A one-dimensional elliptic diffusion coefficient has the adjoint gradient

Put $w(x)=x(1-x)\mathrm{/}2$. Direct differentiation gives $w^{\mathrm{\prime }\mathrm{\prime }}=-1$, $w(0)=w(1)=0$, and [F3] gives $w\in V$. By [F9], $V$ is a real Hilbert space. Holder's inequality [F8] gives $\mathrm{\mid }\mathrm{\ell }(v)\mathrm{\mid }=\mathrm{\mid }{\int }_0^{1}v\mathrm{\mid }\le \mathrm{\parallel }1{\mathrm{\parallel }}_{L^2}\mathrm{\parallel }v{\mathrm{\parallel }}_{L^2}\le \mathrm{\parallel }v{\mathrm{\parallel }}_{H^1}$, so $\mathrm{\ell }\in V^{\ast }$; the objective $J(u,\theta )=\mathrm{\ell }(u)$ is bounded linear in $u$ and independent of $\theta$, hence $C^1$ with $D_{u}J[v]=\mathrm{\ell }(v)$ and $D_{\theta }J=0$. Also $\mathrm{\mid }{a}_{\theta }(u,v)\mathrm{\mid }\le (1+\theta )\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1}\mathrm{\parallel }v{\mathrm{\parallel }}_{H^1}$ for $\theta >-1$, and $A_{\theta }=(1+\theta )A_0$ is operator-norm $C^1$, while ${\mathrm{\ell }}_{\theta }=\mathrm{\ell }$ is constant. For each $v\in V$ and $\epsilon >0$, [F5] supplies $v_{\epsilon }\in C_c^{\mathrm{\infty }}(0,1)$ with $\mathrm{\parallel }v-v_{\epsilon }{\mathrm{\parallel }}_{H^1}<\epsilon$. By the product rule [F7], $(w^{\mathrm{\prime }}{v}_{\epsilon }{)}^{\mathrm{\prime }}=w^{\mathrm{\prime }\mathrm{\prime }}{v}_{\epsilon }+w^{\mathrm{\prime }}{v}_{\epsilon }^{\mathrm{\prime }}$; since $v_{\epsilon }$ vanishes at both endpoints, [F6] gives ${\int }_0^1{w}^{\mathrm{\prime }}{v}_{\epsilon }^{\mathrm{\prime }}={\int }_0^1{v}_{\epsilon }$. By [F8], the difference between the two sides with $v$ in place of $v_{\epsilon }$ is at most $(\mathrm{\parallel }{w}^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^2}+1)\epsilon$, so the weak identity ${\int }_0^1{w}^{\mathrm{\prime }}{v}^{\mathrm{\prime }}={\int }_0^{1}v$ follows as $\epsilon \downarrow 0$. Also, [F4] gives $a_{\theta }(v,v)\ge (1+\theta )\mathrm{\parallel }v{\mathrm{\parallel }}_{H^1}^2\mathrm{/}2$; on a neighborhood of each $\theta >-1$ this is a uniform coercivity bound. Hence $u_{\theta }=w\mathrm{/}(1+\theta )$ solves the state equation and is unique. Finally ${\int }_0^{1}w=1\mathrm{/}12$, so $j(\theta )=1\mathrm{/}(12(1+\theta ))$.

The adjoint equation is $a_{\theta }(v,p)=D_{u}J[v]={\int }_0^{1}v$ for all $v\in V$, so symmetry of $a_{\theta }$ and uniqueness give $p=u_{\theta }$. The direct parameter derivatives of $J$ and $\mathrm{\ell }$ vanish. For the scalar parameter direction $1$, $D_{\theta }a[1](u,p)={\int }_0^1{u}^{\mathrm{\prime }}{p}^{\mathrm{\prime }}dx=(1+\theta {)}^{-2}{\int }_0^1(w^{\mathrm{\prime }}{)}^{2}dx$. Direct integration gives ${\int }_0^1(w^{\mathrm{\prime }}{)}^{2}dx=\frac{1}{4}{\int }_0^1(1-2x{)}^{2}dx=1\mathrm{/}12$, hence $D_{\theta }a[1](u,p)=1\mathrm{/}(12(1+\theta {)}^2)$.

Formula [F1] therefore gives $j^{\mathrm{\prime }}(\theta )=-1\mathrm{/}(12(1+\theta {)}^2)$, equal to the derivative of the explicit reduced objective in step 1.1.