Partial Derivatives on Causal Sets

We will discuss two approaches to estimating partial derivatives and the metric components; one utilizing past work describing a causal set $\mathrm{\square }$ operator, and one using a construction from linear algebra called the Moore-Penrose inverse. After running numerical tests on a causal diamond in ${\mathbb{M}}^2$, we find that the approach using the Moore-Penrose inverse is significantly more accurate. Despite the large variances in the method using the $\mathrm{\square }$ operator, there is reason to believe both approaches should become more accurate at higher densities.