Application: Multigrid / IK

[benrbray]

This is a fantastic list, thanks for putting it together. This topic has been on my mind for a while, too. I have a few suggested additions!

I've noticed that many engineering applications use the "transpose" of a known matrix operator in clever ways, but I don't often see people talk about it from the perspective of adjoints. For instance:

• In multigrid methods, where we use a pair of restriction / prolongation operators to move between coarse and fine computation grids. For the simple case the operators end up just being matrix transposes of one another. Links:

  • Strang 2006, "Mathematical Methods for Engineers" Sec 6.3 Multigrid
  • Wikipedia, Multigrid Method

• A common method for inverse kinematics uses the Jacobian transpose to approximate the desired inverse problem. Links:

  • http://www.math.ucsd.edu/~sbuss/ResearchWeb/ikmethods/iksurvey.pdf
  • http://www.cs.cmu.edu/~15464-s13/lectures/lecture6/IK.pdf
  • https://www.gamasutra.com/blogs/LuisBermudez/20170804/303066/3_Simple_Steps_to_Implement_Inverse_Kinematics.php

Some other things to explore:

• pseudo-inverses
• mathematical definition for transpose of a linear map (without an inner product) in comparison to the definition for adjoint of a linear operator (with inner product)
• Bradley 2019, "PDE-constrained Optimization and the Adjoint Method" and Adjoint State Method