  In this paper we study a nonlinear dual space preconditioning approach for
the relaxed Proximal Point Algorithm (PPA) with application to monotone
inclusions, called anisotropic PPA. The algorithm is an instance of Luque's
nonlinear PPA wherein the nonlinear preconditioner is chosen as the gradient of
a Legendre convex function. Since the preconditioned operator is nonmonotone in
general, convergence cannot be shown using standard arguments, unless the
preconditioner exhibits isotropy (preserves directions) as in existing
literature. To address the broader applicability we show convergence along
subsequences invoking a Bregman version of Fej\`er-monotonicity in the dual
space. Via a nonlinear generalization of Moreau's decomposition for operators,
we provide a dual interpretation of the algorithm in terms of a forward
iteration applied to a $D$-firmly nonexpansive mapping which involves the
Bregman resolvent. For a suitable preconditioner, convergence rates of
arbitrary order are derived under a mild H\"older growth condition. Finally, we
discuss an anisotropic generalization of the proximal augmented Lagrangian
method obtained via the propsed scheme. This has strong connections to
Rockafellar's generalized and sharp Lagrangian functions.
