left( {\overline{{\mathbf {x}}} ^k} - {{{\varvec{\alpha }}}^{k + 1}}\right) ^2}\nonumber \\&\quad = \lambda |\Omega |{\left[ CC\left( P\left( {{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^k}\right) ,Q\left( {{{\varvec{\beta }}}^k}\right) \right) - CC\left( P\left( {{{\varvec{\alpha }}}^k},{{{\mathbf {u}}}^k}\right) ,Q\left( {{{\varvec{\beta }}}^k}\right) \right) \right] ^2}\nonumber \\&\qquad + 2\lambda |\Omega |\left[ 1 - CC\left( P\left( {{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^k}\right) ,Q\left( {{{\varvec{\beta }}}^k}\right) \right) \right] \cdot \left[ CC\left( P\left( {{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^k}\right) ,Q\left( {{{\varvec{\beta }}}^k}\right) \right) \right. \nonumber \\&\qquad \left. - CC\left( P\left( {{{\varvec{\alpha }}}^k},{{{\mathbf {u}}}^k}\right) ,Q\left( {{{\varvec{\beta }}}^k}\right) \right) \right] \nonumber \\&\qquad + \frac{1}{{2{\theta _1}}}{\left( {{{\varvec{\alpha }}}^{k + 1}} - {{{\varvec{\alpha }}}^k}\right) ^2} + \frac{1}{{{\theta _1}}}\left( {{{\varvec{\alpha }}}^{k + 1}} - {{{\varvec{\alpha }}}^k}\right) \left( {\overline{{\mathbf {x}}} ^k} - {{{\varvec{\alpha }}}^{k + 1}}\right) \buildrel \Delta \over = {I_1} + {I_2}, \end{aligned}$$

(A9)

where ${I_1}=\lambda |\Omega |{[\mathrm{CC}(P({{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^k}),Q({{{\varvec{\beta }}}^k})) - \mathrm{CC}(P({{{\varvec{\alpha }}}^k},{{{\mathbf {u}}}^k}),Q({{{\varvec{\beta }}}^k}))]^2} + 2\lambda |\Omega |[1 - \mathrm{CC}(P({{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^k}),Q({{{\varvec{\beta }}}^k}))] \cdot [\mathrm{CC}(P({{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^k}),Q({{{\varvec{\beta }}}^k})) - \mathrm{CC}(P({{{\varvec{\alpha }}}^k},{{{\mathbf {u}}}^k}),Q({{{\varvec{\beta }}}^k}))]$, ${I_2}=\frac{1}{{2{\theta _1}}}{({{{\varvec{\alpha }}}^{k + 1}} - {{{\varvec{\alpha }}}^k})^2} + \frac{1}{{{\theta _1}}}({{{\varvec{\alpha }}}^{k + 1}} - {{{\varvec{\alpha }}}^k})({\overline{{\mathbf {x}}} ^k} - {{{\varvec{\alpha }}}^{k + 1}})$.

Next, we will prove that $\mathrm{CC}(P( \cdot ),Q( \cdot ))$ is two-times differentiable with respect to ${{\varvec{\alpha }}}$.

Because $P = \sum \nolimits _{1 \le s \le p} {{{{\varvec{\alpha }}}_s}K(F({{\mathbf {x}}} + {{\mathbf {u}}}({{\mathbf {x}}})),{{{\mathbf {m}}}_s})}$ is infinitely differentiable with respect to ${{\varvec{\alpha }}}$, $E(P) = \frac{1}{{|\Omega |}}\int _\Omega {Pd{{\mathbf {x}}}}$ is infinitely differentiable with ${{\varvec{\alpha }}}$. Then $\mathrm{Cov}(P,Q) = \frac{1}{{|\Omega |}}\int _\Omega {(P - E(P))(Q - E(Q))d{{\mathbf {x}}}}$ and $\mathrm{Var}(P) = \frac{1}{{|\Omega |}}\int _\Omega {{{(P - E(P))}^2}d{{\mathbf {x}}}}$ are also infinitely differentiable with respect to ${{\varvec{\alpha }}}$. Furthermore, for any ${\mathbf {x}} \in \Omega$, $F({{\mathbf {x}}})\not \equiv c$ and f is a Borel measurable function with finite positive variance, then it can be obtained from the proof by contradiction similar to Theorem 1 that $\mathrm{Var}(P)$ is strictly greater than zero. Similarly, $\mathrm{Var}(Q)$ is also strictly greater than zero. Therefore, $\mathrm{CC}(P( \cdot ),Q( \cdot ))$ is infinitely differentiable with ${\varvec{\alpha }}$.

By Taylor’s formula,

$$\begin{aligned} \mathrm{CC}(P({{{\varvec{\alpha }}}^k},{{{\mathbf {u}}}^k}),Q({{{\varvec{\beta }}}^k}))= & {} \mathrm{CC}(P({{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^k}),Q({{{\varvec{\beta }}}^k})) + {\nabla _{{\varvec{\alpha }}}}\mathrm{CC}(P({{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^k}),Q({{{\varvec{\beta }}}^k}))\nonumber \\&\cdot ({{{\varvec{\alpha }}}^k} - {{{\varvec{\alpha }}}^{k + 1}}) + {{\nabla }^2 _{{{\varvec{\alpha }}}}}\mathrm{CC}(P({\varvec{\vartheta }},{{{\mathbf {u}}}^k}),Q({{{\varvec{\beta }}}^k})){({{{\varvec{\alpha }}}^k} - {{{\varvec{\alpha }}}^{k + 1}})^2}\nonumber \\= & {} \mathrm{CC}(P({{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^k}),Q({{{\varvec{\beta }}}^k}))+A+B, \end{aligned}$$

(A10)

where ${\varvec{\vartheta }}$ is around ${\varvec{\alpha }}$.

