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| <p> $K=f_{K}(v)$ </p> |
| <p> $sim(v_{i},v_{j})=v_{i}^{T}.v_{j}/||v_{i}||||v_{j}||$ </p> |
| <p> $B(M^{p}):=\{1\text{ if }M^{p}>k,0\text{ otherwise}\}$ </p> |
| <p> $v^{*}:=\operatorname*{arg\,min}_{v}E_{x_{0},{\epsilon}\sim N(0,I)}\\ |
| \|{\epsilon}-{\epsilon}_{\theta}(x_{t},t,[c_{\theta}(y),v^{*}]\|^{2}$ </p> |
| <p> $\{v_{b}^{n}\}_{b=1}^{B},_{n=1}^{N}$ </p> |
| <p> $\begin{gathered}L_{PromptCL}=\frac{1}{N}\frac{1}{B}\sum_{\eta=1}^{N}\sum_{i=1}% |
| ^{B}{l^{\eta}_{i,j\in B}},\qquad L_{PromptCL}^{adj}=\frac{1}{NM}\frac{1}{B}% |
| \sum_{\eta=1}^{NM}\sum_{i=1}^{B}{l^{\eta}_{i,j\in B}}\end{gathered}$ </p> |
| <p> $(\tau,\gamma)$ </p> |
| <p> $[v^{*},\ldots,v^{\&}]=[c_{\theta}(p^{*}),\ldots,c_{\theta}(p^{\&})]$ </p> |
| <p> $[y,p^{*},\ldots,p^{\&}]$ </p> |
| <p> $c_{\phi}$ </p> |
| <p> $l^{\eta}_{i,j\in B}=-log(\frac{exp(sim(v^{\eta}_{i},v^{\eta}_{j}))/\tau}{\sum_% |
| {\eta=1}^{N}\sum_{j=1,j\neq{i}}^{B}exp(sim(v^{\eta}_{i},v^{\eta}_{j})/\tau)})$ </p> |
| <p> $\eta\in N$ </p> |
| <p> $\tau=10^{-4}$ </p> |
| <p> $J^{\tau}(\mathbf{\chi}^{k+1},T)$ </p> |
| <p> $q_{1}=1,\ q_{2}=100$ </p> |
| <p> $d=0.1l$ </p> |
| <p> $\tilde{\chi}_{s}$ </p> |
| <p> $\Phi_{h}=(q_{1}-q_{2})G_{\tau}\ast(T_{h}-T^{\ast}_{h})+\gamma\sqrt{\frac{\pi}{% |
| \tau}}G_{\tau}\ast(1-2\chi_{h})+(\kappa_{1}-\kappa_{2})G_{\tau}\ast(\frac{\xi}% |
| {2}\nabla T_{h}\cdot\nabla T_{h}+\nabla T_{h}\cdot\nabla T_{h}^{\ast}).$ </p> |
| <p> $\kappa_{1}=5,\ 10,\ 15$ </p> |
| <p> $\tau=3\times 10^{-5}$ </p> |
| <p> $\Omega\in\mathbb{R}^{d}\ (d=2,3)$ </p> |
| <p> $\mathbf{\chi}_{h}$ </p> |
| <p> $\int_{\Omega}\kappa(\chi)\nabla T\cdot\nabla T\ d\textbf{x}$ </p> |
| <p> $y=1/2$ </p> |
| <p> $\Omega_{1}\subset\Omega$ </p> |
| <p> $(\ref{ad})$ </p> |
| <p> $\frac{q_{1}}{q_{2}}$ </p> |
| <p> $z=0,\ y=1/2$ </p> |
| <p> $\Phi_{h}$ </p> |
| <p> $T^{\ast}_{h}$ </p> |
| <p> $J^{\tau}(\chi,T)=\int_{\Omega}q(\chi)T\ d\textbf{x}+\frac{\xi}{2}\int_{\Omega}% |
| \kappa(\chi)\nabla T\cdot\nabla T\ d\textbf{x}+\gamma\sqrt{\frac{\pi}{\tau}}% |
| \int_{\Omega}\chi G_{\tau}\ast(1-\chi)\ d\textbf{x},$ </p> |
| <p> $\displaystyle\ \ \ \ +\int_{\Omega}\kappa(\chi)\nabla T^{k}\cdot\nabla T^{*k}% |
| \ d\textbf{x}-\int_{\Omega}q(\chi)T^{*k}\ d\textbf{x}.$ </p> |
| <p> $\displaystyle\int_{\Omega}\chi G_{\tau}\ast(1-\chi)\ d\textbf{x}=$ </p> |
| <p> $J^{\tau}(\tilde{\chi}_{s},\tilde{T}_{s})>J^{\tau}(\chi^{k},T^{k})$ </p> |
| <p> $\kappa_{1}=10,\ \kappa_{2}=1,\ q_{1}=1,\ q_{2}=100,\ \tau=1\times 10^{-4}$ </p> |
