Seminar series
Date
Tue, 20 Oct 2026
Time
12:30 -
13:00
Location
L5
Speaker
Francesco Hrobat
Organisation
Mathematical Institute, University of Oxford
We consider a sequence of eigenvalue problems
$$A_i\vec{x}_i=\lambda_i\vec{x}_i,\qquad i=1,\ldots,N,$$
where \(N\) is large and the eigenvalue of interest is usually the largest or smallest one. Often, the matrices are generated by a smooth parameter-dependent matrix function, so that \(A_i=A(p_i)\), with \(A(p)\in\mathbb{C}^{n\times n}\).
Our method uses a small number \(r\ll N\) of parameter values, called \textit{snapshot points}. The eigenvectors computed at these points are used to construct an orthonormal basis \(Q\in\mathbb{C}^{n\times r}\). For each remaining parameter value \(p_i\), we approximate the eigenvector \(\vec{x}_i\) in the subspace spanned by the columns of \(Q\).
To reduce the computational cost, we select a subset of rows of \(A(p_i)\) and use them to form the projection onto this subspace. This is especially useful when the matrices are accessed entry by entry and evaluating an entry is expensive. Once the snapshot points and a suitable set of row indices have been chosen, the same rows can be used for all parameter values in the sequence.