I wanted to generate LU decomposition of large size dense matrices ($N>10^7$), the LU decomposition I'm currently using is based on Adaptive Cross Approximation and is taking very long time to execute for larger $N$. Can anybody suggest a few LU decomposition techniques that can be very well parallelised (using OpenMP) and take a shorter period of time?Note:I write the code in C++ and make use of Xeon Processor(128 threads)and Eigen library.The entries in the matrix are filled through a kernel function ofform $e^{-(x_1-x_2)^2}$.Storage of matrix is not a problem, I'm working on Xeon processor and has enough memory and moreover, I'm not storing the full matrix and whenever  I need to find an entry in matrix, I use the kernel function and generate type-double number for that cell.
LU decomposition of large dense matrices
optimization;algorithms;c++;matrix