Monte Carlo quasi-heat-bath by approximate inversion

When sampling the distribution $P\left(\overrightarrow{\phi }\right)\propto \exp \left(-|A\overrightarrow{\phi }{|}^2\right),$ a global heat bath normally proceeds by solving the linear system $A\overrightarrow{\phi }=\overrightarrow{\eta },$ where $\overrightarrow{\eta }$ is a normal Gaussian vector, exactly. This paper shows how to preserve the distribution $P\left(\overrightarrow{\phi }\right)$ while solving the linear system with arbitrarily low accuracy. Generalizations are presented.