Exact use of higher-order derivatives in large-scale nonlinear solvers

Determine whether nonlinear solvers can utilize higher-order derivative information exactly, rather than relying on approximations of higher-order derivatives, for solving large-scale nonlinear problems.

Background

In the literature, methods that use higher-order derivatives for nonlinear solving are often approximated because computing full higher-order derivatives is believed to be substantially more expensive than evaluating the function itself for large input and output dimensions. This has led to a reliance on approximations rather than exact computations when incorporating higher-order information.

The paper motivates the need to assess the feasibility of exact higher-order derivative usage at scale and proposes Taylor-mode automatic differentiation to compute higher-order directional derivatives efficiently. The explicit open question pertains to whether exact higher-order derivative information can be employed by nonlinear solvers for large-scale problems without resorting to approximations.