Notes for "Posterior consistency

2020-02-10 本文已影响0人 jjx323

H. Kekkonen, M. Lassas, S. Siltanen, Posterior consistency and convergence rates for Bayesian inversion with hypoelliptic operators, Inverse Problems, 32, 2016, 085005

Page 10, line 6, Section 3 Generalised random variables

The connection between $B_V$ and $C_V$ is $B_V = C_V (I-\Delta)^s : H^{s}\rightarrow H^{s}$

Notes:
Consider $\phi, \psi\in H^s(N)$ , then we have
$\langle C_V\phi, \psi \rangle_{H^{s}\times H^{-s}} = \langle \psi, C_V^* \psi\rangle_{H^{s}\times H^{-s}},$
where $C_V^* : H^{s}\rightarrow H^{-s}$ . Since $(B_{V}\phi, \psi)_{H^s} = (\phi, B_V \psi)_{H^s}$ and $(B_{V}\phi, \psi)_{H^s} = \langle C_V\phi, \psi \rangle_{H^{s}\times H^{-s}}$ , we obtain
$\int_{N}(I-\Delta)^{s/2}\phi(x) (I-\Delta)^{s/2}B_V\psi(x) ds = \int_{N}\phi(x)C_V^{*} \psi(x)dx.$
Hence, we find that
$(I-\Delta)^{s}B_V = C_V^{*} : H^{s}\rightarrow H^{-s},$
which implies
$C_V = B_V (I-\Delta)^{s}.$

Page 11, line (from end) 7, Section 3.2

Condition $B_U \in \mathfrak{C}^{1}(H^{\tau})$ guarantees that $\mathbb{E}(U, U)_{H^{\tau}} < \infty$ .

Notes:
Taking $(\lambda_j, \{\varphi_{j}\})_{j=1}^{\infty}$ be eigensystem of $B_U$ on $H^{\tau}$ , we have
$\begin{align} \mathbb{E}(U,U)_{H^{\tau}} & = \mathbb{E}\sum_{j=1}^{\infty}(U,\varphi_{j})(U,\varphi_{j})=\sum_{j=1}^{\infty}\mathbb{E}{((U,\varphi_{j})_{H^{\tau}}(U,\varphi_{j})_{H^{\tau}})} \\ & = \sum_{j=1}^{\infty}(B_U \varphi_{j}, \varphi_{j})_{H^{\tau}} = \sum_{j=1}^{\infty}\lambda_{j} < \infty, \end{align}$
which is just the required estimation.
Here, in this part, we may see $C_U : H^{-\tau} \rightarrow H^{\tau}$ ( $\tau\leq r$ ) and $B_U : H^{\tau} \rightarrow H^{\tau}$ , which coincides with formula (3.3) and (3.4) on page 10. Taking $\nu=\nu_k$ in the first formula on page 12, we have
$k = N(\nu_k) \sim cv_k^{\frac{d}{2(r-\tau)}}(1+O(\nu_k^{-\frac{1}{2(r-\tau)}})).$
Question: The operator $B_U^{-1}$ is a self-adjoint elliptic operator with smooth coefficients (defined on closed manifold), the eigenvalues are irrelevant to the definition function space of the operator $B_U^{-1}$ ?

Proof of Lemma 3, Page 15

$\ldots$ and we can write
$B_0 = B_0 ((A^* A)B_1 - K_2) = B_1 - B_0 K_2.$

Notes:
Here, we assume that $t, t_0 > 0$ . By my understanding, the equality means that for $f\in H^{r+2t_0}\subset H^{r}$ we have
$B_0(f) = B_0((A^* A)B_1)(f) - B_0 K_2(f) = B_0 A^* A B_1(f) - B_0 K_2 (f) .$
Since $B_1(f) \in H^{r}$ , we obtain $B_0 A^* A B_1(f) = B_1 (f)$ and
$B_0(f) = B_1(f) + B_0K_2(f)$
holds for every $f\in H^{r+2t_0}$ . Because for any $r\in \mathbb{R}$ , we can deduce that the corresponding $B_0 : C^{\infty} \rightarrow C^{\infty}$ is continuous. Hence, we find that
$B = B_1 \, \text{mod} \, \Psi^{-\infty}$
holds on $H^{r}(N)$ for any $r\in \mathbb{R}$ . That is to say,
$B = B_1 \, \text{mod} \, \Psi^{-\infty}$
holds on space $\mathcal{D}'$ , which implies $B\in H\Psi^{2t_0, 2t}$ .

2020/02

Notes for "Posterior consistency

Page 10, line 6, Section 3 Generalised random variables

Page 11, line (from end) 7, Section 3.2

Proof of Lemma 3, Page 15

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