| <p> $\displaystyle:=\big{\{}p\in\mathcal{N},\Phi^{k}\leq\sigma_{1},\chi^{k}(p)=0% |
| \big{\}},$ </p> |
| <p> $\tilde{T}_{s}$ </p> |
| <p> $\kappa_{1}=40,\ \kappa_{2}=1$ </p> |
| <p> $\arg\min_{\chi\in\mathcal{B}}\tilde{J}^{\tau,k}(\chi)\in\arg\min_{\chi\in% |
| \mathcal{H}}\tilde{J}^{\tau,k}(\chi).$ </p> |
| <p> $\tilde{B}\subset B$ </p> |
| <p> $A_{1}=\{p\in A:\phi_{A}(p)\leq\sigma_{1}\}\ \ \ \ B_{1}=\{p\in B:\phi_{B}(p)% |
| \leq\sigma_{2}\}$ </p> |
| <p> $H^{1}_{\Gamma_{D}}(\Omega)=\{v\in H^{1}(\Omega)\ |\ v|_{\Gamma_{D}}=0\}$ </p> |
| <p> $(\ref{and})$ </p> |
| <p> $\displaystyle\left\{\begin{aligned} -\nabla\cdot(\kappa(\chi^{k})\nabla T)-q(% |
| \chi^{k})&=0,\ \ &&\rm in\ \ \Omega,\\ |
| T&=0,\ \ &&\rm on\ \ \Gamma_{D},\\ |
| \kappa(\chi^{k})\nabla T\cdot\mathbf{n}&=0,\ \ &&\rm on\ \ \Gamma_{N},\end{% |
| aligned}\right.$ </p> |
| <p> $\displaystyle\mathcal{L}^{\tau,k}_{\chi^{k}}(\chi)$ </p> |
| <p> $\sigma_{1}=\tilde{\phi}_{A}^{k}(n_{s})$ </p> |
| <p> $tol>0$ </p> |
| <p> $\displaystyle\int_{\Omega}(q(\chi^{k})T^{k})\ d\textbf{x}+\frac{\xi}{2}\int_{% |
| \Omega}\kappa(\chi^{k})\nabla T^{k}\cdot\nabla T^{k}\ d\textbf{x}+\gamma\sqrt{% |
| \frac{\pi}{\tau}}\int_{\Omega}\chi^{k}G_{\tau}\ast(1-\chi^{k})\ d\textbf{x}.$ </p> |
| <p> $T\in Q(\chi)$ </p> |
| <p> $\frac{\xi}{2}\int_{\Omega}\kappa(\chi)\nabla T\cdot\nabla T\ d\textbf{x}$ </p> |
| <p> $G_{\tau}=\frac{1}{(4\pi\tau)^{d/2}}\exp\left(-\frac{|\textbf{x}|^{2}}{4\tau}\right)$ </p> |
| <p> $\int_{\Omega}\chi^{k+1}\ d\textbf{x}=V_{0}$ </p> |
| <p> $\chi^{k+1}=\tilde{\chi}_{s}$ </p> |
| <p> $\mathbb{R}^{d}\setminus\overline{\Omega}$ </p> |
| <p> $T^{*k}$ </p> |
| <p> $\displaystyle\tilde{B}_{1}$ </p> |
| <p> $\Omega_{1}\in\Omega$ </p> |
| <p> $\Gamma\colon=\Gamma_{D}\cup\Gamma_{N}$ </p> |
| <p> $\displaystyle:=\big{\{}p\in\mathcal{N},\Phi^{k}>\sigma,\chi^{k}(p)=1\big{\}}.$ </p> |
| <p> $\displaystyle\chi^{k+1}=\begin{cases}1&\ \textrm{if}\ \Phi^{k}\leq\sigma,\\ |
| 0&\ \textrm{otherwise}.\end{cases}$ </p> |
| <p> $\frac{\delta\tilde{J}^{\tau}}{\delta T}=0,\ \ \ \frac{\delta\tilde{J}^{\tau}}{% |
| \delta T^{*}}=0.$ </p> |
| <p> $\chi^{k+1}=\arg\min_{\chi\in\mathcal{B}}\tilde{J}^{\tau,k}(\chi)$ </p> |
| <p> $\tilde{\chi}_{s}=\chi_{A_{1}}+\chi^{k}-\chi_{B_{1}}$ </p> |
| <p> $\kappa_{1},\kappa_{2},q_{1},q_{2}$ </p> |
| <p> $T^{\ast}_{h}\in V_{h}^{0}$ </p> |
| <p> $\Phi^{k}=(q_{1}-q_{2})G_{\tau}\ast(T^{k}-T^{*k})+\gamma\sqrt{\frac{\pi}{\tau}}% |
| G_{\tau}\ast(1-2\chi^{k})+(k_{1}-k_{2})G_{\tau}\ast(\frac{\xi}{2}\nabla T^{k}% |
