This concerns sparse matrices where the sparsity pattern is known beforehand, and where the size is between 5 and up to 50, as the linear solver for a Newton Raphson non linear solver.
For smaller than 5, I use a direct inverse computation, but for problems that are bigger, I currently use a simple dense Householder pivot solver, which get slows really quickly.
I can optimize the layout of the matrix as well, but I'm not sure "usual" sparse matrix solving method could be applied here and if they could be faster than brute-force Householder QR. I feel that knowing the sparsity could be leveraged to solve the problem faster.