Verified Error Bounds for Linear Systems Through the Lanczos Process

This paper discusses a class of two-stage iterations whose outer iterative methods are the block SOR methods for parallel solution of linear systems. Convergence is showed for symmetric positive definite linear systems, and an approximate optimal relaxation factor is defined for block tridiagonal matrices. The numerical results for Poisson model problem are presented.

Some results on convergence of two-stage iterative methods for symmetric positive definite linear systems

Direct optimization of BPX preconditioners

The simulation of flow in fractured media requires handling both a large number of fractures and a complex interconnecting network of these fractures. Networks considered in this paper are three-dimensional domains made up of two-dimensional fractures intersecting each other and randomly generated. Due to the stochastic generation of fractures, intersections can be highly intricate. The numerical method must generate a mesh and define a discrete problem for any discrete fracture network (DFN). A first approach [Erhel, de Dreuzy, and Poirriez, SIAM J. Sci. Comput., 31 (2009), pp. 2688–2705] is to generate a conforming mesh and to apply a mixed hybrid finite element method. However, the resulting linear system becomes very large when the network contains many fractures. Hence a second approach [Pichot, Erhel, and de Dreuzy, Appl. Anal., 89 (2010), pp. 1629–1643] is to generate a nonconforming mesh, using an independent mesh generation for each fracture. Then a Mortar technique applied to the mixed hybrid finite element method deals with the nonmatching grids. When intersections do not cross or overlap, pairwise Mortar relations for each intersection are efficient [Pichot, Erhel, and de Dreuzy, 2010]. But for most random networks, discretized intersections involve more than two fractures. In this paper, we design a new method generalizing the previous one and that is applicable for stochastic networks. The main idea is to combine pairwise Mortar relations with additional relations for the overlapping part. This method still ensures the continuity of fluxes and heads and still yields a symmetric positive definite linear system. Numerical experiments show the efficiency of the method applied to complex stochastic fracture networks. We also study numerical convergence when reducing the mesh step. This method makes it easy to perform mesh optimization and appears to be a very promising tool to simulate flow in multiscale fracture networks.

Many scientific applications require one to solve successively linear systems Ax=b with different right-hand sides b and a symmetric positive definite matrix A. The conjugate gradient method applied to the first system generates a Krylov subspace which can be efficiently recycled thanks to orthogonal projections in subsequent systems. A modified conjugate gradient method is then applied with a specific initial guess and initial descent direction and a modified descent direction during the iterations. This paper gives new theoretical results for this method and proposes a new version. Numerical experiments show the efficacy of our method even for quite different right-hand sides.

Some numerical experiences on convergence criteria for iterative finite element solvers

Alternating cyclic vector extrapolation technique for accelerating nonlinear optimization algorithms and fixed-point mapping applications

A least-squares spectral collocation formulation for the Navier–Stokes problem is presented. By this new approach the well known Babubska–Brezzi condition can be avoided. Here we are able to employ polynomials of the same degree both for the velocity components and for the pressure. The collocation conditions and the boundary conditions lead to a overdetermined system which can be efficiently solved by least-squares. The solution technique will only involve symmetric positive definite linear systems. The numerical simulations confirm the usual exponential rate of convergence for the spectral scheme.

Feedback stabilization of a second-order system: A nonmodal approach

In this paper, we discuss a discontinuous Galerkin finite (DG) element method for linear free surface gravity waves. We prove that the algorithm is unconditionally stable and does not require additional smoothing or artificial viscosity terms in the free surface boundary condition to prevent numerical instabilities on a non-uniform mesh. A detailed error analysis of the full time-dependent algorithm is given, showing that the error in the wave height and velocity potential in the L2-norm is in both cases of optimal order and proportional to O(Δt2+hp+1), without the need for a separate velocity reconstruction, with p the polynomial order, h the mesh size and Δt the time step. The error analysis is confirmed with numerical simulations. In addition, a Fourier analysis of the fully discrete scheme is conducted which shows the dependence of the frequency error and wave dissipation on the time step and mesh size. The algebraic equations for the DG discretization are derived in a way suitable for an unstructured mesh and result in a symmetric positive definite linear system. The algorithm is demonstrated on a number of model problems, including a wave maker, for discretizations with accuracy ranging from second to fourth order.

The two-stage iterative methods for symmetric positive definite matrices

The adaptive s-step CG algorithm is a solver for sparse symmetric positive definite linear systems designed to reduce the synchronization cost per iteration while still achieving a user-specified accuracy requirement. In this work, we improve the adaptive s-step conjugate gradient algorithm by the use of iteratively updated estimates of the largest and smallest Ritz values, which give approximations of the largest and smallest eigenvalues of A, using a technique due to G.Meurant and P.Tichý (2018). The Ritz value estimates are used to dynamically update parameters for constructing Newton or Chebyshev polynomials so that the conditioning of the s-step bases can be continuously improved throughout the iterations. These estimates are also used to automatically set a variable related to the ratio of the sizes of the error and residual, which was previously treated as an input parameter. We show through numerical experiments that in many cases the new algorithm improves upon the previous adaptive s-step approach both in terms of numerical behavior and reduction in number of synchronizations.

During the process of typhoon forecasting, numerious symmetric positive definite linear systems are needed to be solved. They are often solved by conjugate gradient method with preconditioning technique. This paper focuses on the convergence rate analysis of conjugate gradient method. The properties of Chebyshev polynomial and Krylov subspace are utilized. The effect of the right-hand-side vector are considered. Several convergence rate estimations are given. Compared with the existing estimation results, the presented results are more exact. This enable us to construct more efficient preconditioners to forecast typhoon more quickly.

A cell functional minimization scheme for parabolic problem

We present an efficient discrete Hodge decomposition for velocity fields defined over irregular domains in two and three dimensions using a virtual node framework. The method is designed for use in the exact projection discretization of incompressible flow. We leverage the Poisson framework initially developed in [1] and [2]. This approach uses a signed distance function to represent the irregular domain embedded in a Cartesian grid and uses a variational approach to create a symmetric positive definite linear system. We present a novel modification to the previous approach that yields a 5-point stencil (7-point in 3D) across the entire computational domain. The original algorithm required a 9-point stencil (27-point in 3D) near the embedded irregular boundary. We show that this new condensed stencil enables a decomposition of the form A = G T M 1 G, where M is a diagonal weighting matrix and G and D = G T are diagonal scalings of the standard central-difference gradient and divergence operators. We use this factored form as the basis of our discrete Hodge decomposition and show that this can be readily used for exact projection in incompressible flow. Numerical experiments suggest our method is second-order in L ¥ for pressures and first-order in L ¥ , second-order in L 1 for velocities.