When solving linear systems with nonsymmetric Toeplitz or multilevel Toeplitz matrices using Krylov subspace methods, the coefficient matrix may be symmetrized. The preconditioned MINRES method can then be applied to this symmetrized system, which allows rigorous upper bounds on the number of MINRES iterations to be obtained. However, effective preconditioners for symmetrized (multilevel) Toeplitz matrices are lacking. Here, we propose novel ideal preconditioners, and investigate the spectra of the preconditioned matrices. We show how these preconditioners can be approximated and demonstrate their effectiveness via numerical experiments.
A limited-memory block bi-diagonal Toeplitz preconditioner for block lower triangular Toeplitz system from time-space fractional diffusion equation
University of Electronic Science and Technology of China（电子科技大学）；Southwestern University of Finance and Economics（西南财经大学）；Lanzhou University of Technology（兰州理工大学）
In recent years, motivated by computational purposes, the singular value and spectral features of the symmetrization of Toeplitz matrices generated by a Lebesgue integrable function have been studied. Indeed, under the assumptions that $f$ belongs to $L^1([-π,π])$ and it has real Fourier coefficients, the spectral and singular value distribution of the matrix-sequence $\{Y_nT_n[f]\}_n$ has been identified, where $n$ is the matrix-size, $Y_n$ is the anti-identity matrix, and $T_n[f]$ is the Toeplitz matrix generated by $f$. In this note, we consider the multilevel Toeplitz matrix $T_{\bf n}[f]$…
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 thesis we study the preconditioning of square, non-symmetric and real Toeplitz systems. We prove theoretical results, which constitute sufficient conditions for the efficiency of the proposed preconditioners and the fast convergence to the solution of the system, by the Preconditioned Generalized Minimal Residual method (PGMRES) as well as by the Preconditioned Conjugate Gradient method applied to the system of Normal Equations (PCGN). As introduction, in the first chapter, we give the basic definitions and theorems/lemmas that we use to prove the theoretical results of the thesis. The…
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…
Block Schwarz methods and preconditioning strategies using Generalized locally Toeplitz tools - part I: analysis of the preconditioners and numerical validation
National Cheng Kung University（国立成功大学）；National Center for Theoretical Sciences, National Cheng Kung University（国立成功大学国家理论科学中心）；Capital Normal University（首都师范大学）；National Sun Yat-sen University（国立中山大学）
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 |…
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 …
On $τ$-preconditioners for a quasi-compact difference scheme to Riesz fractional diffusion equations with variable coefficients
Hanshan Normal University（韩山师范学院）；Southwestern University of Finance and Economics（西南财经大学）；University of Insubria（因苏布里亚大学）；University of Uppsala（乌普萨拉大学）
Generalized preconditioned conjugate gradients for adaptive FEM with optimal complexity
Sorbonne Université, Université Paris Cité, CNRS, Laboratoire Jacques-Louis Lions（索邦大学，巴黎 Cité 大学，法国国家科学研究中心，雅克-路易·利翁斯实验室）；TU Wien, Institute of Analysis and Scientific Computing（维也纳工业大学，分析与科学计算研究所）
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）
This paper discusses parGeMSLR, a C++/MPI software library for the solution of sparse systems of linear algebraic equations via preconditioned Krylov subspace methods in distributed-memory computing environments. The preconditioner implemented in parGeMSLR is based on algebraic domain decomposition and partitions the symmetrized adjacency graph recursively into several non-overlapping partitions via a p-way vertex separator, where p is an integer multiple of the total number of MPI processes. From a numerical perspective, parGeMSLR builds a Schur complement approximate inverse preconditioner a…