This implies,

$$\begin{aligned} {I_1}= & {} \lambda |\Omega |{(A + B)^2} - 2\lambda |\Omega |[1 - \mathrm{CC}(P({{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^k}),Q({{{\varvec{\beta }}}^k}))] \cdot (A + B)\nonumber \\\ge & {} - 2\lambda |\Omega |[1 - \mathrm{CC}(P({{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^k}),Q({{{\varvec{\beta }}}^k}))] \cdot (A + B)\nonumber \\= & {} - 2\lambda |\Omega |[1 - \mathrm{CC}(P({{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^k}),Q({{{\varvec{\beta }}}^k}))]{\nabla _{{\varvec{\alpha }}}}\mathrm{CC}(P({{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^k}),Q({{{\varvec{\beta }}}^k}))({{{\varvec{\alpha }}}^k} - {{{\varvec{\alpha }}}^{k + 1}})\nonumber \\&- 2\lambda |\Omega |[1 - \mathrm{CC}(P({{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^k}),Q({{{\varvec{\beta }}}^k}))]{\nabla ^2}_{{\varvec{\alpha }}}\mathrm{CC}(P({{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^k}),Q({{{\varvec{\beta }}}^k})){({{{\varvec{\alpha }}}^k} - {{{\varvec{\alpha }}}^{k + 1}})^2}\nonumber \\\ge & {} - 2\lambda |\Omega |[1 - \mathrm{CC}(P({{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^k}),Q({{{\varvec{\beta }}}^k}))]{\nabla _{{\varvec{\alpha }}}}\mathrm{CC}(P({{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^k}),Q({{{\varvec{\beta }}}^k}))({{{\varvec{\alpha }}}^k} - {{{\varvec{\alpha }}}^{k + 1}})\nonumber \\&- {{\overline{M}} _1}{({{{\varvec{\alpha }}}^k} - {{{\varvec{\alpha }}}^{k + 1}})^2}, \end{aligned}$$

(A11)

where $[1 - \mathrm{CC}(P({{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^k}),Q({{{\varvec{\beta }}}^k}))] \in [0,1], {{\overline{M}} _1} = 2\lambda |\Omega |{\left\| {{\nabla ^2 _{{\varvec{\alpha }}}}\mathrm{CC}(P((\cdot ),Q(\cdot ))} \right\| _\infty }$.

It follows from (A8), (A9), (A11) that

(A12)

where $b_1=\frac{1}{{2{\theta _1}}} - {{\overline{M}} _1}>0$ because of the fact that $0< {\theta _1} < \frac{1}{{4\lambda |\Omega |{{\left\| {{{\nabla }^2 _{{\varvec{\alpha }}}}\mathrm{CC}(P( \cdot ),Q( \cdot ))} \right\| }_\infty }}}$.

In addition,

$$\begin{aligned}&{E_1}({{{\varvec{\alpha }}}^{k + 1}},{\overline{{\mathbf {x}}} ^k},{{{\varvec{\beta }}}^k},{\overline{{\mathbf {y}}} ^k},{{{\mathbf {u}}}^k}) - {E_1}({{{\varvec{\alpha }}}^{k + 1}},{\overline{{\mathbf {x}}} ^{k + 1}},{{{\varvec{\beta }}}^k},{\overline{{\mathbf {y}}} ^k},{{{\mathbf {u}}}^k})\nonumber \\&\quad = \frac{1}{{2{\theta _1}}}{({\overline{{\mathbf {x}}} ^k} - {{{\varvec{\alpha }}}^{k + 1}})^2} - \frac{1}{{2{\theta _1}}}{({\overline{{\mathbf {x}}} ^{k + 1}} - {{{\varvec{\alpha }}}^{k + 1}})^2}\nonumber \\&\quad = \frac{1}{{2{\theta _1}}}{({\overline{{\mathbf {x}}} ^k} - {\overline{{\mathbf {x}}} ^{k + 1}})^2} + \frac{1}{{{\theta _1}}}({\overline{{\mathbf {x}}} ^k} - {\overline{{\mathbf {x}}} ^{k + 1}})({\overline{{\mathbf {x}}} ^{k + 1}} - {{{\varvec{\alpha }}}^{k + 1}}). \end{aligned}$$

(A13)

