prior knowledge

$$ \begin{aligned} \dot{D}_q^0\left(t,\xi\right) \thickapprox \sum_{k=1}^K \sum_{l=1}^L a_{qkl} X_k\left(\xi\right) & T_l\left(t-t_k\right) \quad;\quad a_{qkl}~\sim~\overline{D} \\ \int X_k\left(\mathbf{\xi}\right) ~d\mathbf{\xi} = \Delta \xi_1 \Delta \xi_2 \quad &; \quad \int ~ T_l\left(t-t_k\right) ~ ~dt = 1 \end{aligned} $$

$$ \begin{aligned} f(\mathbf{x}+\Delta\mathbf{x})& \approx f(\mathbf{x}) + J(\mathbf{x})\Delta \mathbf{x} +\frac{1}{2} \Delta\mathbf{x}^\mathrm{T} H(\mathbf{x}) \Delta\mathbf{x}\\ & \mathbf{J}\,\textit { is the Jacobian matrix }\\ & \mathbf{H}\,\textit { is the Hessian matrix } \\ \end{aligned} $$

$$ \begin{aligned} H(f) &= \begin{bmatrix} \dfrac{\partial^2 f}{\partial x_1^2} & \dfrac{\partial^2 f}{\partial x_1\,\partial x_2} & \cdots & \dfrac{\partial^2 f}{\partial x_1\,\partial x_n} \\[2.2ex] \dfrac{\partial^2 f}{\partial x_2\,\partial x_1} & \dfrac{\partial^2 f}{\partial x_2^2} & \cdots & \dfrac{\partial^2 f}{\partial x_2\,\partial x_n} \\[2.2ex] \vdots & \vdots & \ddots & \vdots \\[2.2ex] \dfrac{\partial^2 f}{\partial x_n\,\partial x_1} & \dfrac{\partial^2 f}{\partial x_n\,\partial x_2} & \cdots & \dfrac{\partial^2 f}{\partial x_n^2} \end{bmatrix}. \end{aligned} $$

$$ \vec{f}=\begin{pmatrix}{f_1(\xi_1,\xi_2) \\ f_2(\xi_1,\xi_2) }\end{pmatrix} \iff \mathbf{H}\vec{f}=\begin{pmatrix}{\mathbf{H}f_1(\xi_1,\xi_2) \\ \mathbf{H}f_2(\xi_1,\xi_2) }\end{pmatrix} = \mathbf{0} \\ ~\\ \begin{bmatrix} \dfrac{\partial^2 f_1}{\partial \xi_1^2} & \dfrac{\partial^2 f_1}{\partial \xi_1\,\partial \xi_2} \\ \dfrac{\partial^2 f_1}{\partial \xi_2\,\partial \xi_1} & \dfrac{\partial^2 f_1}{\partial \xi_2^2} \\ \end{bmatrix} = \mathbf{0}; \quad \begin{bmatrix} \dfrac{\partial^2 f_2}{\partial \xi_1^2} & \dfrac{\partial^2 f_2}{\partial \xi_1\,\partial \xi_2} \\ \dfrac{\partial^2 f_2}{\partial \xi_2\,\partial \xi_1} & \dfrac{\partial^2 f_2}{\partial \xi_2^2} \\ \end{bmatrix} = \mathbf{0} \\ ~ \\ \textit{suppose}\, f_1,\,f_2\,\textit{independent, so we can quantify as follows(not unique)} \\ ~ \\ \iff \left[ \left(\dfrac{\partial^2 f_j}{\partial \xi_1^2}\right)^2 + \left(\dfrac{\partial^2 f_j}{\partial \xi_1\,\partial \xi_2}\right)^2 + \left(\dfrac{\partial^2 f_j}{\partial \xi_2\,\partial \xi_1}\right)^2 + \left(\dfrac{\partial^2 f_j}{\partial \xi_2^2}\right)^2 \right] = 0 \\ ~ \\ ~ \\ \mathbb{R}_\xi f_j(\xi) \overset{def}{\iff} \left(\frac{\partial^2 f_j(\xi)}{\partial \xi^2}\right)^2 \overset{def}{\iff} \left[\left(\frac{\partial^2 f_j(\xi)}{\partial \xi^2_1}\right)^2+2\left(\frac{\partial^2 f_j(\xi)}{\partial\xi_1\partial\xi_2}\right)^2+\left(\frac{\partial^2 f_j(\xi) }{\partial \xi^2_2}\right)^2\right] \\ \mathbb{R}_\tau f_j(t) \overset{def}{\iff} \left(\frac{\partial^2 f_j(t)}{\partial t^2}\right)^2 \\ \textit {if base function of time choose trianle instead 4-oder B-spine, then: }\\ \mathbb{R}_\tau f_j(t) \overset{def}{\iff} \left(\frac{\partial f_j(t)}{\partial t}\right)^2 \\ $$

