I believe I’ve been solving large and sparse linear systems most of my career, both as the end solution of a problem or in the calculation of non-linear least-squares iterations. I imagine a lot of people share my experience, and I am looking for suggestions for improving my work, and potentially do some collaboration developing tools for this kind of problem.

The greatest techniques I have learned in the past decades to deal with such problems must have been to use PCG, sparse matrices, and AD. Now all of those seem to be already well-supported in Julia, and I’m having a great time using the available tools. I may be just overlooking some project, but there seems to be something missing to help you actually deal with large models that result in large but sparse Jacobian matrices.

I imagine most people solving sparse systems are working with FEM. Myself, I work mostly with mobile robotics and problems such as SLAM and SFM and large regularization problems.

In robotics, we have seen the development of tools such as g2o, that allow you to represent your problem as a graph from which a gradient and Hessian matrix can be calculated. Some people have been giving this the special name of *graph optimization* even though in my perhaps limited understanding it is “nothing but” a way to help you assemble that sparse Jacobian. And I am not sure building such a graph is the optimal way of obtaining the Jacobian-updating-procedure that is what you are really interested in having.

There seems to be a nice group of people doing robotics in Julia. Is RoME.jl the project I should look for to help me? Is JuMP adequate for such a problem?

One of the reasons I wanted to open the question to the general community is because I am curious to hear what people from other areas have to say about these problems. AD is great, but there is a lot of extra work necessary to update a sparse Jacobian, and do it optimally. What do people in FEM usually do, for instance?

Also, I’ve been only using ForwardDiff, but there’s a lot of talking out there about other approaches to AD, not only alternatives to dual numbers (like Cassette) but also alternatives to forward-mode AD. I am curious to hear, what possibilities could I try to explore in this kind of sparse-Jacobian optimization problems?

How could do we best go from an abstract, declarative description of a problem with a model that has “N” control points and “M” observations, and come up with an efficient procedure to update a sparse Jacobian matrix? Is this not perhaps a great opportunity for metaprogramming?