Computing gaussian \& exponential measures of semi-algebraic sets

Abstract

We provide a numerical scheme to approximate as closely as desired the Gaussian or exponential measure $μ(\om)$ 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 $(\overlineω\_d)$, $(\underlineω\_d)$, $d\in\N$, each converging to $μ(\om)$ as $d\to\infty$. For each $d$, computing $\overlineω\_d$ or $\underlineω\_d$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.