Data-sparse approximation on the computation of a weakly singular Fredholm equation: A stellar radiative transfer application

Abstract

Data-sparse representation techniques are emerging on computing approximate solutions for large scale problems involving matrices with low numerical rank. This representation provides both low memory requirements and cheap computational costs.In this work we consider the numerical solution of a large dimensional problem resulting from a finite rank discretization of an integral radiative transfer equation in stellar atmospheres. The integral operator, defined through the first exponential-integral function, is of convolution type and weakly singular.Hierarchically semiseparable representation of the matrix operator with low-rank blocks is built and data-sparse matrix computations can be performed with almost linear complexity. This representation of the original fully populated matrix is an algebraic multilevel structure built from a specific hierarchy of partitions of the matrix indices.Numerical tests illustrate the benefits of this matrix technique compared to standard storage schemes, dense and sparse, in terms of computational cost as well as memory requirements.This approach is particularly useful when a fine discretization of the integral equation is required and the resulting linear system of equations is of large dimension and numerically difficult to solve.