  In this paper, we develop a new type of accelerated algorithms to solve some
classes of maximally monotone equations as well as monotone inclusions. Instead
of using Nesterov's accelerating approach, our methods rely on a so-called
Halpern-type fixed-point iteration in [32], and recently exploited by a number
of researchers, including [24, 70]. Firstly, we derive a new variant of the
anchored extra-gradient scheme in [70] based on Popov's past extra-gradient
method to solve a maximally monotone equation $G(x) = 0$. We show that our
method achieves the same $\mathcal{O}(1/k)$ convergence rate (up to a constant
factor) as in the anchored extra-gradient algorithm on the operator norm $\Vert
G(x_k)\Vert$, , but requires only one evaluation of $G$ at each iteration,
where $k$ is the iteration counter. Next, we develop two splitting algorithms
to approximate a zero point of the sum of two maximally monotone operators. The
first algorithm originates from the anchored extra-gradient method combining
with a splitting technique, while the second one is its Popov's variant which
can reduce the per-iteration complexity. Both algorithms appear to be new and
can be viewed as accelerated variants of the Douglas-Rachford (DR) splitting
method. They both achieve $\mathcal{O}(1/k)$ rates on the norm $\Vert
G_{\gamma}(x_k)\Vert$ of the forward-backward residual operator
$G_{\gamma}(\cdot)$ associated with the problem. We also propose a new
accelerated Douglas-Rachford splitting scheme for solving this problem which
achieves $\mathcal{O}(1/k)$ convergence rate on $\Vert G_{\gamma}(x_k)\Vert$
under only maximally monotone assumptions. Finally, we specify our first
algorithm to solve convex-concave minimax problems and apply our accelerated DR
scheme to derive a new variant of the alternating direction method of
multipliers (ADMM).
