Abstract : Solving vast sparse or structured linear systems is one of the major tasks when solving partial differential equation (PDE) problems in many applications in science and engineering. Iterative solvers such as Krylov subspace methods and the multigrid methods, with effective preconditioners, are often used for solving these linear systems efficiently. This minisymposium brings various researchers and experts in these areas together to present the latest development of iterative methods, preconditioners, and linear algebra software.
[02883] Multigrid Methods for Saddle-Point Matrices with Structured Blocks
Format : Talk at Waseda University
Author(s) :
Isabella Furci (University of Genoa)
Matthias Bolten (University of Wuppertal)
Marco Donatelli (University of Insubria)
Paola Ferrari (University of Wuppertal)
Abstract : We consider efficient multigrid methods for large linear systems with particular saddle-point structures. Often, powerful smoothers are used to take into account the special coupling. Alternatively, Notay recently proposed an algebraic approach that analyzes properly preconditioned saddle-point problems.
We provide theoretical tools to analyze the latter procedure when applied to saddle-point systems whith (multilevel) block-Toeplitz blocks. As a test problem, we consider the linear system stemming from the Finite Element approximation of the Stokes equation.
[05299] New Linear Solvers Features and Improvements in Trilinos
Format : Online Talk on Zoom
Author(s) :
Jennifer Ann Loe (Sandia National Laboratories)
Ichitaro Yamazaki (SNL)
Sivasankaran Rajamanickam (Sandia National Laboratories)
Heidi Thornquist (Sandia National Laboratories)
Christian Glusa (Sandia National Laboratories)
Abstract : Trilinos is a large open-source mathematical software library which includes algorithms for discretization, optimization, preconditioners, non-linear solvers, and linear solvers. Its Tpetra linear algebra backend allows it to run effectively on highly parallel computers and various GPU accelerators. In this talk, we discuss recent improvements and additions to the Trilinos linear solver capabilities. One addition allows mixed precision solver and preconditioner combinations. Another recent improvement provides an abstract interface for small dense matrices in the linear solvers, eliminating extra data movement for GPU-based computers. We will demonstrate potential performance gains with use of the new linear solvers features.
[04769] A rational preconditioner for multi-dimensional Riesz fractional diffusion equations
Format : Online Talk on Zoom
Author(s) :
Mariarosa Mazza (University of Insubria)
Lidia Aceto (University of Eastern Piedmont)
Abstract : Starting from a rational approximation of the Riesz operator expressed as the integral of the standard heat diffusion semigroup, we propose a rational preconditioner for solving linear systems arising from the finite difference/element discretization of multi-dimensional Riesz fractional diffusion equations. We show that, despite the lack of clustering just as for the Laplacian, for fractional orders close to 1 our preconditioner provides better results than the Laplacian itself, while sharing the same computational complexity.
[02906] Some step-size independent theoretical bounds for preconditioning techniques of discrete PDEs
Format : Talk at Waseda University
Author(s) :
Xuelei Lin (Harbin Institute of Technology Shenzhen)
Abstract : It is well-known that the condition number of coefficient matrices arising from discretization of differential equations increases as the grid gets refined, because of which iterative solvers converge slowly for the linear systems when the grid is dense. In this talk, preconditioning techniques for discretization of some differential equations are introduced, with which some Krylov subspace solvers for the preconditioned systems are proven to have a linear convergence rate independent of the stepsizes. Numerical results are also reported.
[04605] A single-sided all-at-once preconditioning for linear system from a non-local evolutionary equation with weakly singular kernels
Format : Talk at Waseda University
Author(s) :
Xuelei Lin (Harbin Institute of Technology Shenzhen)
Jiamei Dong (Hong Kong Baptist University)
Sean Hon (Hong Kong Baptist University)
Abstract : We propose a preconditioning technique for the multilevel Toeplitz all-at-once linearsystem arising from a time-space fractional diffusion equation. The preconditioning tech-nique is based on replacing the spatial discretization matrix with aτ-matrix, due to whichthe preconditioner can be fast inverted. Theoretically, we show that the condition numberof the intermediate two-sided preconditioned matrix is bounded by 3. And the norm of ourpreconditioner residual is bounded by the residual of intermediate preconditioner .
[04933] Fast algorithms for space fractional Cahn-Hilliard equations
Format : Online Talk on Zoom
Author(s) :
Xin Huang (Huazhong University of Science and Technology)
Abstract : In this talk, the space fractional Cahn-Hilliard (CH) equation is considered. Combining the scale auxiliary variable (SAV) technique with the leapfrog scheme, an unconditional energy-stable, non-couple and linearly implicit numerical scheme is derived. The fully-discrete scheme gives rise to an ill-conditioned system. The Krylov subspace method combing with the preconditioning technique is adopted to solve the resulting system. Numerical results are given to show the efficiency of the proposed method.
[05262] A parallel preconditioner for the all-at-once linear system from evolutionary PDEs with Crank-Nicolson discretization in time
Format : Online Talk on Zoom
Author(s) :
Xian-Ming Gu (Southwestern University of Finance and Economics)
Yong-Liang Zhao (Sichuan Normal University)
Abstract : The Crank-Nicolson (CN) method is a fashionable time integrator for evolutionary partial differential equations (PDEs) arisen in many areas of applied mathematics, however since the solution at any time depends on the solution at previous time steps, thus the CN method will be inherently difficult to parallelize. In this talk, we consider a parallel approach for the solution of evolutionary PDEs with the CN scheme. Using an allat-once approach, we can solve for all time steps simultaneously using a parallelizable over time preconditioner within a standard iterative method. Due to the diagonalization of the proposed preconditioner, we can minutely prove that most eigenvalues of preconditioned matrices are equal to 1 and the others $z\in\mathbb{C}$ have the model with 1/(1 + α) < |z| < 1/(1 - α), where 0 < α < 1 is a free parameter. Meanwhile, the efficient and parallel implementation of this proposed preconditioner is described in details. Finally, we will verify our theoretical findings via numerical experiments.
[05468] A preconditioned MINRES method for optimal control of wave equations
Format : Talk at Waseda University
Author(s) :
Sean Hon (Hong Kong Baptist University)
Abstract : In this work, we propose a novel preconditioned Krylov subspace method for solving an optimal control problem of wave equations, after explicitly identifying the asymptotic spectral distribution of the involved sequence of linear coefficient matrices from the optimal control problem. Namely, we first show that the all-at-once system stemming from the wave control problem is associated to a structured coefficient matrix-sequence possessing an eigenvalue distribution. Then, based on such a spectral distribution of which the symbol is explicitly identified, we develop an ideal preconditioner and two parallel-in-time preconditioners for the saddle point system composed of two block Toeplitz matrices. For the ideal preconditioner, we show that the eigenvalues of the preconditioned matrix-sequence all belong to the set $\left(-\frac{3}{2},-\frac{1}{2}\right)\bigcup \left(\frac{1}{2},\frac{3}{2}\right)$ well separated from zero, leading to mesh-independent convergence when the minimal residual method is employed. The proposed parallel-in-time preconditioners can be implemented efficiently using fast Fourier transforms or discrete sine transforms, and their effectiveness is theoretically shown in the sense that the eigenvalues of the preconditioned matrix-sequences are clustered around $\pm 1$, which leads to rapid convergence. When these parallel-in-time preconditioners are not fastly diagonalizable, we further propose modified versions which can be efficiently inverted. Several numerical examples are reported to verify our derived localization and spectral distribution result and to support the effectiveness of our proposed preconditioners.