For fixed ${{{\varvec{\alpha }}}^k},{{{\varvec{\beta }}}^k},{\overline{{\mathbf {y}}} ^k},{{{\mathbf {u}}}^k}$, when ${\overline{{\mathbf {x}}} ^{k+1}}={{{\varvec{\alpha }}}^{k + 1}}$, $E_1$ reaches the minimum. Then

$$\begin{aligned}&{E_1}({{{\varvec{\alpha }}}^{k + 1}},{\overline{{\varvec{x}}} ^k},{{{\varvec{\beta }}}^k},{\overline{{\mathbf {y}}} ^k},{{{\mathbf {u}}}^k}) - {E_1}({{{\varvec{\alpha }}}^{k + 1}},{\overline{{\mathbf {x}}} ^{k + 1}},{{{\varvec{\beta }}}^k},{\overline{{\mathbf {y}}} ^k},{{{\mathbf {u}}}^k})\nonumber \\&\quad = \frac{1}{{2{\theta _1}}}{({\overline{{\mathbf {x}}} ^k} - {\overline{{\mathbf {x}}} ^{k + 1}})^2}. \end{aligned}$$

(A14)

Similarly, we can obtain that

(A15)

and

$$\begin{aligned}&{E_1}({{{\varvec{\alpha }}}^{k + 1}},{\overline{{\mathbf {x}}} ^{k + 1}},{{{\varvec{\beta }}}^{k + 1}},{\overline{{\mathbf {y}}} ^k},{{{\mathbf {u}}}^k}) - {E_1}({{{\varvec{\alpha }}}^{k + 1}},{\overline{{\mathbf {x}}} ^{k + 1}},{{{\varvec{\beta }}}^{k + 1}},{\overline{{\mathbf {y}}} ^{k + 1}},{{{\mathbf {u}}}^k})\nonumber \\&\quad = \frac{1}{{2{\theta _1}}}{({\overline{{\mathbf {y}}} ^k} - {\overline{{\mathbf {y}}} ^{k + 1}})^2}, \end{aligned}$$

(A16)

where ${{\overline{M}} _2} = 2\lambda |\Omega |{\left\| {{\nabla ^2}_{{\varvec{\beta }}}\mathrm{CC}(P(\cdot ),Q(\cdot ))} \right\| _\infty }$, ${b_2} \buildrel \Delta \over = \frac{1}{{2{\theta _1}}} - {{\overline{M}} _2} >0$ because of the fact that $0< {\theta _1} < {\frac{1}{{4\lambda |\Omega |{{\left\| {{{\nabla ^2}_{{\varvec{\beta }}}} \mathrm{CC}(P( \cdot ),Q( \cdot ))} \right\| }_\infty }}}}$.

Furthermore, ${\mathbf {u}}$ satisfies the following formula:

$$\begin{aligned}&\int _\Omega { - 2\lambda \left[ 1 - \mathrm{CC}(P({{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^{k + 1}}),Q({{{\varvec{\beta }}}^{k + 1}}))\right] H({{{\varvec{\alpha }}}^{k + 1}},{{{\varvec{\beta }}}^{k + 1}},{{{\mathbf {u}}}^{k + 1}})} {\varvec{\eta }}({{\mathbf {x}}})d{{\mathbf {x}}}\nonumber \\&\quad + 2\Theta \int _\Omega {\left( \frac{{\partial u_1^{k + 1}}}{{\partial {x_1}}} - \frac{{\partial u_2^{k + 1}}}{{\partial {x_2}}}\right) \left( \frac{{\partial {\eta _1}}}{{\partial {x_1}}} - \frac{{\partial {\eta _2}}}{{\partial {x_2}}}\right) } d{{\mathbf {x}}}\nonumber \\&\quad + 2\Theta \int _\Omega {\left( \frac{{\partial u_1^{k + 1}}}{{\partial {x_2}}} + \frac{{\partial u_2^{k + 1}}}{{\partial {x_1}}}\right) \left( \frac{{\partial {\eta _1}}}{{\partial {x_2}}} + \frac{{\partial {\eta _2}}}{{\partial {x_1}}}\right) } d{{\mathbf {x}}}\nonumber \\&\quad +\delta \int _\Omega {{\nabla ^\alpha }{{\mathbf {u}}}^{k+1} \cdot } {\nabla ^\alpha }{\varvec{\eta }}d{{\mathbf {x}}} = 0 \qquad \quad \forall {\varvec{\eta }}\in {[H_0^\alpha (\Omega )]^2}, \end{aligned}$$

(A17)

where $\int _\Omega {{\nabla ^\alpha }{{\mathbf {u}}} \cdot } {\nabla ^\alpha }{\varvec{\eta }}d{{\mathbf {x}}} = \int _\Omega {\sum \nolimits _{i,j = 1}^2 {\frac{{{\partial ^\alpha }{u_i}}}{{\partial x_j^\alpha }}} } \frac{{{\partial ^\alpha }{\eta _i}}}{{\partial x_j^\alpha }}d{{\mathbf {x}}}$.

