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2 added names of king's coauthors

This is a rapidly developing area, and there are many short-cuts if all you want to do is compute the homology of sub-level sets of $f$. To answer your main question, as Liviu has already mentioned: there is no standard way. However, you should be able to compute the homology you desire as follows.

My impression is that you have three jobs, in chronological order:

1. Construct a simplicial complex $X_M$ homologically faithful to $M$.
2. Construct a discrete Morse function $\mu:X_M \to \mathbb{R}$ which approximates $f$, and
3. Compute homology of everything in sight.

First, the easiest way to build a simplicial approximation if you know your $M$ is to embed it in some suitable $\mathbb{R}^n$ and sample the hell out of it. Since you are working on data analysis, this should not be too drastic a step. Given a point sample $P$ coming from a submanifold of Euclidean space, for each radius $\epsilon$ you can construct a Cech complex of radius $\epsilon$ around $P$. Precise bounds on how many points $P$ should have and how large $\epsilon$ can be in order for the Cech complex to recover the homology of $M$ with high confidence are available in the work of Niyogi, Smale and Weinberger here in the case when $P$ is uniformly sampled. These bounds are in terms of the injectivity radius of the embedding of $M$ into Euclidean space, and of course once these bounds are satisfied it doesn't hurt to add your known critical points to $P$. You have your homologically faithful Cech complex $X_M$.

Next, for 2, you can easily infer a discrete Morse function on an entire simplicial complex just from knowing its values on the vertices using the work of King, Knudson and Mramor. You may be required to perturb $f$ slightly so that its restriction to $P$ is injective, but this is easy and generically true. You have $\mu$!

And finally, I have written software to handle 3 if you already have a $\mu:X_M \to \mathbb{R}$: you can input a filtered simplicial complex and compute not just homology at each sub-level set of $\mu$ but the persistent homology across all level-sets in the case of field coefficients. Meaning, instead of just knowing the homology of the subcomplexes $X_M^c$ consisting of all simplices with $\mu$-value less than or equal to $c$, you also recover the morphism on homology groups induced by including $X_M^c$ into $X_M^d$ whenever $c \leq d$.

All the best with your computations.

1

This is a rapidly developing area, and there are many short-cuts if all you want to do is compute the homology of sub-level sets of $f$. To answer your main question, as Liviu has already mentioned: there is no standard way. However, you should be able to compute the homology you desire as follows.

My impression is that you have three jobs, in chronological order:

1. Construct a simplicial complex $X_M$ homologically faithful to $M$.
2. Construct a discrete Morse function $\mu:X_M \to \mathbb{R}$ which approximates $f$, and
3. Compute homology of everything in sight.

First, the easiest way to build a simplicial approximation if you know your $M$ is to embed it in some suitable $\mathbb{R}^n$ and sample the hell out of it. Since you are working on data analysis, this should not be too drastic a step. Given a point sample $P$ coming from a submanifold of Euclidean space, for each radius $\epsilon$ you can construct a Cech complex of radius $\epsilon$ around $P$. Precise bounds on how many points $P$ should have and how large $\epsilon$ can be in order for the Cech complex to recover the homology of $M$ with high confidence are available in the work of Niyogi, Smale and Weinberger here in the case when $P$ is uniformly sampled. These bounds are in terms of the injectivity radius of the embedding of $M$ into Euclidean space, and of course once these bounds are satisfied it doesn't hurt to add your known critical points to $P$. You have your homologically faithful Cech complex $X_M$.

Next, for 2, you can easily infer a discrete Morse function on an entire simplicial complex just from knowing its values on the vertices using the work of King. You may be required to perturb $f$ slightly so that its restriction to $P$ is injective, but this is easy and generically true. You have $\mu$!

And finally, I have written software to handle 3 if you already have a $\mu:X_M \to \mathbb{R}$: you can input a filtered simplicial complex and compute not just homology at each sub-level set of $\mu$ but the persistent homology across all level-sets in the case of field coefficients. Meaning, instead of just knowing the homology of the subcomplexes $X_M^c$ consisting of all simplices with $\mu$-value less than or equal to $c$, you also recover the morphism on homology groups induced by including $X_M^c$ into $X_M^d$ whenever $c \leq d$.

All the best with your computations.