| \cdot\nabla T^{k}+\nabla T^{k}\cdot\nabla T^{\ast k})$ </p> |
| <p> $\displaystyle\tilde{J}^{\tau,k}(\chi)$ </p> |
| <p> $\displaystyle\int_{\Omega}\left[G_{\tau/2}\ast\chi\right]\left[G_{\tau/2}\ast(% |
| 1-\chi)\right]\ d\textbf{x}$ </p> |
| <p> $\kappa_{1}=10,\ \kappa_{2}=1,\ q_{1}=1,\ q_{2}=100$ </p> |
| <p> $\kappa_{1}=40,\kappa_{2}=1$ </p> |
| <p> $\int_{\Omega}\chi\ d\textbf{x}=V_{0}$ </p> |
| <p> $T^{k+1}$ </p> |
| <p> $\chi^{0}$ </p> |
| <p> $\tilde{\phi}_{A}^{k}$ </p> |
| <p> $\displaystyle\tilde{A}_{2}$ </p> |
| <p> $\chi^{k+1}=Proj_{[0,1]}\left(\chi^{k}-s\left.\frac{\delta J^{\tau}}{\delta\chi% |
| }\right|_{\chi^{k}}\right),$ </p> |
| <p> $\kappa(\chi)$ </p> |
| <p> $0.1l$ </p> |
| <p> $\displaystyle\chi^{k+1}=\chi_{A}+\chi^{k}-\chi_{B}.$ </p> |
| <p> $J^{\tau}(\chi^{k+1},T^{k+1})\leq J^{\tau}(\chi^{k+1},T^{k}),$ </p> |
| <p> $\kappa_{2}=1$ </p> |
| <p> $\displaystyle\kappa(\chi)=\kappa_{1}G_{\tau}\ast\chi+\kappa_{2}G_{\tau}\ast(1-% |
| \chi),$ </p> |
| <p> $\tilde{J}^{\tau}(\chi,T,T^{*})=J^{\tau}(\chi,T)+\int_{\Omega}(-\nabla\cdot(% |
| \kappa(\chi)\nabla T)-q(\chi))\cdot T^{*}d\textbf{x}.$ </p> |
| <p> $q_{2}=100$ </p> |
| <p> $\chi^{1},T^{1},T^{*1},\chi^{2},T^{2},T^{*2},\cdots,\chi^{k},T^{k},T^{*k},\cdots$ </p> |
| <p> $\tilde{J}^{\tau}(\mathbf{\chi}^{k+1},T^{k})\leq\tilde{J}^{\tau}(\mathbf{\chi}^% |
| {k},T^{k}),$ </p> |
| <p> $J^{\tau}(\chi,T)$ </p> |
| <p> $\displaystyle\left\{\begin{aligned} \nabla\cdot(\kappa(\chi^{k})\nabla T^{*})&% |
| =q(\chi^{k})-\xi(\nabla\cdot(\kappa(\chi^{k})\nabla T)),\ \ &&\rm in\ \ \Omega% |
| ,\\ |
| T^{*}&=0,\ \ &&\rm on\ \ \Gamma_{D},\\ |
| \kappa(\chi^{k})\nabla T^{*}\cdot\mathbf{n}&=0,\ \ &&\rm on\ \ \Gamma_{N}\end{% |
| aligned}\right.$ </p> |
| <p> $J^{\tau}(\mathbf{\chi}^{k+1},T^{k+1})\leq J^{\tau}(\mathbf{\chi}^{k},T^{k})$ </p> |
| <p> $\displaystyle:=\big{\{}p\in\mathcal{N},\Phi^{k}>\sigma_{2},\chi^{k}(p)=1\big{% |
| \}},$ </p> |
| <p> $\chi^{k+1}\in\mathcal{B}$ </p> |
| <p> $\|\chi^{k+1}-\chi^{k}\|_{2}>tol$ </p> |
| <p> $\displaystyle:=\big{\{}p\in\mathcal{N},\Phi^{k}\in(\sigma_{1},\sigma),\chi^{k}% |
| (p)=0\big{\}},$ </p> |
| <p> $\kappa=\kappa_{1}\chi_{\Omega_{1}}+\kappa_{2}\chi_{\Omega_{2}}$ </p> |
| <p> $\displaystyle\ J^{\tau}(\mathbf{\chi}^{k+1},T^{k+1})\leq J^{\tau}(\mathbf{\chi% |
| }^{k},T^{k}).$ </p> |
| <p> $\xi=1\times 10^{-5}$ </p> |
| <p> $600\times 600$ </p> |
| <p> $P_{1}(K)$ </p> |
| <p> $T^{k},T^{*k}$ </p> |
| <p> $\int_{\Omega}\chi^{k+1}(\textbf{x})d\textbf{x}=V_{0}$ </p> |
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