$$ \begin{aligned} \nabla^2 \dot{D}(t,\xi)+e_{\xi} =0 \overset{1}{\iff} & \mathbf{S_{\xi }a+e_{\xi }=0} \overset{2}{\iff} \mathbf{G_{\xi }=S_{\xi }^TS_{\xi }} ; \quad \mathbf{G}_{\xi},\mathbf{S}_\xi: M \times M\\ \frac{\partial^2}{\partial t^2}\dot{D}(t,\xi)+e_{\tau} =0 \iff & \mathbf{S_{\tau}a+e_{\tau}=0} \iff \mathbf{G_{\tau}=S_{\tau}^TS_{\tau}} ; \quad \mathbf{G}_{\tau}, \mathbf{S}_\tau: M \times M \\ \end{aligned} $$

$$ \begin{aligned} \xi &= \xi^1 ~,~ \xi^2 \cdots \xi^k \cdots \xi^{k-1} ~,~ \xi^K \quad ? \\ t &= t^1 ~,~ ~ t^2 ~ \cdots t^l ~ \cdots ~ t^{l-1} ~,~ t^L \quad ? \end{aligned} $$

$$ \begin{aligned} & \qquad \dot{D}_q^0\left(t,\xi\right) \thickapprox \sum_{k=1}^K \sum_{l=1}^L a_{qkl} X_k\left(\xi\right)T_l\left(t-t_k\right) \\ \int X_k& \left(\mathbf{\xi}\right) ~d\mathbf{\xi} = \Delta \xi_1 \Delta \xi_2 \quad ; \quad \int ~ T_l\left(t-t_k\right) ~ ~dt = 1 \quad;\quad \int M_{4,j+2}(s)~ds=\frac{1}{4};\\ m_q(\xi)&=\int_t \sum_{k=1}^K \sum_{l=1}^L a_{qkl}X_k(\xi)T_l(t-\tau_k)~dt ~ = \sum_{k=1}^K \sum_{l=1}^L a_{qkl}X_k(\xi) \\ m_q(t) &=\int_\Sigma \sum_{k=1}^K \sum_{l=1}^L a_{qkl}X_k(\xi)T_l(t-\tau_k)~d\xi = \sum_{k=1}^K \sum_{l=1}^L a_{qkl} \Delta \xi_1 \Delta \xi_2 T_l(t-\tau_k) \\ \textit{rewrite} & \\ m_{ql}(\xi)&=\sum_{k=1}^K a_{qkl}X_k(\xi) \quad;\quad m_{qk}(t)=\Delta \xi_1 \Delta \xi_2 \sum_{l=1}^L a_{qkl}T_l(t-\tau_k) \\ \end{aligned} $$

$$ \begin{aligned} X_k(\xi)&=16\Delta \xi_1 \Delta \xi_2 M_{4,i+2}(\xi_1)M_{4,j+2}(\xi_2) \,;\quad k=(j-1)*I+i \\ T_l(t)&=4M_{4,l+2}(t) \\ \end{aligned} \\ $$

