Numerical study of a double preconditioning strategy

Two classes of SSOR-type incomplete block factorization methods are proposed for preconditioning of linear algebraic systems of equations with block banded matrices of complex structure. Correctness conditions are derived for these methods in application to M-matrices and their efficiency is demonstrated by numerical experiments with linear algebraic systems obtained by discretization of the three-dimensional Poisson equation using quadratic and cubic serendipity finite elements.

The purpose of the article is to present and evaluate usefulness of a new preconditioning technique (for the Gauss-Seidel algorithm), namely incomplete WZ factorization, for iterative solving of sparse and singular linear equations systems, which arise during modeling with Markov chains. The incomplete WZ factorization proposed here will be compared with the incomplete LU factorization in respect of amount of fill-in (newly created non-zeros) and in respect of the accuracy improvement of preconditioned algorithms in relation to not preconditioned ones. In the paper, the results of some numerical experiments will be presented, which were conducted for various matrices representing Markov chains. The experiments show that the incomplete WZ factorization can be a real alternative - because it is faster than incomplete LU factorization and the fill-in generated in the process is smaller (the output matrices are sparser).

The boundary element method (BEM) is an efficient numerical method that approximates solutions of various boundary value problems. Despite its success little research has been performed on the conditioning of the linear systems that appear in the BEM. For a Laplace equation with Dirichlet boundary conditions a remarkable phenomenon is observed; the corresponding boundary integral equation (BIE) is singular for a certain critical size of the 2D domain. As a consequence the discrete counterpart of the BIE, the linear system, is singular too, or at least ill-conditioned. This is reflected by the condition number of the system matrix, which is infinitely large, or at least very large. When the condition number of the BEM-matrix is large, the linear system is difficult to solve and the solution of the system is very sensible to perturbations in the boundary data. For a Laplace equation with mixed boundary conditions a similar phenomenon is observed. The corresponding BEM-matrix consists of two blocks; one block originates from the BEM-matrix belonging to the Dirichlet problem, the other block originates from the BEM-matrix belonging to the Neumann problem. The composite matrix inherits the solvability problems from the Dirichlet block. In other words, for the Laplace equation with mixed boundary conditions there exists also a critical size of the 2D domain for which the BEM-matrix has an infinitely large condition number. Hence the size and shape of the domain affects the solvability of the BEM problem. The critical size of the domain for which the BIE becomes singular is related to the logarithmic capacity of the domain. The logarithmic capacity is a positive real number that is a function of the size and shape of the domain. If this logarithmic capacity is equal to one, the domain is a critical domain, and for this domain the BIE becomes singular. Thus by computing the logarithmic capacity we can a-priori determine whether the BIE will be singular or not. The logarithmic capacity depends linearly on the scale of the domain, and thus a domain with logarithmic capacity equal to one can always be found by rescaling the domain. Unfortunately the logarithmic capacity can only be computed analytically for a few simple domains; for more involved domains the logarithmic capacity can be estimated though. There are several possibilities to avoid large condition numbers, i.e. singular BIEs that appear at critical domains. The first option is to rescale the domain such that the logarithmic capacity is unequal to one. One can also add a supplementary condition to the BIE and the linear system. A drawback of this option is that the linear system has more equations than unknowns and different techniques are required to solve the system. A third option is to slightly modify the fundamental solution of the Laplace operator. This fundamental solution directly appears in the BIE and it can be shown that a suitable modification yields BIEs that do not become singular. The critical domains for which the BIEs become singular do not restrict to Laplace equations only. Also for BIEs applied to the biharmonic equation or the elastostatic equations and the Stokes equations such critical domains exist. As the last two equations are vectorial equations, also the corresponding BIE consists of two equations. As a consequence two critical domains can be found for which these BIEs become singular. To obtain nonsingular BIEs techniques similar to the Laplace case can be used. Unfortunately we cannot a-priori determine the sizes for which the BIEs becomes singular, and thus do not know to what size we should rescale the domain to obtain nonsingular BIEs. The existence of critical domains is in essence caused by the logarithmic term in the fundamental solutions for the elliptic boundary value problems in 2D. This logarithmic term does not depend linearly on the size of the domain. When a domain is scaled, i.e. multiplied by a scale factor, the argument of the logarithm is also multiplied by this scale factor, but the logarithm turns this into an additive term. Thus the logarithm transforms multiplication into addition. This affects the BIEs in such a way that critical domains can appear. The fundamental solutions of boundary value problems in 3D do not contain a logarithmic term. Hence scaling of the domain does not affect the fundamental solution, and consequently also the BIE is not affected. Hence we may safely rescale 3D domains without the risk to encounter a critical domain. An example in which a domain takes many different sizes and shapes is the blowing problem. In this problem a viscous fluid is blown to a desired shape. Typically the time is discretised into a set of discrete time steps, and at each step the shape of the fluid is computed by solving the Stokes equations. When attempting to simulate this problem in 2D, we meet a large number of 2D domains, and we risk that one of these domains is equal to or approaches a critical domain. In such a case the BEM will have difficulties with solving the Stokes equations for that particular domain. When simulating the blowing problem in 3D, no critical domains are encountered. It turns out that the BEM is a very efficient numerical method for this particular 3D problem with a moving boundary. As we are merely interested in the shape of the fluid, we only need to know the flow of its boundary. The BEM does exactly that; it does not compute the flow at the interior of the fluid. Furthermore it is rather easy to include other effects from the blowing problem in the model, such as gravity, surface tension and friction from the contact of the fluid with a wall. As only the boundary of the fluid is discretised, the system matrices that appear in the BEM are smaller than the system matrices that appear when solving the problem with a finite element method, for example. Though the BEM-matrices are dense, while the finite element matrices are sparse, the computational effort for the BEM is relatively low. In short, the BEMis a very appropriate numerical method when solving blowing problems.

