  We study approximations to the Moreau envelope -- and infimal convolutions
more broadly -- based on Laplace's method, a classical tool in analysis which
ties certain integrals to suprema of their integrands. We believe the
connection between Laplace's method and infimal convolutions is generally
deserving of more attention in the study of optimization and partial
differential equations, since it bears numerous potentially important
applications, from proximal-type algorithms to solving Halmiton-Jacobi
equations.