$$ \begin{aligned} \nabla^2 m_{ql}(\xi)+e_{\xi} =0\,;\quad\xi &= \xi^1 ~,~ \xi^2 \cdots \xi^k \cdots \xi^{k-1} ~,~ \xi^K \quad \\ \frac{\partial^2}{\partial t^2}m_{qk}(t)+e_{\tau} =0\,;\quad t &= t^1 ~,~ ~ t^2 ~ \cdots t^l ~ \cdots ~ t^{l-1} ~,~ t^L \quad\\ \end{aligned} $$

$$ \begin{matrix} \nabla^2 m_{ql}(\xi)+e_{\xi} = 0 \iff & \sum_{k=1}^Ka_{qkl} \nabla^2 X_k\left(\xi\right) & \iff \mathbf{S}_\xi^{ql} ~ \mathbf{a}^{ql} + \mathbf{e}_\xi =\mathbf{0} \\ \frac{\partial^2}{\partial t^2}m_{qk}(t)+e_{\tau} \iff & \sum_{l=1}^L a_{qkl} \Delta \xi_1\Delta \xi_2 \frac{\partial^2}{\partial t^2}T_l\left(t-\tau_k\right) & \iff \mathbf{S}_\tau^{qk} ~ \mathbf{a}^{qk} + \mathbf{e}_\tau=\mathbf{0} \\ \end{matrix} $$

$$ \begin{matrix} \mathbf{G}_{ij}^{ql} &\mapsto & \mathbf{G}_{_{IJ}} \\ \mathbf{a}_i^{ql}\mathbf{G}_{ij}^{ql}\mathbf{a}_j^{ql} &=& \mathbf{a}_{_I} \mathbf{G}_{_{IJ}} \mathbf{a}_{_J} \\ \mathbf{G}_{ij}^{ql} & \mapsto& \mathbf{G}_{_{(q,i,l)\,,\,(q,j,l)}} \end{matrix} $$

$$ \mathbf{G}_\xi =\bigoplus_{\overset{q=1}{l=1}}^{Q,L}G_{\xi}^{q\,l} ; \qquad \vec{\mathbf{a}} \textit{ arranged as }\, a_{qlk} \\ \mathbf{G}_\tau=\bigoplus_{\overset{q=1}{k=1}}^{Q,K}G_{\tau}^{qk} ; \qquad \vec{\mathbf{a}} \textit{ arranged as }\, a_{qkl} $$

$$ \begin{aligned} {\color{red}\because} \quad & \mu \int_\Sigma \int_t ~ \dot{D}_q^0(t,\xi) ~dt~d\xi \thickapprox \mu \int_\Sigma \int_t ~ \sum_{k=1}^K \sum_{l=1}^L a_{qkl}X_k(\xi)T_l(t-t_k) ~dt~d\xi \\ &=\sum_{k=1}^K \sum_{l=1}^L \mu a_{qkl} \Delta \xi_1 \Delta \xi_2 \iff a_{qkl} ~ \sim \overline{D} \iff \sum_{k=1}^K \sum_{l=1}^L \mu \overline{D} \Delta \xi_1 \Delta \xi_2 \\ {\color{red}\therefore} \quad & \mathbf{S} \quad \textit{can be assigned as:}\quad \textit{diag(1)},\quad \textit{diag(-1 2 -1)},\quad \textit{diag(-1 -1 4 -1 -1)} \cdots\\ \textit{at}&\textit{least diag(1) can be reasonable, to minimize moment.} \end{aligned} $$

$$ \begin{aligned} \int_\Sigma \mathbb{R}_\xi~ \left(\int_t \sum_{k=1}^K \sum_{l=1}^L a_{qkl}X_k\left(\xi\right)T_l\left(t-\tau_k\right)~dt\right) ~d\xi = r_\xi \iff \mathbf{G}_{\xi}: M \times M\\ \int_t \mathbb{R}_\tau~\left(\int_\Sigma\sum_{k=1}^K\sum_{l=1}^L a_{qkl}X_k\left(\xi\right)T_l\left(t-\tau_k\right)~d\xi\right) ~dt= r_\tau \iff \mathbf{G}_{\tau}: M \times M \end{aligned} $$

