Pluripotential Numerics

Abstract

We introduce numerical methods for the approximation of the main (global) quantities in Pluripotential Theory as the \emph{extremal plurisubharmonic function} $V_E^*$ of a compact $\mathcal L$-regular set $E\subset \C^n$, its \emph{transfinite diameter} $δ(E),$ and the \emph{pluripotential equilibrium measure} $μ_E:=\ddcn{V_E^*}.$ The methods rely on the computation of a \emph{polynomial mesh} for $E$ and numerical orthonormalization of a suitable basis of polynomials. We prove the convergence of the approximation of $δ(E)$ and the uniform convergence of our approximation to $V_E^*$ on all $\C^n;$ the convergence of the proposed approximation to $μ_E$ follows. Our algorithms are based on the properties of polynomial meshes and Bernstein Markov measures. Numerical tests are presented for some simple cases with $E\subset \R^2$ to illustrate the performances of the proposed methods.