By letting ${\varvec{\eta }}({{\mathbf {x}}}) = {{{\mathbf {u}}}^k} - {{{\mathbf {u}}}^{k + 1}}$, we obtain that

$$\begin{aligned}&L\left( {{{\mathbf {u}}}^k},{{{\mathbf {u}}}^{k + 1}}\right) \nonumber \\&\quad = \int _\Omega { - 2\lambda [1 - CC\left( P\left( {{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^{k + 1}}\right) ,Q\left( {{{\varvec{\beta }}}^{k + 1}}\right) \right) ] H\left( {{{\varvec{\alpha }}}^{k + 1}},{{{\varvec{\beta }}}^{k + 1}},{{{\mathbf {u}}}^{k + 1}}\right) } \left( {{{\mathbf {u}}}^k} - {{{\mathbf {u}}}^{k + 1}}\right) d{{\mathbf {x}}}\nonumber \\&\qquad +\{ 2\Theta \int _\Omega {\left( \frac{{\partial u_1^{k + 1}}}{{\partial {x_1}}} - \frac{{\partial u_2^{k + 1}}}{{\partial {x_2}}}\right) \left( \frac{{\partial u_1^k}}{{\partial {x_1}}} - \frac{{\partial u_2^k}}{{\partial {x_2}}}\right) } d{{\mathbf {x}}}\nonumber \\&\quad + 2\Theta \int _\Omega {\left( \frac{{\partial u_1^{k + 1}}}{{\partial {x_2}}} + \frac{{\partial u_2^{k + 1}}}{{\partial {x_1}}}\right) \left( \frac{{\partial u_1^k}}{{\partial {x_2}}} + \frac{{\partial u_2^k}}{{\partial {x_1}}}\right) } d{{\mathbf {x}}}\nonumber \\&\qquad {\mathbf { - }}2\Theta \int _\Omega {{{\left( \frac{{\partial u_1^{k + 1}}}{{\partial {x_1}}} - \frac{{\partial u_2^{k + 1}}}{{\partial {x_2}}}\right) }^2}} \mathrm{d}{\mathbf {x - }}2\Theta \int _\Omega {{{\left( \frac{{\partial u_1^{k + 1}}}{{\partial {x_2}}} + \frac{{\partial u_2^{k + 1}}}{{\partial {x_1}}}\right) }^2}} d{{\mathbf {x}}}\}\nonumber \\&\qquad + \delta \int _\Omega {{\nabla ^\alpha }{{\mathbf {u}}}^{k+1} \cdot } {\nabla ^\alpha }\left( {{{\mathbf {u}}}^k} - {{{\mathbf {u}}}^{k + 1}}\right) d{{\mathbf {x}}} = {I_3} + {I_4} + {I_5}= 0. \end{aligned}$$

(A18)

Then, we have

$$\begin{aligned}&{E_1}\left( {{{\varvec{\alpha }}}^{k + 1}},{\overline{{\mathbf {x}}} ^{k + 1}},{{{\varvec{\beta }}}^{k + 1}},{\overline{{\mathbf {y}}} ^{k + 1}},{{{\mathbf {u}}}^k}\right) - {E_1}\left( {{{\varvec{\alpha }}}^{k + 1}},{\overline{{\mathbf {x}}} ^{k + 1}},{{{\varvec{\beta }}}^{k + 1}},{\overline{{\mathbf {y}}} ^{k + 1}},{{{\mathbf {u}}}^{k + 1}}\right) \nonumber \\&\quad = \lambda |\Omega |{\left[ 1 - CC\left( P\left( {{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^k}\right) ,Q\left( {{{\varvec{\beta }}}^{k + 1}}\right) \right) \right] ^2}\nonumber \\&- \lambda |\Omega |{\left[ 1 - CC\left( P\left( {{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^{k + 1}}\right) ,Q\left( {{{\varvec{\beta }}}^{k + 1}}\right) \right) \right] ^2}\nonumber \\&\qquad + \delta {\int _\Omega {|{{\nabla ^\alpha }{{{\mathbf {u}}}^k}} |} ^2}d{{\mathbf {x}}} - \delta {\int _\Omega {|{{\nabla ^\alpha }{{{\mathbf {u}}}^{k + 1}}} |} ^2}d{{\mathbf {x}}} + {I_4}\nonumber \\&\quad = \lambda |\Omega |\left[ 2 - CC\left( P\left( {{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^k}\right) ,Q\left( {{{\varvec{\beta }}}^{k + 1}}\right) \right) - CC\left( P\left( {{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^{k + 1}}\right) ,Q\left( {{{\varvec{\beta }}}^{k + 1}}\right) \right) \right] \nonumber \\&\qquad \cdot \left[ CC\left( P\left( {{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^{k + 1}}\right) ,Q\left( {{{\varvec{\beta }}}^{k + 1}}\right) \right) - CC\left( P\left( {{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^k}\right) ,Q\left( {{{\varvec{\beta }}}^{k + 1}}\right) \right) \right] \nonumber \\&\qquad + \delta \int _\Omega {\left( {\nabla ^\alpha }{{{\mathbf {u}}}^k} + {\nabla ^\alpha }{{{\mathbf {u}}}^{k + 1}}\right) \left( {\nabla ^\alpha }{{{\mathbf {u}}}^k} - {\nabla ^\alpha }{{{\mathbf {u}}}^{k + 1}}\right) } d{{\mathbf {x}}} + {I_4}\nonumber \\&\quad = {I_6} + {I_7} + {I_4}. \end{aligned}$$

(A19)

Since F is piecewise two-times differentiable, one can use the similar way of proving the differentiability of $\mathrm{CC}(P(\cdot ),Q(\cdot ))$ with respect to ${\varvec{\alpha }}$ to get that $\mathrm{CC}(P(\cdot ),Q(\cdot ))$ is also two-times differentiable with respect to ${\mathbf {u}}$.