$$ \begin{aligned} \int_\Sigma \int_t \mathbb{R}_\xi~ \left( \sum_{k=1}^K \sum_{l=1}^L a_{jkl}X_k\left(\xi\right)T_l\left(t-\tau_k\right)\right)~dt ~d\xi = r_\xi \iff \mathbf{G}_{\xi}: M \times M\\ \int_t \int_\Sigma \mathbb{R}_\tau~\left( \sum_{k=1}^K \sum_{l=1}^L a_{jkl}X_k\left(\xi\right)T_l\left(t-\tau_k\right)\right)~d\xi ~dt= r_\tau \iff \mathbf{G}_{\tau}: M \times M \end{aligned} $$

$$ \begin{aligned} r_\xi&= \sum_{j=1}^2\int_\Sigma\int_t \begin{pmatrix} \left( \sum_{k=1}^K\sum_{l=1}^L a_{jkl} \partial^2 \frac{X_k(\xi) }{\partial \xi_1^2} T_l(t-\tau_k) \right)^2 \\ 2 \left( \sum_{k=1}^K\sum_{l=1}^L a_{jkl} \partial^2 \frac{X_k(\xi) }{\partial \xi_1 \partial \xi_2} T_l(t-\tau_k) \right)^2 \\ \left( \sum_{k=1}^K\sum_{l=1}^L a_{jkl} \partial^2 \frac{X_k(\xi) }{\partial \xi_2^2} T_l(t-\tau_k) \right)^2 \\ \end{pmatrix} ~dt~d\xi \\ &=\sum_{j=1}^2\sum_{k=1}^K\sum_{l=1}^L\sum_{p=1}^K\sum_{q=1}^L a_{jkl} \int_\Sigma\int_t \begin{pmatrix} \frac{\partial^2 x_k}{\partial^2 \xi_1 }T_l(t-\tau_k)\frac{\partial^2 x_p}{\partial^2 \xi_1 }T_q(t-\tau_p) \\ 2\frac{\partial^2 x_k}{\partial \xi_1\xi_2}T_l(t-\tau_k)\frac{\partial^2 x_p}{\partial \xi_1\xi_2}T_q(t-\tau_p) \\ \frac{\partial^2 x_k}{\partial^2 \xi_2 }T_l(t-\tau_k)\frac{\partial^2 x_p}{\partial^2 \xi_2 }T_q(t-\tau_p) \\ \end{pmatrix} ~dt~d\xi~a_{jpq} \\ &=\sum_{j=1}^2\sum_{k=1}^K\sum_{l=1}^L\sum_{p=1}^K\sum_{q=1}^L a_{jkl} \int_\Sigma \begin{pmatrix} \frac{\partial^2 x_k}{\partial^2 \xi_1 }\frac{\partial^2 x_p}{\partial^2 \xi_1 } \\ 2\frac{\partial^2 x_k}{\partial \xi_1\xi_2}\frac{\partial^2 x_p}{\partial \xi_1\xi_2} \\ \frac{\partial^2 x_k}{\partial^2 \xi_2 }\frac{\partial^2 x_p}{\partial^2 \xi_2 } \\ \end{pmatrix}~d\xi~\int_t~T_l(t-\tau_k)T_q(t-\tau_p) ~dt~a_{jpq} \\ &=\sum_{j=1}^2\sum_{\overset{k_d=1}{k_s=1}}^{\overset{k_d=K_d}{k_s=K_s}}\sum_{l=1}^L\sum_{\overset{p_d=1}{p_s=1}}^{\overset{p_d=K_d}{p_s=K_s}}\sum_{q=1}^L a_{jkl} \\ &\qquad\qquad\times \int_{\xi_1} \int_{\xi_2} \begin{pmatrix} \frac{\partial^2 N_{k_s}(\xi_1)}{\partial^2 \xi_1 }\frac{\partial^2 N_{p_s}(\xi_1)}{\partial^2 \xi_1 }N_{k_d}(\xi_2)N_{p_d}(\xi_2) \\ 2\frac{\partial N_{k_s}(\xi_1)}{\partial \xi_1}\frac{\partial N_{p_s}(\xi_1)}{\partial \xi_1}\frac{\partial N_{k_d}(\xi_1)}{\partial \xi_2}\frac{\partial N_{p_d}(\xi_2)}{\partial \xi_2} \\ N_{k_s}(\xi_1)N_{p_s}(\xi_1)\frac{\partial^2 N_{k_d}(\xi_2)}{\partial^2 \xi_2 }\frac{\partial^2 N_{p_d}}{\partial^2 \xi_2 } \\ \end{pmatrix}~d\xi_1~d\xi_2~\\ &\qquad\qquad\times\int_t~T_l(t-\tau_k)T_q(t-\tau_p) ~dt~a_{jpq} \\ &=\sum_{j=1}^2\sum_{k=1}^{K}\sum_{l=1}^L\sum_{p=1}^K\sum_{q=1}^L a_{jkl} R_{klpq} a_{jpq} \qquad, k=(k_d-1)\times K_s+k_s;p=(p_d-1)\times K_s+p_s \\ \end{aligned} $$