On incomplete block factorization methods of generalized SSOR type for H-matrices

A numerical study on fractional order financial system with chaotic and Lyapunov stability analysis

Introduction. Computer Program Packages. Typical Problems. Classification of Equations. Discrete Methods. Finite Differences and Computational Molecules. Finite Difference Operators. Method of Weighted Residuals. Finite Elements. Method of Lines. Errors. Stability and Convergence. Irregular Boundaries. Choice of Discrete Network. Dimensionless Forms. Parabolic Equations: Introduction. Properties of a Simple Explicit Method. Fourier Stability Method. Implicit Methods. Additional Stability Considerations. Matrix Stability Analysis. Extension of Matrix Stability Analysis. Consistency, Stability, And Convergence. Pure Initial Value Problems. Variable Coefficients. Examples of Equations with Variable Coefficients. General Concepts of Error Reduction. Methods of Lines (MOL) for Parabolic Equations. Weighted Residuals and the Method of Lines. Bubnov-Galerkin Scheme for Parabolic Equations. Finite Elements and Parabolic Equations. Hermite Basis. Finite Elements and Parabolic Equations. General Basis Fucntion. Finite Elements and Parabolic Equations. Special Basis Functions. Explicity Finite Difference Methods for Nonlinear Problems. Further Applications on One Dimentions. Asymmetric Approximations. Elliptic Equations: Introduction. Simple Finite Difference Schemes. Direct Methods. Iterative Methods. Linear Elliptic Equations. Some Poit Iterative Methods. Convergence of Point Iterative Methods. Rates of Convergence. Accelerations. Conjugate Gradient Method. Extensions of SOR. Auliative Examples of Over-Relaxation. Other Point Iterative Methods. Block Iterative Methods. Alternating Direction Methods. Summary of ADI Results. Triangular Elements. Boundary Element Method (BEM). Spectral Methods. Some Nonlinear Examples. Hyperbolic Equations: Introduction. The Quasilinear System. Introductory Examples. Method of Characteristics. Constant States and Simple Waves. Typical Application of Characteristics. Finite Differences for First-Order Equations. Lax-Wendroff Methods and Other Algorithms Dissipation and Dispersion. Explicity Finite Difference Methods. Attenuation. Implicit Methods for Second-Order Equations. Time Quasilinear Examples. Simultaneous First-Order Equations. Explicit Methods. An Implicity Method for First-Order Equations. Hybrid Methods for First-Order Equations. Finite Elements and the Wave Equation. Spectral Methods and Periodic Systems. Gas Dyunamics in One Space Variable. Eulerian Difference Equations. Lagrangian Difference Equations. Hopscotch Methods for Conservation Laws. Explicity-Implicity Schemes for Conservation Laws. Special Topics: Introduction. Singularities. Shocks. Eigenvale Problems. Parabolic Equations in Segeral Space Variables. Additional Comments on Elliptic Equations. Hyperbolic Equations in Higher Dimensions. Mixed Systems. Higher Order Equations in Elasticity and Vibrations. Computational Fluid Mechanics. Stream Function. Vorticity Method for Fluid Mechanics. Primitive Variable Methods for Fluid Mechanics. Vector Potential Methods for Fluid Mechanics. Introduction to Monte Carlo Mehtods. Fast Fourier Transform and Applications. Method of Fractional Steps. Applications of Group Theory in Computation. Computational Ocean Acoustics. Enclosure Methods. Chapter References. Author Index. Subject Index.