By Taylors formula, there holds

$$\begin{aligned}&CC\left( P\left( {{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^k}\right) ,Q\left( {{{\varvec{\beta }}}^{k + 1}}\right) \right) \nonumber \\&\quad = CC\left( P\left( {{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^{k + 1}}\right) ,Q\left( {{{\varvec{\beta }}}^{k + 1}}\right) \right) \nonumber \\&\qquad + {\nabla _{{\mathbf {u}}}}CC\left( P\left( {{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^{k+1}}\right) ,Q\left( {{{\varvec{\beta }}}^{k + 1}}\right) \right) \left( {{{\mathbf {u}}}^k} - {{{\mathbf {u}}}^{k + 1}}\right) \nonumber \\&\qquad + \left( {{{\mathbf {u}}}^k} - {{{\mathbf {u}}}^{k + 1}}\right) H\left( \sigma ,\tau \right) {\left( {{{\mathbf {u}}}^k} - {{{\mathbf {u}}}^{k + 1}}\right) ^T}\nonumber \\&\quad = CC\left( P\left( {{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^{k + 1}}\right) ,Q\left( {{{\varvec{\beta }}}^{k + 1}}\right) \right) + I + J, \end{aligned}$$

(A20)

where $H(\sigma ,\tau )$ is Hessian matrix of $F(\cdot )$ around ${\mathbf {u}}^{k+1}$. Therefore,

$$\begin{aligned} \ {I_6}&= \lambda |\Omega |{\left( I + J\right) ^2} - 2\lambda |\Omega |\left[ 1 - CC\left( P\left( {{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^{k + 1}}\right) ,Q\left( {{{\varvec{\beta }}}^{k + 1}}\right) \right) \right] \left( I + J\right) \nonumber \\&\ge - 2\lambda |\Omega |\left[ 1 - CC\left( P\left( {{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^{k + 1}}\right) ,Q\left( {{{\varvec{\beta }}}^{k + 1}}\right) \right) \right] {\nabla _{{\mathbf {u}}}}CC\left( P\left( {{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^{k + 1}}\right) ,Q\left( {{{\varvec{\beta }}}^{k + 1}}\right) \right) \nonumber \\&\quad \cdot \left( {{{\mathbf {u}}}^k} - {{{\mathbf {u}}}^{k + 1}}\right) - {{\overline{M}} _3}\left\| {{{{\mathbf {u}}}^k} - {{{\mathbf {u}}}^{k + 1}}} \right\| _{{{\left[ {L^2}\left( \Omega \right) \right] }^2}}^2, \end{aligned}$$

(A21)

where ${\nabla _{{\mathbf {u}}}}\mathrm{CC}(P({{{\varvec{\alpha }}}^{k + 1}},{{{\mathbf {u}}}^{k + 1}}),Q({{{\varvec{\beta }}}^{k + 1}})) = \frac{1}{{|\Omega |}} H({{{\varvec{\alpha }}}^{k + 1}},{{{\varvec{\beta }}}^{k + 1}},{{{\mathbf {u}}}^{k + 1}}), {{\overline{M}} _3} = 2\lambda |\Omega |{\left\| {\nabla _{{\mathbf {u}}}^2\mathrm{CC}(P( \cdot ),Q( \cdot ))} \right\| _\infty }$.

It follows from (A18), (A19), (A21) that

(A22)

where b is a constant and $b_3>0$ because of the fact that $\delta > 2b\lambda |\Omega |{\left\| {\nabla _{{\mathbf {u}}}^2CC\left( P\left( \cdot \right) ,Q\left( \cdot \right) \right) } \right\| _\infty }$.

It follows from (A12), (A14), (A15), (A16) and (A22) that

$$\begin{aligned}&{E_1}\left( {{{\varvec{\alpha }}}^k},{\overline{{\mathbf {x}}} ^k},{{{\varvec{\beta }}}^k},{\overline{{\mathbf {y}}} ^k},{{{\mathbf {u}}}^k}\right) - {E_1}\left( {{{\varvec{\alpha }}}^{k + 1}},{\overline{{\mathbf {x}}} ^{k + 1}},{{{\varvec{\beta }}}^{k + 1}},{\overline{{\mathbf {y}}} ^{k + 1}},{{{\mathbf {u}}}^{k + 1}}\right) \nonumber \\&\quad \ge {b_1}{\left( {{{\varvec{\alpha }}}^k} - {{{\varvec{\alpha }}}^{k + 1}}\right) ^2} + {b_2}{\left( {{{\varvec{\beta }}}^k} - {{{\varvec{\beta }}}^{k + 1}}\right) ^2} + \frac{1}{{2{\theta _1}}}{\left( {\overline{{\mathbf {x}}} ^k} - {\overline{{\mathbf {x}}} ^{k + 1}}\right) ^2}\nonumber \\&\qquad + \frac{1}{{2{\theta _1}}}{\left( {\overline{{\mathbf {y}}} ^k} - {\overline{{\mathbf {y}}} ^{k + 1}}\right) ^2} + {b_3}\left\| {{{{\mathbf {u}}}^k} - {{{\mathbf {u}}}^{k + 1}}} \right\| _{{{\left[ H_0^\alpha \left( \Omega \right) \right] }^2}}^2. \end{aligned}$$

(A23)