$$ \dot{D}_q^0(t,\xi) \thickapprox \sum_{k=1}^K \sum_{l=1}^L a_{qkl} X_k(\xi)T_l(t-\tau_k)=\sum_{k=1}^K \sum_{l=1}^L a_{qkl}\widetilde{D}_{kl}(\xi,t) \\ \nabla^2 \dot{D}(t,\xi)+e =0 \overset{1}{\iff} \mathbf{Sa+e=0} \overset{2}{\iff} \mathbf{G=S^TS} \\ \mathbf{S}\mathbf{a}=\mathbf{0} \quad;\quad \begin{pmatrix} \mathbf{S}_{kl} & 0 \\ 0 & \mathbf{S}_{kl} \end{pmatrix} \begin{pmatrix} \mathbf{a}_{1_{kl}} \\ \mathbf{a}_{2_{kl}} \end{pmatrix} \quad;\quad S_{ij}a_j=S_{qkl,q'k'l'}a_{q'k'l'} \\ $$

$$ \begin{aligned} \mathbb{P}=\mathbf{a}^T\mathbf{S}^T\mathbf{S}\mathbf{a} &= \sum_{q=1}^2\sum_{k'=1}^K\sum_{l'=1}^L \left[\sum_{k=1}^K\sum_{l=1}^L a_{qkl} \nabla^2 \widetilde{D}_{kl}(\xi_{k'},t_{l'}) \right]^2 \\ \mathbb{R}_{ij}=\mathbf{a}^T\mathbf{G}\mathbf{a}&=\sum_{q=1}^2 ~ \int_\xi ~ \int _t \left[ \sum_{k=1}^K \sum_{l=1}^L a_{qkl} \nabla_{ij}^2 \widetilde{D}_{kl}(\;\,\xi\;,\;t\;)\right]^2 ~ dt d\xi \\ \nabla_{ij}^2 &= \begin{bmatrix} \dfrac{\partial^2 }{\partial \xi_1^2} & \dfrac{\partial^2 }{\partial \xi_1\,\partial \xi_2} \\ \dfrac{\partial^2 }{\partial \xi_2\,\partial \xi_1} & \dfrac{\partial^2 }{\partial \xi_2^2} \\ \end{bmatrix} \nabla_{t}^2=\dfrac{\partial^2 }{\partial t^2} .or. \dfrac{d^2 }{d t^2} \\ \therefore \quad & \\ \mathbb{P}_{ij} & \overset{def}{\iff} \sum_{q=1}^2\sum_{k=1'}^K\sum_{l=1'}^L \left[\sum_{k=1}^K\sum_{l=1}^L a_{qkl} \nabla_{ij}^2 \widetilde{D}_{kl}(\xi_{k'},t_{l'}) \right]^2 \\ \end{aligned} $$

