Two absolute-value based preconditioned MINRES methods for Crank-Nicolson discretization of the viscoelastic equation

SummaryPreconditioned Krylov subspace (KSP) methods are widely used for solving large‐scale sparse linear systems arising from numerical solutions of partial differential equations (PDEs). These linear systems are often nonsymmetric due to the nature of the PDEs, boundary or jump conditions, or discretization methods. While implementations of preconditioned KSP methods are usually readily available, it is unclear to users which methods are the best for different classes of problems. In this work, we present a comparison of some KSP methods, including GMRES, TFQMR, BiCGSTAB, and QMRCGSTAB, coupled with three classes of preconditioners, namely, Gauss–Seidel, incomplete LU factorization (including ILUT, ILUTP, and multilevel ILU), and algebraic multigrid (including BoomerAMG and ML). Theoretically, we compare the mathematical formulations and operation counts of these methods. Empirically, we compare the convergence and serial performance for a range of benchmark problems from numerical PDEs in two and three dimensions with up to millions of unknowns and also assess the asymptotic complexity of the methods as the number of unknowns increases. Our results show that GMRES tends to deliver better performance when coupled with an effective multigrid preconditioner, but it is less competitive with an ineffective preconditioner due to restarts. BoomerAMG with a proper choice of coarsening and interpolation techniques typically converges faster than ML, but both may fail for ill‐conditioned or saddle‐point problems, whereas multilevel ILU tends to succeed. We also show that right preconditioning is more desirable. This study helps establish some practical guidelines for choosing preconditioned KSP methods and motivates the development of more effective preconditioners.

We consider the bilinear optimal control of an advection-reaction-diffusion system, where the control arises as the velocity field in the advection term. Such a problem is generally challenging from both theoretical analysis and algorithmic design perspectives, mainly because the state variable depends nonlinearly on the control variable and, an additional divergence-free constraint on the control is coupled together with the state equation. Mathematically, the proof of the existence of optimal solutions is delicate, and, up to now, only some results have been known for a few special cases where additional restrictions are imposed on the space dimension and the regularity of the control. We prove the existence of optimal controls and derive the first-order optimality conditions in general settings without any extra assumptions. Computationally, the well-known conjugate gradient (CG) method can be applied conceptually. However, due to the additional divergence-free constraint on the control variable and the nonlinear relation between the state and control variables, it is challenging to compute the gradient and the optimal stepsize at each CG iteration, and thus nontrivial to implement the CG method. To address these issues, we advocate a fast inner preconditioned CG method to ensure the divergence-free constraint and an efficient inexactness strategy to determine an appropriate stepsize. An easily implementable nested CG method is thus proposed for solving such a complicated problem. For the numerical discretization, we combine finite difference methods for the time discretization and finite element methods for the space discretization. Efficiency of the proposed nested CG method is promisingly validated by the results of some preliminary numerical experiments.

Thermal analysis is crucial for determining the propagation of heat and tracking the formation of hot spots in advanced integrated circuit technologies. At the core of the thermal analysis for integrated circuits is the numerical solution of the heat equation. Prior academic thermal analysis tools typically compute temperature by applying finite difference methods on uniform grids with time integration methods having fixed time step size. Additionally, the linear systems arising from the discretized heat equation are solved using direct methods based on matrix factorizations. Direct methods, however, do not scale well as the problem size increases. Moreover, most of the tools support only 2-D or a limited number of 3-D technologies. To address these issues, this paper presents a novel thermal analyzer with the ability to model both 2-D and 3-D circuit technologies. The analyzer solves the heat equation using the finite element method for the spatial discretization coupled with implicit time integration methods for advancing the solution in time. It also offers fully adaptive spatio-temporal refinement features for improved accuracy and computational efficiency. The resulting linear systems are solved by a multigrid preconditioned Krylov subspace iterative method, which gives superior performance for 3-D transient analyses. The analyzer is shown to accurately capture the propagation of heat in both the horizontal and vertical directions of integrated systems.

We study linear systems of equations arising from a stochastic Galerkin finite element discretization of saddle point problems with random data and its iterative solution. We consider the Stokes flow model with random viscosity described by the exponential of a correlated random process and shortly discuss the discretization framework and the representation of the emerging matrix equation. Due to the high dimensionality and the coupling of the associated symmetric, indefinite, linear system, we resort to iterative solvers and problem-specific preconditioners. As a standard iterative solver for this problem class, we consider the block diagonal preconditioned MINRES method and further introduce the Bramble-Pasciak conjugate gradient method as a promising alternative. This special conjugate gradient method is formulated in a non-standard inner product with a block triangular preconditioner. From a structural point of view, such a block triangular preconditioner enables a better approximation of the original problem than the block diagonal one. We derive eigenvalue estimates to assess the convergence behavior of the two solvers with respect to relevant physical and numerical parameters and verify our findings by the help of a numerical test case. We model Stokes flow in a cavity driven by a moving lid and describe the viscosity by the exponential of a truncated Karhunen-Loeve expansion. Regarding iteration counts, the Bramble-Pasciak conjugate gradient method with block triangular preconditioner is superior to the MINRES method with block diagonal preconditioner in the considered example.

