Matrix-Less Methods for Computing Eigenvalues of Large Structured Matrices

  • Datum:
  • Plats: 2446 ITC, Lägerhyddsvägen 2, hus 2, Uppsala
  • Doktorand: Ekström, Sven-Erik
  • Om avhandlingen
  • Arrangör: Avdelningen för beräkningsvetenskap
  • Kontaktperson: Ekström, Sven-Erik
  • Disputation

When modeling natural phenomena with linear partial differential equations, the discretized system of equations is in general represented by a matrix. To solve or analyze these systems, we are often interested in the spectral behavior of these matrices.

Whenever the matrices of interest are Toeplitz, or Toeplitz-like, we can use the theory of Generalized Locally Toeplitz (GLT) sequences to study the spectrum (eigenvalues). A central concept in the theory of GLT sequences is the so-called symbol, that is, a function associated with a sequence of matrices of increasing size. When sampling the symbol and when the related matrix sequence is Hermitian (or quasi-Hermitian), we obtain an approximation of the spectrum of a matrix of a fixed size and we can therefore see its general behavior. However, the so-computed approximations of the eigenvalues are often affected by errors having magnitude of the reciprocal of the matrix size.

In this thesis we develop novel methods, which we call "matrix-less" since they neither store the matrices of interest nor depend on matrix-vector products, to estimate these errors. Moreover, we exploit the structures of the considered matrices to efficiently and accurately compute the spectrum.

We begin by considering the errors of the approximate eigenvalues computed by sampling the symbol on a uniform grid, and we conjecture the existence of an asymptotic expansion for these errors. We devise an algorithm to approximate the expansion by using a small number of moderately sized matrices, and we show through numerical experiments the effectiveness of the algorithm. We also show that the same algorithm works for preconditioned matrices, a result which is important in practical applications. Then, we explain how to use the approximated expansion on the whole spectrum for large matrices, whereas in earlier works its applicability was restricted only to certain matrix sizes and to a subset of the spectrum. Next, we demonstrate how to use the so-developed techniques to investigate, solve, and improve the accuracy in the eigenvalue computations for various differential problems discretized by the isogeometric analysis (IgA) method. Lastly, we discuss a class of non-monotone symbols for which we construct the sampling grid yielding exact eigenvalues and eigenvectors.

To summarize, we show, both theoretically and numerically, the applicability of the presented matrix-less methods for a wide variety of problems.