  We investigate a family of approximate multi-step proximal point methods,
accelerated by implicit linear discretizations of gradient flow. The resulting
methods are multi-step proximal point methods, with similar computational cost
in each update as the proximal point method. We explore several optimization
methods where applying an approximate multistep proximal points method results
in improved convergence behavior. We argue that this is the result of the
lowering of truncation error in approximating gradient flow
