Pluripotential Numerics
For researchers in pluripotential theory and computational complex analysis, this provides the first numerical schemes with convergence guarantees for these fundamental objects.
The paper introduces numerical methods for approximating key quantities in Pluripotential Theory, including the extremal plurisubharmonic function, transfinite diameter, and pluripotential equilibrium measure, with proven convergence. Numerical tests on simple 2D sets demonstrate the methods' performance.
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.