Computing gaussian \& exponential measures of semi-algebraic sets

TL;DR

This paper introduces a numerical method to accurately approximate Gaussian and exponential measures of semi-algebraic sets using converging bounds derived from semidefinite programming, applicable to various measures with known moments.

Contribution

The authors develop a novel numerical scheme that provides converging upper and lower bounds for measures of semi-algebraic sets, extending to measures satisfying Carleman's condition.

Findings

Bounds converge to the true measure as degree increases

Method successfully applied in small-dimensional computational experiments

Applicable to any measure with known moments satisfying Carleman's condition

Abstract

We provide a numerical scheme to approximate as closely as desired the Gaussian or exponential measure $\mu(\om)$ of (not necessarily compact) basic semi-algebraic sets$\om\subset\R^n$. We obtain two monotone (non increasing and non decreasing) sequences of upper and lower bounds $(\overline{\omega}\_d)$, $(\underline{\omega}\_d)$, $d\in\N$, each converging to $\mu(\om)$ as $d\to\infty$. For each $d$, computing $\overline{\omega}\_d$ or $\underline{\omega}\_d$reduces to solving a semidefinite program whose size increases with $d$. Some preliminary (small dimension) computational experiments are encouraging and illustrate thepotential of the method. The method also works for any measure whose moments are known and which satisfies Carleman's condition.