a trial function that does not solve the equation, and then iteratively deforms it to obtain functions that solve the equation to a better and better approximation. There are two basic strategies for how to relax the metric. If we iteratively replace $g_{\mu\nu}$ with $g_{\mu\nu} + h_{\mu\nu}$, then one can either deform in the direction of $-R^H_{\mu\nu}$, $$\label{RF}
h_{\mu\nu} = -\epsilon R^H_{\mu\nu}\,,$$ or solve the linearized equation, $$\label{Newton}
h_{\mu\nu} = -\Delta_H^{-1}R^H_{\mu\nu}\,.$$ The first is closely related to Ricci-flow, and the second is a Newton method. In this section we will discuss these two approaches in turn, trying to understand their general properties (rather than the specifics of how to implement them). Our discussion should not be considered as exhaustive, and these strategies can be generalized, refined, and combined in various ways. We will for the most part work in the continuum limit, although it should be understood that in practice the metric and operators will be discretized in some form. The choice of discretization scheme—finite differencing, finite element, spectral, etc.—is a separate issue that we will not address in this section. (For the application discussed in Section 4, we have chosen finite differencing, because it is the easiest to implement. 