Let $e_k \buildrel \Delta \over = {E_1}({{{\varvec{\alpha }}}^k},{\overline{{\mathbf {x}}} ^k},{{{\varvec{\beta }}}^k},{\overline{{\mathbf {y}}} ^k},{{{\mathbf {u}}}^k}) \ge 0$, then $\{ {e_k}\}$ is a decreasing sequence with lower bound. Therefore, $\mathop {\lim }\nolimits _{k \rightarrow + \infty } ({e_k} - {e_{k+1}})= \mathop {\lim }\nolimits _{k \rightarrow + \infty } {E_1}({{{\varvec{\alpha }}}^k},{\overline{{\mathbf {x}}} ^k},{{{\varvec{\beta }}}^k},{\overline{{\mathbf {y}}} ^k},{{{\mathbf {u}}}^k}) - {E_1}({{{\varvec{\alpha }}}^{k + 1}},{\overline{{\mathbf {x}}} ^{k + 1}},{{{\varvec{\beta }}}^{k + 1}},{\overline{{\mathbf {y}}} ^{k + 1}},{{{\mathbf {u}}}^{k + 1}})= 0$. That is, there exists an e, such that $\mathop {\lim }\nolimits _{k \rightarrow + \infty } {e_k} = e$. This implies that

$$\begin{aligned}&{({{{\varvec{\alpha }}}^k} - {{{\varvec{\alpha }}}^{k + 1}})^2}_{{{\mathbb {R}}^p}} {\mathop {\longrightarrow }\limits ^{k}} 0,{({\overline{{\mathbf {x}}} ^k} - {\overline{{\mathbf {x}}} ^{k + 1}})^2}_{{{\mathbb {R}}^p}}{\mathop {\longrightarrow }\limits ^{k}} 0, {({{{\varvec{\beta }}}^k} - {{{\varvec{\beta }}}^{k + 1}})^2}_{{{\mathbb {R}}^l}}{\mathop {\longrightarrow }\limits ^{k}} 0, \nonumber \\&\quad {({\overline{{\mathbf {y}}} ^k} - {\overline{{\mathbf {y}}} ^{k + 1}})^2}_{{{\mathbb {R}}^l}} {\mathop {\longrightarrow }\limits ^{k}} 0, \left\| {{{{\mathbf {u}}}^k} - {{{\mathbf {u}}}^{k + 1}}} \right\| _{{{[H_0^\alpha (\Omega )]}^2}}^2 {\mathop {\longrightarrow }\limits ^{k}}0.\quad \quad \qquad \end{aligned}$$

(A24)

Since ess ${\sup _{\Omega \backslash \Delta }}|F({{\mathbf {x}}})|\le {\overline{M}} < + \infty$ and f is bounded and continuous, this implies that

$$\begin{aligned} |P \left( {{\varvec{\alpha }}}^k, {\mathbf {u}}^k\right) |\buildrel \Delta \over = |f\left( F\left( {{\mathbf {x}}} + {{\mathbf {u}}^k}\left( {{\mathbf {x}}}\right) \right) \right) |\le {\widetilde{M}}\quad {\widetilde{M}} \in \left( 0, +\infty \right) , \end{aligned}$$

and for any $k \in {\mathbb {N}}$, we have

$$\begin{aligned} |K(F({{\mathbf {x}}} + {{\mathbf {u}}^k}({{\mathbf {x}}})),{m_s})|= \frac{1}{{\sqrt{2\pi } \delta }}{e^{ - \frac{{{{(F({{\mathbf {x}}} + {{\mathbf {u}}}^k({{\mathbf {x}}})) - {m_s})}^2}}}{{2{\delta ^2}}}}} \ge \frac{1}{{\sqrt{2\pi } \delta }}{e^{-\widetilde{M_1}}}\quad \widetilde{M_1} \in (0, +\infty ). \end{aligned}$$

Then, for any $k \in {\mathbb {N}}$, we have

$$\begin{aligned}{{\widetilde{M}}} \ge |P({{{\varvec{\alpha }}}^k},{{{\mathbf {u}}}^k})|= |{{{\varvec{\alpha }}}^k} \cdot K(F({{\mathbf {x}}} + {{{\mathbf {u}}}^k}({{\mathbf {x}}})),{{\varvec{m}}})|\ge \frac{1}{{\sqrt{2\pi } \delta }}{e^{ - \widetilde{{M_1}}}}|{{{\varvec{\alpha }}}^k}|, \end{aligned}$$

where ${{\varvec{m}}} = {({m_s})_{1 \times p}}$. Therefore,

$$\begin{aligned} |{{{\varvec{\alpha }}}^k}|\le \sqrt{2\pi } \delta {{\widetilde{M}}}{e^{\widetilde{{M_1}}}} \buildrel \Delta \over = \overline{{\overline{M}}} < + \infty . \end{aligned}$$

