Calculating the minimal/maximal eigenvalue of symmetric parametrized matrices using projection

In applications of linear algebra including nuclear physics and structural dynamics, there is a need to deal with uncertainty in the matrices. We focus on matrices that depend on a set of parameters $ω$ and we are interested in the minimal eigenvalue of a matrix pencil $(A,B)$ with $A,B$ symmetric and $B$ positive definite. If $ω$ can be interpreted as the realisation of random variables, one may be interested in statistical moments of the minimal eigenvalue. In order to obtain statistical moments, we need a fast evaluation of the eigenvalue as a function of $ω$. Since this is costly for large matrices,we are looking for a small parametrized eigenvalue problem whose minimal eigenvalue makes a small error with the minimal eigenvalue of the large eigenvalue problem. The advantage, in comparison with a global polynomial approximation (on which, e.g., the polynomial chaos approximation relies), is that we do not suffer from the possible non-smoothness of the minimal eigenvalue. The small scale eigenvalue problem is obtained by projection of the large scale problem. Our main contribution is that for constructing the subspace we use multiple eigenvectors as well as derivatives of eigenvectors.We provide theoretical results and document numerical experiments regarding the beneficial effect of adding multiple eigenvectors and derivatives.