Spectral approximation with matrices issued from discretized operators

In this book, the numerical solution of a large eigenvalue problem is considered, in which the integral operator comes from a radiative transfer problem. It is considered the use of hierarchical matrices, an efficient data-sparse representation of matrices, especially useful for large dimensional problems. It is discussed the use of the hierarchical matrix technique in the numerical solution of a large scale eigenvalue problem arising from a finite rank discretization of an integral operator. HLIB (Hierarchical matrices LIBrary) and SLEPc (Scalable Library for Eigenvalue Problem Computations) are accessed. Moreover, some routines from HLIB are incorporated into SLEPc in order to exploit the available algorithms to solve eigenvalue problems. It is also developed analytical expressions for the approximate degenerate kernels and deducted error upper bounds for these approximations. The numerical results illustrates the efficiency of the techniques developed and implemented in this work.