$$ \begin{aligned} \mathbb{R}_{ij}&=\sum_{q=1}^2 ~ \int_\xi ~ \int _t \left[ \sum_{k=1}^K \sum_{l=1}^L a_{qkl} \nabla_{ij}^2 \widetilde{D}_{kl}(\;\,\xi\;,\;t\;)\right]^2 ~ dt d\xi \\ \mathbb{P}_{ij} &= \sum_{q=1}^2\sum_{k=1'}^K\sum_{l=1'}^L \left[\sum_{k=1}^K\sum_{l=1}^L a_{qkl} \nabla_{ij}^2 \widetilde{D}_{kl}(\xi_{k'},t_{l'}) \right]^2 \\ \mathbb{R}_{ij}&=\sum_{q=1}^2 ~ \int_\xi ~ \int _t \left[\sum_{k=1}^K\sum_{l=1}^L \sum_{\bar{k}=1}^K \sum_{\bar{l}=1}^L a_{qkl} \nabla_{ij}^2 \widetilde{D}_{kl}(\;\,\xi\;,\;t\;) \cdot a_{q\bar{k}\bar{l}} \nabla_{ij}^2 \widetilde{D}_{\bar{k}\bar{l}}(\;\,\xi\;,\;t\;) \right] ~ dt d\xi \\ \mathbb{P}_{ij} &= \sum_{q=1}^2\sum_{k=1'}^K\sum_{l=1'}^L \left[\sum_{k=1}^K\sum_{l=1}^L \sum_{\bar{k}=1}^K \sum_{\bar{l}=1}^L a_{qkl} \nabla_{ij}^2 \widetilde{D}_{kl}(\xi_{k'},t_{l'}) \cdot a_{q\bar{k}\bar{l}} \nabla_{ij}^2 \widetilde{D}_{\bar{k}\bar{l}}(\xi_{k'},t_{l'}) \right] \\ \mathbb{G}_{kl,\bar{k}\bar{l}}^{q,R_{ij}} &=~ \int_\xi ~ \int _t \left[ \nabla_{ij}^2 \widetilde{D}_{kl}(\;\,\xi\;,\;t\;) \cdot \nabla_{ij}^2 \widetilde{D}_{\bar{k}\bar{l}}(\;\,\xi\;,\;t\;) \right] ~ dt d\xi \\ \mathbb{G}_{kl,\bar{k}\bar{l}}^{q,P_{ij}} &=\sum_{k=1'}^K\sum_{l=1'}^L \left[ \nabla_{ij}^2 \widetilde{D}_{kl}(\xi_{k'},t_{l'}) \cdot \nabla_{ij}^2 \widetilde{D}_{\bar{k}\bar{l}}(\xi_{k'},t_{l'}) \right] \\ \end{aligned} $$

