Posterior consistency and convergence rates for Bayesian inversion with hypoelliptic operators Article Swipe
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· 2016
· Open Access
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· DOI: https://doi.org/10.1088/0266-5611/32/8/085005
· OA: W1786480885
Bayesian approach to inverse problems is studied in the case where the\nforward map is a linear hypoelliptic pseudodifferential operator and\nmeasurement error is additive white Gaussian noise. The measurement model for\nan unknown Gaussian random variable $U(x,\\omega)$ is \\begin{eqnarray*}\nM(y,\\omega) = A(U(x,\\omega) )+ \\delta\\hspace{.2mm}\\mathcal{E}(y,\\omega),\n\\end{eqnarray*} where $A$ is a finitely many times smoothing linear\nhypoelliptic operator and $\\delta>0$ is the noise magnitude. The covariance\noperator $C_U$ of $U$ is $2r$ times smoothing, self-adjoint, injective and\nelliptic pseudodifferential operator.\n If $\\mathcal{E}$ was taking values in $L^2$ then in Gaussian case solving the\nconditional mean (and maximum a posteriori) estimate is linked to solving the\nminimisation problem \\begin{eqnarray*} T_\\delta(M) = \\text{argmin}_{u\\in H^r}\n \\big\\{\\|A u-m\\|_{L^2}^2+ \\delta^2\\|C_U^{-1/2}u\\|_{L^2}^2 \\big\\}.\n\\end{eqnarray*} However, Gaussian white noise does not take values in $L^2$ but\nin $H^{-s}$ where $s>0$ is big enough. A modification of the above approach to\nsolve the inverse problem is presented, covering the case of white Gaussian\nmeasurement noise. Furthermore, the convergence of conditional mean estimate to\nthe correct solution as $\\delta\\rightarrow 0$ is proven in appropriate function\nspaces using microlocal analysis. Also the contraction of the confidence\nregions is studied.\n
