In this work, we perform a spectral analysis of flipped multilevel Toeplitz sequences, i.e., we study the asymptotic spectral behaviour of $\{Y_{\boldsymbol{n}}T_{\boldsymbol{n}}(f)\}_{\boldsymbol{n}}$, where $T_{\boldsymbol{n}}(f)$ is a real, square multilevel Toeplitz matrix generated by a function $f\in L^1([-π,π]^d)$ and $Y_{\boldsymbol{n}}$ is the exchange matrix, which has $1$s on the main anti-diagonal. In line with what we have shown for unilevel flipped Toeplitz matrix sequences, the asymptotic spectrum is determined by a $2\times 2$ matrix-valued function whose eigenvalues are $\pm |f|$. Furthermore, we characterize the eigenvalue distribution of certain preconditioned flipped multilevel Toeplitz sequences with an analysis that covers both multilevel Toeplitz and circulant preconditioners. Finally, all our findings are illustrated by several numerical experiments.
In a series of recent papers the spectral behavior of the matrix sequence $\{Y_nT_n(f)\}$ is studied in the sense of the spectral distribution, where $Y_n$ is the main antidiagonal (or flip matrix) and $T_n(f)$ is the Toeplitz matrix generated by the function $f$, with $f$ being Lebesgue integrable and with real Fourier coefficients. This kind of study is also motivated by computational purposes for the solution of the related large linear systems using the (preconditioned) MINRES algorithm. Here we complement the spectral study with more results holding both asymptotically and for a fixed dim…
Block Schwarz methods and preconditioning strategies using Generalized locally Toeplitz tools - part I: analysis of the preconditioners and numerical validation
Asymptotic spectral properties and preconditioning of an approximated nonlocal Helmholtz equation with Caputo fractional Laplacian and variable coefficient wave number $μ$
In this paper, we analyze the spectra of the preconditioned matrices arising from discretized multi-dimensional Riesz spatial fractional diffusion equations. The finite difference method is employed to approximate the multi-dimensional Riesz fractional derivatives, which will generate symmetric positive definite ill-conditioned multi-level Toeplitz matrices. The preconditioned conjugate gradient method with a preconditioner based on the sine transform is employed to solve the resulting linear system. Theoretically, we prove that the spectra of the preconditioned matrices are uniformly bounded …
We begin by showing that any $n \times n$ matrix can be decomposed into a sum of $n$ circulant matrices with periodic relaxations on the unit circle. This decomposition is orthogonal with respect to a Frobenius inner product, allowing recursive iterations for these circulant components. It is also shown that the dominance of a few circulant components in the matrix allows sparse similarity transformations using Fast-Fourier-transform (FFT) operations. This enables the evaluation of all eigenvalues of dense Toeplitz, block-Toeplitz, and other periodic or quasi-periodic matrices, to a reasonable…
Circulant preconditioners for discrete ill-posed Toeplitz systems
Kent State University（肯特州立大学）；University School, Hunting Valley（亨廷谷大学学校）；SAPIENZA Università di Roma（罗马大学）；Dipartimento di Matematica “Guido Castelnuovo”, SAPIENZA Università di Roma（罗马大学“吉多·卡斯特尔努沃”数学系）
In this paper, we study a parallel-in-time (PinT) algorithm for all-at-once system from a non-local evolutionary equation with weakly singular kernel where the temporal term involves a non-local convolution with a weakly singular kernel and the spatial term is the usual Laplacian operator with variable coefficients. We propose to use a two-sided preconditioning technique for the all-at-once discretization of the equation. Our preconditioner is constructed by replacing the variable diffusion coefficients with a constant coefficient to obtain a constant-coefficient all-at-once matrix. We split a…
Under appropriate technical assumptions, the simple-loop theory allows to deduce various types of asymptotic expansions for the eigenvalues of Toeplitz matrices $T_{n}(f)$ generated by a function $f$, unfortunately, such a theory is not available in the preconditioning setting, that is for matrices of the form $T_{n}^{-1}(g)T_{n}(l)$ with $l,g$ real-valued, $g$ nonnnegative and not identically zero almost everywhere. Independently and under the milder hypothesis that $f=\frac{l}{g}$ is even and monotonic over $[0,π]$, matrix-less algorithms have been developed for the fast eigenvalue computati…
It is well known that the discretization of fractional diffusion equations (FDEs) with fractional derivatives $α\in(1,2)$, using the so-called weighted and shifted Grünwald formula, leads to linear systems whose coefficient matrices show a Toeplitz-like structure. More precisely, in the case of variable coefficients, the related matrix sequences belong to the so-called Generalized Locally Toeplitz (GLT) class. Conversely, when the given FDE have constant coefficients, using a suitable discretization, we encounter a Toeplitz structure associated to a nonnegative function $\mathcal{F}_α$, called…
Analysis of Block Jacobi/Gauss-Seidel and additive/multiplicative Schwarz preconditioning through the theory of GLT sequences, with applications to domain decomposition discretizations
University of Rome Tor Vergata（罗马托尔维加塔大学）；University Mohammed VI Polytechnic（穆罕默德六世polytechnic大学）；University of Insubria（因苏布里亚大学）；Uppsala University（乌普萨拉大学）
