Pluripotential Numerics

Abstract

We introduce numerical methods for the approximation of the main (global) quantities in Pluripotential Theory as the extremal plurisubharmonic function V * E of a compact L-regular set E ⊂ C n , its transfinite diameter δ(E), and the pluripotential equilibrium measure µ E := dd c V * E n .Date: October 7, 2018. 1991 Mathematics Subject Classification. MSC 65E05 and MSC 41A10 and MSC 32U35 and MSC 42C05.

Markov inequalities, Dubiner distance, norming meshes and polynomial optimization on convex bodies

“…Note that the problem of finding unisolvent interpolation arrays with slowly increasing (e.g., polynomial in the degree) Lebesgue constant on a given compact set K ⊂ R d is very hard to attack numerically, even for small values of d > 1. Lastly, we mention that polynomial meshes are the key ingredient for the approximation algorithms proposed in [24], where the numerical approximation of the main quantities of pluripotential theory (a non linear potential theory in…”

Near G-optimal Tchakaloff designs

“…Moreover, polynomial meshes contain extremal subsets of Fekete and Leja type for polynomial interpolation, and have been applied in polynomial optimization and in pluripotential numerics; cf., e.g., [5,19,23]. The class of compact sets which admit (constructively) a polynomial mesh is very wide.…”

Near optimal polynomial regression on norming meshes

“…In addition, norming meshes play a role in the computation of good interpolations sets of Fekete and Leja type, and have been applied in the fields of polynomial optimization and pluripotential numerics; cf., e.g., [5], [16], [18], [19].…”