We consider a generalized Stokes equation with problem parameters ξ⩾0 (size of the reaction term) and ν>0 (size of the diffusion term). We apply a standard finite element method for discretization. The main topic of the paper is a study of efficient iterative solvers for the resulting discrete saddle point problem. We investigate a coupled multigrid method with Braess–Sarazin and Vanka‐type smoothers, a preconditioned MINRES method and an inexact Uzawa method. We present a comparative study of these methods. An important issue is the dependence of the rate of convergence of these methods on the mesh size parameter and on the problem parameters ξ and ν. We give an overview of the main theoretical convergence results known for these methods. For a three‐dimensional problem, discretized by the Hood–Taylor 𝒫2–𝒫1 pair, we give results of numerical experiments. Copyright © 2007 John Wiley & Sons, Ltd.

We develop an implicit discretization method for pricing European and American options when the underlying asset is driven by an infinite activity Lévy process. For processes of finite variation, quadratic convergence is obtained as the mesh and timestep are refined. For infinite variation processes, better than first-order accuracy is achieved. The jump component in the neighborhood of log jump size zero is specially treated by using a Taylor expansion approximation and the drift term is dealt with using a semi-Lagrangian scheme. The resulting partial integro-differential equation is then solved using a preconditioned BiCGSTAB method coupled with a fast Fourier transform. Proofs of fully implicit timestepping stability and monotonicity are provided. The convergence properties of the BiCGSTAB scheme are discussed and compared with a fixed point iteration. Numerical tests showing the convergence and performance of this method for European and American options under processes of finite and infinite variation are presented.

Problem statement: We consider the numerical solvers for the linearized Navier-Stokes problem. Both the Stokes problem and Oseen problems are considered. Approach: We used the Mark and Cell (MAC) discretization method to discretize the Navier-Stokes equations. We used preconditioned Krylov subspace methods to solve the resulting linear systems. Results: Numerical experimental results are performed to compare the different preconditioners. Conclusion: The choice of the preconditioner is highly problem dependent and we give the suggestions for individual cases.

SUMMARYRobust computational procedures for the solution of non‐hydrostatic, free surface, irrotational and inviscid free‐surface water waves in three space dimensions can be based on iterative preconditioned defect correction (PDC) methods. Such methods can be made efficient and scalable to enable prediction of free‐surface wave transformation and accurate wave kinematics in both deep and shallow waters in large marine areas or for predicting the outcome of experiments in large numerical wave tanks. We revisit the classical governing equations are fully nonlinear and dispersive potential flow equations. We present new detailed fundamental analysis using finite‐amplitude wave solutions for iterative solvers. We demonstrate that the PDC method in combination with a high‐order discretization method enables efficient and scalable solution of the linear system of equations arising in potential flow models. Our study is particularly relevant for fast and efficient simulation of non‐breaking fully nonlinear water waves over varying bottom topography that may be limited by computational resources or requirements. To gain insight into algorithmic properties and proper choices of discretization parameters for different PDC strategies, we study systematically limits of accuracy, convergence rate, algorithmic and numerical efficiency and scalability of the most efficient known PDC methods. These strategies are of interest, because they enable generalization of geometric multigrid methods to high‐order accurate discretizations and enable significant improvement in numerical efficiency while incuring minimal storage requirements. We demonstrate robustness using such PDC methods for practical ranges of interest for coastal and maritime engineering, that is, from shallow to deep water, and report details of numerical experiments that can be used for benchmarking purposes. Copyright © 2014 John Wiley & Sons, Ltd.

A time-derivative preconditioned system of equations suitable for the numerical simulation of single and multicomponent inviscid flows at all speeds is formulated. For low speed multicomponent flow, a preconditioned nonconservative discretization method is described which preserves pressure and velocity equilibrium across fluid interfaces. This method is extended to transonic and supersonic flows using a hybrid approach that combines conservative and nonconservative formulations. Both multicomponent (multiple fluids) and multiphase (same fluid in different phases) can be solved using the proposed methods.

A finite element penalty method for the linearized viscoelastic Oldroyd fluid motion equations

On convergence of the inexact Rayleigh quotient iteration with MINRES

We consider saddle point problems that result from the finite element discretization of stationary and nonstationary Stokes equations. Three efficient iterative solvers for these problems are treated, namely, the preconditioned conjugate gradient method introduced by Bramble and Pasciak, the preconditioned MINRES method, and a method due to Bank et al. We give a detailed overview of algorithmic aspects and theoretical convergence results. For the method of Bank et al. a new convergence analysis is presented. A comparative study of the three methods for a three-dimensional Stokes problem discretized by the Hood--Taylor P2-P1 finite element pair is given.

In this paper, we present and discuss the results of our numerical studies of preconditioned MinRes methods for solving the optimality systems arising from the multiharmonic finite element approximations to time-periodic eddy current optimal control problems in different settings including different observation and control regions, different tracking terms, as well as box constraints for the Fourier coefficients of the state and the control. These numerical studies confirm the theoretical results published by the first author in a recent paper.

A sine transform based preconditioned MINRES method for all-at-once systems from constant and variable-coefficient evolutionary PDEs

We present a novel monolithic divergence-conforming HDG scheme for a linear fluid-structure interaction problem with a thick structure. A pressure-robust optimal energy-norm estimate is obtained for the semidiscrete scheme. When combined with a Crank--Nicolson time discretization, our fully discrete scheme is energy stable and produces an exactly divergence-free fluid velocity approximation. The resulting linear system, which is symmetric and indefinite, is solved using a preconditioned MinRes method with a robust block algebraic multigrid preconditioner.