Similarly, we can obtain the uniform boundedness of $\overline{{\mathbf {x}}}^k$, $\overline{{\mathbf {y}}}^k$ and ${{\varvec{\beta }}}^k$. Therefore, there exists $({{\varvec{\alpha }}},\overline{{\mathbf {x}}} ,{{\varvec{\beta }}},\overline{{\mathbf {y}}} ,{{\mathbf {u}}}) \in {{\mathbb {R}}^p} \times {{\mathbb {R}}^p} \times {{\mathbb {R}}^l} \times {{\mathbb {R}}^l} \times {[H_0^\alpha (\Omega )]^2}$ such that

$$\begin{aligned}&{{{\varvec{\alpha }}}^k} {\mathop {\longrightarrow }\limits ^{k}} {{\varvec{\alpha }}},\quad {\overline{{\mathbf {x}}} ^k} {\mathop {\longrightarrow }\limits ^{k}} \overline{{\mathbf {x}}} \quad \text{ in } \quad {{{\mathbb {R}}^p}},\nonumber \\&{{{\varvec{\beta }}}^k} {\mathop {\longrightarrow }\limits ^{k}} {{\varvec{\beta }}},\quad {\overline{{\mathbf {y}}} ^k} {\mathop {\longrightarrow }\limits ^{k}} \overline{{\mathbf {y}}}\quad \text{ in } \quad {{{\mathbb {R}}^l}}, \nonumber \\&{{{\mathbf {u}}}^k} {\mathop {\longrightarrow }\limits ^{k}} {{\mathbf {u}}}\quad \text{ in } \quad {[H_0^\alpha (\Omega )]^2}, \end{aligned}$$

(A25)

because of the fact that ${{{\mathbb {R}}^p}}$, ${{{\mathbb {R}}^l}}$, $H_0^\alpha (\Omega )$ are Banach spaces.

This concludes claim (A7).

Step 2: In this step, we will proof that $({{\varvec{\alpha }}},\overline{{\mathbf {x}}} ,{{\varvec{\beta }}},\overline{{\mathbf {y}}} ,{{\mathbf {u}}})$ is a solution of (A1).

Because ess ${\sup _{\Omega \backslash \Delta }}|F({{\mathbf {x}}})|\le {\overline{M}}$, $\Delta$ is a zero measure set and f is continuous and bounded, by the definition of P in model (40), P is bounded in $\Omega$. Furthermore, since $P( \cdot ) = \sum \nolimits _{1 \le s \le p} {{\alpha _s}} K(F({{\mathbf {x}}} + {{\mathbf {u}}}({{\mathbf {x}}})),{m_s})$ is continuous with respect to ${\varvec{\alpha }}$, then similar to the continuity proof process of ${\mathbf {u}}$ in Theorem 1, we can get the continuity of ${\varvec{\alpha }}$. That is, $\mathrm{CC}(P({{\mathbf {u}}},{{{\varvec{\alpha }}}^k}),Q({{\varvec{\beta }}})) {\mathop {\longrightarrow }\limits ^{k}} \mathrm{CC}(P({{\mathbf {u}}},{{{\varvec{\alpha }}}}),Q({{\varvec{\beta }}})), \mathrm{Var}(P({{\mathbf {u}}},{{{\varvec{\alpha }}}^k})) {\mathop {\longrightarrow }\limits ^{k}} \mathrm{Var}(P({{\mathbf {u}}},{{\varvec{\alpha }}})), \quad \mathrm{Cov}(P({{\mathbf {u}}},{{{\varvec{\alpha }}}^k}), \cdot ) {\mathop {\longrightarrow }\limits ^{k}} \mathrm{Cov}(P({{\mathbf {u}}},{{{\varvec{\alpha }}}}), \cdot )$.

Then, based on the conclusion

$$\begin{aligned} {({{{\varvec{\alpha }}}^k} - {{{\varvec{\alpha }}}^{k + 1}})^2}_{{{\mathbb {R}}^p}} {\mathop {\longrightarrow }\limits ^{k}} 0, \end{aligned}$$

we can obtain that

$$\begin{aligned}&\mathop {\lim }\limits _{k \rightarrow + \infty } - 2\lambda |\Omega |(1 - \mathrm{CC}(P({{\mathbf {u}}},{{{\varvec{\alpha }}}^k}),Q({{\varvec{\beta }}}))) \cdot \mathrm{Var}{^{ - \frac{3}{2}}}(P({{\mathbf {u}}},{{{\varvec{\alpha }}}^k}))\mathrm{Var}{^{ - \frac{1}{2}}}(Q({{\varvec{\beta }}}))\\&\qquad \cdot [\mathrm{Cov}({S_s}({{\mathbf {u}}}),Q({{\varvec{\beta }}}))\mathrm{Var}(P({{\mathbf {u}}},{{{\varvec{\alpha }}}^k})) - \mathrm{Cov}(P({{\mathbf {u}}},{{{\varvec{\alpha }}}^k}),{S_s}({{\mathbf {u}}}))\cdot \mathrm{Cov}(P({{\mathbf {u}}},{{{\varvec{\alpha }}}^k}),Q({{\varvec{\beta }}}))] \\&\qquad - \frac{1}{\theta }(\overline{{\mathbf {x}}} - {{{\varvec{\alpha }}}^k})\\&\quad = - 2\lambda |\Omega |(1 - \mathrm{CC}(P({{\mathbf {u}}},{{\varvec{\alpha }}}),Q({{\varvec{\beta }}}))) \cdot \mathrm{Var}{^{ - \frac{3}{2}}}(P({{\mathbf {u}}},{{\varvec{\alpha }}}))\mathrm{Var}{^{ - \frac{1}{2}}}(Q({{\varvec{\beta }}}))\\&\qquad \cdot [\mathrm{Cov}({S_s}({{\mathbf {u}}}),Q({{\varvec{\beta }}}))\mathrm{Var}(P({{\mathbf {u}}},{{\varvec{\alpha }}})) - \mathrm{Cov}(P({{\mathbf {u}}},{{\varvec{\alpha }}}),{S_s}({{\mathbf {u}}}))\cdot \mathrm{Cov}(P({{\mathbf {u}}},{{\varvec{\alpha }}}),Q({{\varvec{\beta }}}))] \\&\qquad - \frac{1}{\theta }(\overline{{\mathbf {x}}} - {{{\varvec{\alpha }}}})= 0. \end{aligned}$$

