[00727] Recent Advances in Fast Iterative Methods for PDE Problems
Session Date & Time :
00727 (1/3) : 4C (Aug.24, 13:20-15:00)
00727 (2/3) : 4D (Aug.24, 15:30-17:10)
00727 (3/3) : 4E (Aug.24, 17:40-19:20)
Type : Proposal of Minisymposium
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.
Xian-Ming Gu (Southwestern University of Finance and Economics)
Isabella Furci (The University of Wuppertal)
Sean Hon (Hong Kong Baptist University)
Xuelei Lin (Harbin Institute of Technology)
Jiamei Dong (Hong Kong Baptist University)
Xin Huang (Huazhong University of Science and Technology)
Talks in Minisymposium :
[02883] Multigrid Methods for Saddle-Point Matrices with Structured Blocks
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.
[02906] Some step-size independent theoretical bounds for preconditioning techniques of discrete PDEs
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.
[03484] A block α-circulant based preconditioned MINRES method for evolutionary PDEs
Author(s) :
Sean Hon (Hong Kong Baptist University)
Abstract : In this work, we propose a novel parallel-in-time preconditioner for an all-at-once system arising from the numerical solution of evolutionary partial differential equations (PDEs). Namely, considering the wave equation as a model problem, our main result concerns a block $\alpha$-circulant matrix based preconditioner that can be fast diagonalized via fast Fourier transforms, whose effectiveness is theoretically supported for the modified block Toeplitz system arising from discretizing the concerned wave equation. Namely, after first transforming the original all-at-once linear system into a symmetric one, we develop the desired preconditioner based on the spectral information of the modified matrix. Then, we show that the eigenvalues of the preconditioned matrix are clustered around $1$ without any extreme outlier far away from the clusters. In other words, mesh-independent convergence is theoretically guaranteed when the minimal residual method is employed. Moreover, our proposed solver is further generalized to a full block triangular Toeplitz system which arises when a high order discretization scheme is used. Numerical experiments are given to support the effectiveness of our preconditioner, showing that the expected optimal convergence can be achieved.
This is joint work with Xuelei Lin (Harbin Institute of Technology).
[04605] A single-sided all-at-once preconditioning for linear system from a non-local evolutionary equation with weakly singular kernels
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 .
[04769] A rational preconditioner for multi-dimensional Riesz fractional diffusion equations
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.
[04933] Fast algorithms for space fractional Cahn-Hilliard equations
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
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.
[05299] New Linear Solvers Features and Improvements in Trilinos
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.