$$ \begin{aligned} m_q(\xi)&=\int_t \sum_{k=1}^K \sum_{l=1}^L a_{qkl}X_k(\xi)T_l(t-\tau_k)~dt ~ = \sum_{l=1}^L \bbox[2pt,#F5A9F2]{\sum_{k=1}^K a_{qkl}X_k(\xi)} = \sum_{l=1}^L \bbox[2pt,#F5A9F2]{m_{ql}(\xi)}\\ m_q(t) &=\int_\Sigma \sum_{k=1}^K \sum_{l=1}^L a_{qkl}X_k(\xi)T_l(t-\tau_k)~d\xi = \sum_{k=1}^K \bbox[2pt,#F5A9F2]{\sum_{l=1}^L a_{qkl} \Delta \xi_1 \Delta \xi_2 T_l(t-\tau_k)} = \sum_{k=1}^K \bbox[2pt,#F5A9F2] {m_{qk}(t)}\\ \mathbb{R}_{\xi,ij} &=\sum_{q=1}^2 \sum_{l=1}^L ~ \int_\xi ~ \left[ \nabla_{ij}^2 \sum_{k=1}^K a_{qkl} X_k(\xi) \right]^2 d\xi \\ &=\sum_{q=1}^2 \sum_{l=1}^L ~ \int_\xi ~ \left[ \sum_{k=1}^K \sum_{\bar{k}=1}^K a_{qkl} \nabla_{ij}^2 X_k(\xi) \cdot \nabla_{ij}^2 X_\bar{k}(\xi) a_{q\bar{k}l} \right] d\xi \\ {\color{#8A0808} \mathbb{G}_{_{k,\bar{k}}}^{\mathbb{R}^{^{q\,l}}_{\xi,ij}}}&{\color{#8A0808} = \int_\xi ~ \nabla_{ij}^2 X_k(\xi) \cdot \nabla_{ij}^2 X_\bar{k}(\xi) ~ d\xi}\\ \mathbb{P}_{\xi,ij} &=\sum_{q=1}^2 \sum_{l=1}^L ~ \sum_{k'=1}^K \left[ \sum_{k=1}^K a_{qkl} \nabla_{ij}^2 X_k(\xi_{k'}) \right]^2 \\ &=\sum_{q=1}^2 \sum_{l=1}^L ~ \sum_{k'=1}^K \left[ \sum_{k=1}^K \sum_{\bar{k}=1}^K a_{qkl} \nabla_{ij}^2 X_k(\xi_{k'}) \cdot \nabla_{ij}^2 X_\bar{k}(\xi_{k'}) a_{q\bar{k}l} \right] \\ {\color{#0404B4} \mathbb{G}_{_{k,\bar{k}}}^{\mathbb{P}^{^{q\,l}}_{\xi,ij}}}& {\color{#0404B4} = \sum_{k'=1}^K \nabla_{ij}^2 X_k(\xi_{k'}) \cdot \nabla_{ij}^2 X_\bar{k}(\xi_{k'})} \\ \mathbb{R}_{\tau,t} &=\sum_{q=1}^2 \sum_{k=1}^K ~ \int_t ~ \left[ \nabla_{t}^2 \sum_{l=1}^L a_{qkl} \Delta\xi_1 \Delta\xi_2 T_l(t-\tau_k)\right]^2 dt \\ &=\sum_{q=1}^2 \sum_{k=1}^K ~ \int_t ~ \left[ \sum_{l=1}^L \sum_{\bar{l}=1}^L a_{qkl} \nabla_{t}^2 T_l(t-\tau_k) \cdot \nabla_{t}^2 T_\bar{l}(t-\tau_k) a_{qk\bar{l}} \right] dt \cdot \Delta \xi_1^2 \Delta \xi_2^2 \\ {\color{#8A0808} \mathbb{G}_{l,\bar{l}}^{\mathbb{R}^{q\,k}_{\tau,t} } }&{\color{#8A0808} = \Delta \xi_1^2 \Delta \xi_2^2 \cdot \int_t \nabla_{t}^2 T_l(t-\tau_k) \cdot \nabla_{t}^2 T_\bar{l}(t-\tau_k)~dt} \\ \mathbb{P}_{\tau,t} &=\sum_{q=1}^2 \sum_{k=1}^K ~ \sum_{l'=1}^L \left[ \sum_{l=1}^L a_{qkl} \Delta\xi_1 \Delta\xi_2 \nabla_{t}^2 T_l(t_{l'}-\tau_k) \right]^2 \\ &=\sum_{q=1}^2 \sum_{k=1}^K ~ \sum_{l'=1}^L \left[ \sum_{l=1}^L \sum_{\bar{l}=1}^L a_{qkl} \nabla_{t}^2 T_l(t_{l'}-\tau_k) \cdot \nabla_{t}^2 T_\bar{l}(t_{l'}-\tau_k) a_{qk\bar{l}} \right] \cdot \Delta\xi_1^2 \Delta\xi_2^2 \\ {\color{#0404B4} \mathbb{G}_{l,\bar{l}}^{\mathbb{P}^{q\,k}_{\tau,t} }} &{\color{#0404B4} =\Delta\xi_1^2 \Delta\xi_2^2 \cdot \sum_{l'=1}^L \nabla_{t}^2 T_l(t_{l'}-\tau_k) \cdot \nabla_{t}^2 T_\bar{l}(t_{l'}-\tau_k)} \\ \end{aligned} $$