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SUMMARY:Statistical analysis of a non-linear inverse problem for an ellipt
 ic PDE - Sven Wang\, University of Cambridge
DTSTART:20190130T160000Z
DTEND:20190130T170000Z
UID:TALK119449@talks.cam.ac.uk
CONTACT:Angeliki Menegaki
DESCRIPTION:The talk will be about some statistical inverse problems arisi
 ng in the context of second order elliptic PDEs. Concretely\, suppose $f$ 
 is the unknown coefficient function of a second order elliptic partial dif
 ferential operator $L_f$ on some bounded domain $\\mathcal O\\subseteq \\R
 ^d$\, and the unique solution $u_f$ of a corresponding boundary value prob
 lem is observed\, corrupted by additive Gaussian white noise. Concrete exa
 mples include $L_fu=\\Delta u-2fu$ (Schr\\"odinger equation with attenuati
 on potential $f$) and $L_fu=\\text{div} (f\\nabla u)$ (divergence form equ
 ation with conductivity $f$). I will present some recent results on the co
 nvergence rates for Tikhonov-type penalised least squares estimators $\\ha
 t f$ for $f$ in these contexts. The penalty functionals are of squared Sob
 olev-norm type and thus $\\hat f$ can also be interpreted as a Bayesian `M
 AP'-estimator corresponding to some Gaussian process prior. We derive rate
 s of convergence of $\\hat f$ and of $u_{\\hat f}$\, to $f\, u_f$\, respec
 tively. The rates obtained are minimax-optimal in prediction loss.
LOCATION:MR14\, Centre for Mathematical Sciences
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