  We consider statistical linear inverse problems in separable Hilbert spaces
and filter-based reconstruction methods of the form $\hat f_\alpha = q_\alpha
\left(T^*T\right)T^*Y$, where $Y$ is the available data, $T$ the forward
operator, $\left(q_\alpha\right)_{\alpha \in \mathcal A}$ an ordered filter,
and $\alpha > 0$ a regularization parameter. Whenever such a method is used in
practice, $\alpha$ has to be appropriately chosen. Typically, the aim is to
find or at least approximate the best possible $\alpha$ in the sense that mean
squared error (MSE) $\mathbb E [\Vert \hat f_\alpha - f^\dagger\Vert^2]$
w.r.t.~the true solution $f^\dagger$ is minimized. In this paper, we introduce
the Sharp Optimal Lepski\u{\i}-Inspired Tuning (SOLIT) method, which yields an
a posteriori parameter choice rule ensuring adaptive minimax rates of
convergence. It depends only on $Y$ and the noise level $\sigma$ as well as the
operator $T$ and the filter $\left(q_\alpha\right)_{\alpha \in \mathcal A}$ and
does not require any problem-dependent tuning of further parameters. We prove
an oracle inequality for the corresponding MSE in a general setting and derive
the rates of convergence in different scenarios. By a careful analysis we show
that no other a posteriori parameter choice rule can yield a better performance
in terms of the order of the convergence rate of the MSE. In particular, our
results reveal that the typical understanding of Lepski\u\i-type methods in
inverse problems leading to a loss of a log factor is wrong. In addition, the
empirical performance of SOLIT is examined in simulations.
