Eigenvalue computations in the context of data-sparse approximations of integral operators

In this work, we consider the numerical solution of a large eigenvalue problem resulting from a finite rank discretization of an integral operator. We are interested in computing a few eigenpairs, with an iterative method, so a matrix representation that allows for fast matrix-vector products is required. Hierarchical matrices are appropriate for this setting, and also provide cheap LU decompositions required in the spectral transformation technique. We illustrate the use of freely available software tools to address the problem, in particular SLEPc for the eigensolvers and HLib for the construction of H-matrices. The numerical tests are performed using an astrophysics application. Results show the benefits of the data-sparse representation compared to standard storage schemes, in terms of computational cost as well as memory requirements.

Cognitive intervention protocol for age-related memory impairments

With the number of elderly people increasing tremendously worldwide, comes the need for effective methods to maintain or improve older adults' cognitive performance. Using continuous neurofeedback, through the use of EEG techniques, people can learn how to train and alter their brain electrical activity. A software platform that puts together the proposed rehabilitation methodology has been developed: a digital game protocol that supports neurofeedback training of alpha and theta rhythms, by reading the EEG activity and presenting it back to the subject, interleaved with neurocognitive tasks such as n-Back and Corsi Block-Tapping. This tool will be used as a potential rehabilitative platform for age-related memory impairments.

Error bounds for low-rank approximations of the first exponential integral kernel

A hierarchical matrix is an efficient data-sparse representation of a matrix, especially useful for large dimensional problems. It consists of low-rank subblocks leading to low memory requirements as well as inexpensive computational costs. In this work, we discuss 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. The operator is of convolution type, it is defined through the first exponential-integral function and, hence, it is weakly singular. We develop analytical expressions for the approximate degenerate kernels and deduce error upper bounds for these approximations. Some computational results illustrating the efficiency and robustness of the approach are presented.