CHAPTER 7 - Iterative Solution of Finite Element Equations

Introduction: Three-dimensional (3D)-printed custom pelvic implants have become a clinically viable option for patients undergoing pelvic cancer surgery with resection of the hip joint. However, increased clinical utilization has also necessitated improved implant durability, especially with regard to the compression screws used to secure the implant to remaining pelvic bone. This study evaluated six different finite element (FE) screw modeling methods for predicting compression screw pullout and fatigue failure in a custom pelvic implant secured to bone using nine compression screws. Methods: Three modeling methods (tied constraints (TIE), bolt load with constant force (BL-CF), and bolt load with constant length (BL-CL)) generated screw axial forces using functionality built into Abaqus FE software; while the remaining three modeling methods (isotropic pseudo-thermal field (ISO), orthotropic pseudo-thermal field (ORT), and equal-and-opposite force field (FOR)) generated screw axial forces using iterative physics-based relationships that can be implemented in any FE software. The ability of all six modeling methods to match specified screw pretension forces and predict screw pullout and fatigue failure was evaluated using an FE model of a custom pelvic implant with total hip replacement. The applied hip contact forces in the FE model were estimated at two locations in a gait cycle. For each of the nine screws in the custom implant FE model, likelihood of screw pullout failure was predicted using maximum screw axial force, while likelihood of screw fatigue failure was predicted using maximum von Mises stress. Results: The three iterative physics-based modeling methods and the non-iterative Abaqus BL-CL method produced nearly identical predictions for likelihood of screw pullout and fatigue failure, while the other two built-in Abaqus modeling methods yielded vastly different predictions. However, the Abaqus BL-CL method required the least computation time, largely because an iterative process was not needed to induce specified screw pretension forces. Of the three iterative methods, FOR required the fewest iterations and thus the least computation time. Discussion: These findings suggest that the BL-CL screw modeling method is the best option when Abaqus is used for predicting screw pullout and fatigue failure in custom pelvis prostheses, while the iterative physics-based FOR method is the best option if FE software other than Abaqus is used.

This paper deals with the class of singular continuous systems. By decomposing the singular linear system with time-delay into slow and fast subsystems, LMI-based delay-dependent stability and stabilization conditions are established, and an LMI-based algorithm to design a memoryless state feedback control that stabilizes the system is provided. Two numerical examples are worked out to show the usefulness and validness of the theoretical results.

Tropical countries are prone to slope instability due to the presence of residual soil, especially under rainfall loading. Past studies only depended solely on laboratory testing or numerical analyses or field instrumentation or two combinations between either of them to assess the stability of residual soil slope. The objective of this study is to develop the integrated framework combining instrumentation reading, laboratory testing, and numerical modelling based on unsaturated soil mechanics principles to assess the residual soil slope stability related to shallow slope failure. The investigation reveals that instrumentation readings align closely with numerical seepage analysis, with the wetting front occurring at a depth of 0.55m after 72 hours. Stability analysis, evaluated using PLAXIS 2D (finite element) and GeoStudio (limit equilibrium), indicates a variance of 6.25% before and 2.97% after rainfall between two methods. Numerical analyses with unsaturated soil mechanics are required to ensure stability and deformation meet the requirement.

