CodPy: a tutorial

A class of mesh-free algorithms for mathematical finance, machine learning and fluid dynamics

“…Dimension one. We present a rather basic test, while referring to [16] for further numerical examples. We consider the role of kernels and we illustrate graphically the extrapolation operator (2.16) as well as the discrepancy error and functional norm computed from the kernels.…”

“…This minimization problem relies on the Helmholtz-Hodge decomposition, which is computed using a kernel (cf. [16] for further details). We emphasize that this algorithm assumes that the initial map is positive in the sense that its Jacobian is positive, that is, ∇ x h ≥ 0.…”

“…To this purpose, a reordering algorithm is required (and is described in [18]). In particular, the operator (∇ x ) −1 is presented in [16].…”

“…Each algorithm is illustrated by numerical experiments which demonstrate the efficiency of the proposed approach. While the present paper describes the algorithms and contains only basic numerical tests, the companion papers [16]- [19] are devoted to the implementation of our algorithms in a Python code (called CodPy, standing for the "Curse of dimensionality in Python").…”