Smooth Morse function from Forman's discrete Morse function Let $M$ be a smooth manifold and $K$ a triangulation of $M$, so $K$ is a regular CW-complex and in particular a simplicial complex. Assume that $M$ is compact so $K$ is finite. Let $f\colon K \to \mathbb{R}$ be a discrete Morse function (in the sense of Forman). Is is possible to define a smooth Morse function $f'\colon M \to \mathbb{R}$ with the same critical points as $f$ (and satisfying a correspondence between the indexes of the critical points)? Is it possible to do it in "an algorithmic way" (I mean that the proof is constructive)? 
As far as I know, the converse was addressed by Gallais and Benedetti, am I right? 
I apologize in advance if the questions are to vague or the answers are well-known. Thanks in advance for your time.
 A: You can do the next best think. To a  Forman-Morse function $f$ one can associate  a flow on the manifold whose stationary points are precisely  the barycenters  of the faces  of your simplicial decomposition.    The Conley index of the barycenter of a critical face has the homotopy type of a sphere of the dimension of that face. The Conley index of the barycenter of a non-critical face is homotopically trivial.
Additionally, one can construct a continuous function $\tilde{f}$ on the manifold that decreases along the trajectories of this flow and whose value at a barycenter is equal to the value of $f$ on the corresponding face. As  Mike Miller correctly pointed out,  a critical face is filled out by the  trajectories exiting the barycenter.
For details  see Chapter 11 of this paper.  The faces of the barycentric subdivision of your simplicial complex are invariant  sets of this flow, and on  such a face the flow is depicted in Figure 2, p.16 of the above paper.
It took me a while to realize that in Morse theory  the gradient flow associated to a Morse function is more important than the function itself.  The function plays a sort of accounting role and  the Morse condition  restricts the nature of the stationary points of the gradient flows.
Remark  A while ago I asked this question on MathOverflow that is related to the abundance of discrete Morse functions. They are extremely rare as opposed to the  usual smooth Morse functions that are generic.