Application of modified iterative method to simulate rainfall infiltration in unsaturated soils

The execution of iteration methods, including iteration methods for time-dependent differential equations, is one of the most time-consuming tasks in scientific computing, especially when large systems have to be solved, large time scales have to be computed or solutions of high quality are required. To decrease the computation time, parallel machines can be used but a detailed analysis of the potential parallelism is needed for an efficient implementation as data dependencies can strongly limit the performance. This article describes and compares the potential parallelism of various iterative solution methods for linear and non-linear equation systems and ordinary differential equations. Methods that have been especially designed for parallel execution as well as parallel implementations of modern efficient sequential solution methods are considered.

Coupling Mortar Finite Element and Boundary Element Methods for 2D Navier-Stokes Equations Courant Element: Before and After Iterative Methods for Solving Stiff Elliptic Problems Straight and Curved Finite Elements of Class C1 and Some Applications to Thin Shell Problems Exact Controllability to Solve the Helmholtz Equation with Absorbing Boundary Conditions Cubic Version of FEM in Elliptic Problems with Interfaces and Singularities Least Squares Mixed Finite Elements Turbulence Modelling in Finite Element Industrial Applications Necessary and Sufficient Conditions for the Numerical Approximation of a Partial Differential Equation Depending on a Small Parameter Efficient Solution Methods for Compressible Flow Computations Parallel Finite Volume Algorithms for Solving the Time-Domain Maxwell Equations on Nonstructured Meshes Coupling Between Nonlinear Maxwell and Heat Equations for an Induction Heating Problem: Modelling and Numerical Methods Solving the 3D Harmonic Maxwell Equations with Finite Elements, Lagrange Multipliers, and Iterative Methods Some Applications of the Hierarchic High Order MITC Finite Elements for Reissner-Mindlin Plates Domain Decomposition for Immiscible Displacement in Single Porosity Systems An Error Estimator for Nonconforming Approximations of a Nonlinear Problem Some Observations on Raviart-Thomas Mixed Finite Elements in p Extension for Parabolic Problems Mixed Finite Element Methods in Fluid Structure Systems A Black-Box Solver for the Solution of General Nonlinear Functional Equations by Mixed FEM A Remark on the Asymptotic Behaviour of Parabolic Variational Inequalities and Their Finite Element Approximation by the Courant Element Domain Decomposition vs. Adaptivity Material Optimization of Composites. Part Contents.

Iterative methods for linear systems of algebraic equations arising from the finite element discretization of nonsymmetric and indefinite elliptic problems are considered. Methods previously known to work well for positive definite, symmetric problems are extended to certain nonsymmetric problems, which can also have some eigenvalues in the left half plane. This paper presents an additive Schwarz method applied to linear, second order, symmetric or nonsymmetric, indefinite elliptic boundary value problems in two and three dimensions. An alternative linear system, which has the same solution as the original problem, is derived and this system is then solved by using GMRES, an iterative method of conjugate gradient type. In each iteration step, a coarse mesh finite element problem and a number of local problems are solved on small, overlapping subregions into which the original region is subdivided. The rate of convergence is shown to be independent of the number of degrees of freedom and the number of local problems if the coarse mesh is fine enough. The performance of the method in two dimensions is illustrated by results of several numerical experiments. Two other iterative methods for solving the same class of elliptic problems in two dimensions is also considered. Using an observation of Dryja and Widlund, it is shown that the rate of convergence of certain iterative substructuring methods deteriorates only quite slowly when the local problems increase in size. A similar result is established for Yserentant's hierarchical basis method.

The aim of this paper is to develop a family of iterative methods with accelerated convergence for solving restricted linear system of equations $$Ax = b$$ , $$x\in T$$ , where $$A \in \mathbb {C}^{m\times n}$$ , $$b \in \mathbb {C}^{m}$$ and T is a subspace of $$\mathbb {C}^{n}$$ . Necessary and sufficient convergence conditions along with the estimation of error bounds of derived approximations are established. The proposed family of iterative methods is applied to solve restricted linear system of equations for different subspaces T. In particular, least-squares solution of an overdetermined linear system and Krylov solution of a square singular linear system of equations are inspected. In addition, the iterative family is tested on a number of numerical examples, including randomly generated dense matrices, sparse singular matrices, and standard test matrices from the Matrix Computational Toolbox. The obtained results are compared with the existing iterative methods and with direct methods, such as the singular value decomposition technique and the QR factorization method.