I am searching for a faster method to calculate an approximate inverse of a large matrix (up to 32000x32000) resulting from a discrete non-linear system of partial differential equations. I'm using C++ together with the Eigen library (MKL activated). Unfortunately the resulting matrix is not sparse as it can be seen in:
The blue entries denotes negative and the orange positive numbers. Currently I'm using a block matrix inversion scheme and then a partialPivLu() decomposition from Eigen to calculate the necessary inverses. The matrix has no specific properties like definiteness. Is there the possibility of a faster algorithm for a matrix with such a structure than to calculate the inverse or rather to solve the linear system of equations $Ax=b$ with the standard implementation of a Housholder or Lu decomposition?