Similarly, we can obtain that ${\varvec{\beta }}$ and ${\mathbf {u}}$ satisfy the Euler–Lagrange equation of (A4) and (A6), respectively. Furthermore, since $\overline{{\mathbf {x}}}$ and $\overline{{\mathbf {y}}}$ are the minimizer of (A3) and (A5), respectively, $({{\varvec{\alpha }}},\overline{{\mathbf {x}}} ,{{\varvec{\beta }}},\overline{{\mathbf {y}}} ,{{\mathbf {u}}})$ is a solution of (A1).

Finally, based on the conclusion of step 1 and step 2, we can obtain that the convergence of sequences $({\overline{{\mathbf {x}}} ^k},{\overline{{\mathbf {y}}} ^k},{{{\mathbf {u}}}^k})$ obtained from subproblems (53)–(55). $\square$

Remark 5

${\varvec{\alpha }}$ and ${\varvec{\beta }}$ are introduced to prove the convergence of sequences $({\overline{{\mathbf {x}}} ^k},{\overline{{\mathbf {y}}} ^k},{{{\mathbf {u}}}^k})$ obtained from subproblems (53)–(55), which is not used in numerical implementation because of the fact that $\overline{{\mathbf {x}}} = {{\varvec{\alpha }}}$ and $\overline{{\mathbf {y}}} = {{\varvec{\beta }}}$ are the minimizer of (A3) and (A5), respectively.

Appendix B: The discretization of $R_0({\mathbf {u}})$

For ${{\mathbf {x}}} = ({x_1},{x_2}) \in \Omega ,i = 1,2$, $\alpha >2$, and function ${g_1}:\Omega \rightarrow {\mathbb {R}}$, define

where ${\nabla ^\alpha }{{\mathbf {u}}} = {(\frac{{{\partial ^\alpha }{u_i}}}{{\partial x_j^\alpha }})_{2 \times 2}},\Phi (s) = \int _0^{ + \infty } {{t^{s - 1}}{e^{ - t}}} \mathrm{d}t,[ \cdot ]$ is a round-down function, $g_1^{(1)}({{\mathbf {x}}},t) = {g_1}(t,{x_2}),g_1^{(2)}({{\mathbf {x}}},t) = {g_1}({x_1},t)$, and $|{\nabla ^\alpha }{{\mathbf {u}}}|= \sqrt{\sum \nolimits _{i,j = 1}^2 {{{\left( \frac{{{\partial ^\alpha }{u_i}}}{{\partial x_j^\alpha }}\right) }^2}} }$.

Using Grunwald approximation (Hou et al. 2017; Tian et al. 2015), $\frac{{{\partial ^\alpha }{g_1}({{\mathbf {x}}})}}{{\partial x_i^\alpha }},\frac{{{\partial ^{{\alpha _*}}}{g_1}({{\mathbf {x}}})}}{{\partial x_i^{{\alpha _*}}}}$ ($i=1,2$) are discretized by

$$\begin{aligned} \frac{{{\partial ^\alpha }{g_1}({{{\mathbf {P}}}_{p,q}})}}{{\partial x_i^\alpha }} = \delta _{i - }^\alpha {g_1}({{{\mathbf {P}}}_{p,q}}) + O({h_i}),\frac{{{\partial ^{{\alpha _*}}}{g_1}({{{\mathbf {P}}}_{p,q}})}}{{\partial x_i^{{\alpha _*}}}} = \delta _{i + }^\alpha {g_1}({{{\mathbf {P}}}_{p,q}}) + O({h_i}), \end{aligned}$$

where $\delta _{1 - }^\alpha {g_1}_{i,j} = \frac{1}{{h_1^\alpha }}\sum \nolimits _{l = 1}^{i + 1} {\rho _l^{(\alpha )}} {g_1}_{i - l + 1,j},\delta _{1 + }^\alpha {g_1}_{i,j} = \frac{1}{{h_1^\alpha }}\sum \nolimits _{l = 1}^{{N_1} - i + 2} {\rho _l^{(\alpha )}} {g_1}_{i + l - 1,j}, \delta _{2 - }^\alpha {g_1}_{i,j} = \frac{1}{{h_2^\alpha }}\sum \nolimits _{m = 1}^{j + 1} {\rho _m^{(\alpha )}} {g_1}_{i,j - m + 1}, \delta _{2 + }^\alpha {g_1}_{i,j} = \frac{1}{{h_2^\alpha }}\sum \nolimits _{m = 1}^{{N_2} - j + 2} {\rho _m^{(\alpha )}} {g_1}_{i,j + m - 1}, \rho _0^{(\alpha )} = 1,\rho _l^{(\alpha )} = (1 - \frac{{1 + \alpha }}{l})\rho _{l - 1}^{(\alpha )}$.