Optimal Convergence Properties Research Articles

Recursive Krylov‐based multigrid cycles

AbstractWe consider multigrid (MG) cycles based on the recursive use of a two‐grid method, in which the coarse‐grid system is solved by µ⩾1 steps of a Krylov subspace iterative method. The approach is further extended by allowing such inner iterations only at the levels of given multiplicity, whereas V‐cycle formulation is used at all other levels. For symmetric positive definite systems and symmetric MG schemes, we consider a flexible (or generalized) conjugate gradient method as Krylov subspace solver for both inner and outer iterations. Then, based on some algebraic (block matrix) properties of the V‐cycle MG viewed as a preconditioner, we show that the method can have optimal convergence properties if µ is chosen to be sufficiently large. We also formulate conditions that guarantee both, optimal complexity and convergence, bounded independently of the number of levels. Our analysis shows that the method is, at least, as effective as the standard W‐cycle, whereas numerical results illustrate that it can be much faster than the latter, and actually more robust than predicted by the theory. Copyright © 2007 John Wiley & Sons, Ltd.

An application of multigrid methods for a discrete elastic model for epitaxial systems

We apply an efficient and fast algorithm to simulate the atomistic strain model for epitaxial systems, recently introduced by Schindler et al. [Phys. Rev. B 67, 075316 (2003)]. The discrete effects in this lattice statics model are crucial for proper simulation of the influence of strain for thin film epitaxial growth, but the size of the atomistic systems of interest is in general quite large and hence the solution of the discrete elastic equations is a considerable numerical challenge. In this paper, we construct an algebraic multigrid method suitable for efficient solution of the large scale discrete strain model. Using this method, simulations are performed for several representative physical problems, including an infinite periodic step train, a layered nanocrystal, and a system of quantum dots. The results demonstrate the effectiveness and robustness of the method and show that the method attains optimal convergence properties, regardless of the problem size, the geometry and the physical parameters. The effects of substrate depth and of invariance due to traction-free boundary conditions are assessed. For a system of quantum dots, the simulated strain energy density supports the observations that trench formation near the dots provides strain relief.

Optimal convergence properties of the FETI domain decomposition method

AbstractIn this paper an original variant of the FETI domain decomposition method is introduced for heterogeneous media. This method uses new absorbing interface conditions in place of the Neumann interface conditions defined in the classical FETI method. The optimal convergence properties of the classical FETI method and of its variant are first demonstrated, both in the case of homogeneous and heterogeneous media. Secondly, novel and efficient absorbing interface conditions, which avoid rigid body motions, are investigated and analysed. Numerical experiments illustrate the dependence of the proposed method upon several parameters, and confirm the robustness and efficiency of this method when equipped with such absorbing interface conditions. Copyright © 2006 John Wiley & Sons, Ltd.

Reduced Finite Element Discretizations of the Stokes and Navier-Stokes Equations

ABSTRACT If finite element spaces for the velocity and pressure do not satisfy the Babuška-Brezzi condition, a stable conforming discretization of the Stokes or Navier-Stokes equations can be obtained by enriching the velocity space by suitable functions. Writing any function from the enriched space as a sum of a function from the original space and a function from the supplementary space, the discretization will contain a number of additional terms compared with a conforming discretization for the original pair of spaces. We show that not all these terms are necessary for the solvability of the discrete problem and for optimal convergence properties of the discrete solutions, which is useful for saving computer memory and for establishing a connection to stabilized methods.

ALGEBRAIC WAY TO DERIVE ABSORBING BOUNDARY CONDITIONS FOR THE HELMHOLTZ EQUATION

This paper is dedicated to the optimal convergence properties of a domain decomposition method involving two-Lagrange multipliers at the interface between the subdomains and additional augmented interface operators. Most methods for optimizing these augmented interface operators are based on the discretization of continuous approximations of the optimal transparent operators.1–5 Such approach is strongly linked to the continuous equation, and to the discretization scheme. At the discrete level, the optimal transparent operator can be proved to be equal to the Schur complement of the outer subdomain. Our idea consists of approximating directly the Schur complement matrix with purely algebraic techniques involving local condensation of the subdomain degree of freedom on small patch defined on the interface between the subdomains. The main advantage of such approach is that it is much more easy to implement in existing code without any information on the geometry of the interface and of the finite element formulation used. Such technique leads to a so-called "black box" for the users. Convergence results and parallel efficiency of this new and original algebraic optimization technique of the interface operators are presented for acoustics applications.

Edge stabilization for the generalized Stokes problem: A continuous interior penalty method

In this note we introduce and analyze a stabilized finite element method for the generalized Stokes equation. Stability is obtained by adding a least squares penalization of the gradient jumps across element boundaries. The method can be seen as a higher order version of the Brezzi–Pitkäranta penalty stabilization [F. Brezzi, J. Pitkäranta, On the stabilization of finite element approximations of the Stokes equations, in: W. Hackbusch (Ed.), Efficient Solution of Elliptic Systems, Vieweg, 1984], but gives better resolution on the boundary for the Stokes equation than does classical Galerkin least-squares formulation. We prove optimal and quasi-optimal convergence properties for Stokes’ problem and for the porous media models of Darcy and Brinkman. Some numerical examples are given.

Cross-layer optimization for OFDM wireless networks-part I: theoretical framework

In this paper, we provide a theoretical framework for cross-layer optimization for orthogonal frequency division multiplexing (OFDM) wireless networks. The utility is used in our study to build a bridge between the physical layer and the media access control (MAC) layer and to balance the efficiency and fairness of wireless resource allocation. We formulate the cross-layer optimization problem as one that maximizes the average utility of all active users subject to certain conditions, which are determined by adaptive resource allocation schemes. We present necessary and sufficient conditions for utility-based optimal subcarrier assignment and power allocation and discuss the convergence properties of optimization. Numerical results demonstrate a significant performance gain for the cross-layer optimization and the gain increases with the number of active users in the networks.

Convergence Properties of the Normal Mode Optimization and Its Combination with Molecular Geometry Constraints

Optimization in the normal mode coordinates has been established as a useful tool for modeling of vibrational spectra (J. Chem. Phys. 2002, 117, 4126). In this work the algorithm is extended with the aid of harmonic penalty functions to allow for multiple restraints of geometry parameters, such as bond lengths, bond and dihedral angles, and for simultaneous optimization of more molecules. Additionally, geometry optimization when atomic nuclei are maintained on the constant electrostatic potential surface was implemented and its applications for solvent models are discussed. Model systems include small molecules, water cluster, antiparallel β-sheet peptide containing intermolecular hydrogen bonds, periodic α-helix and a parallel β-sheet segments. The normal mode method provided better numerical stability than the conventional redundant internal coordinates, especially for weakly hydrogen-bonded systems, while the speed of the optimization was found similar as for the Cartesian coordinates.

Optimal Superconvergence Orders of Iterated Collocation Solutions for Volterra Integral Equations with Vanishing Delays

In this paper we analyze the optimal convergence properties of collocation approximations to solutions of Volterra integral equations of the second kind with vanishing variable delays. The focus of the analysis is on the superconvergence of the (iterated) collocation approximation corresponding to collocation in the space of (discontinuous) piecewise polynomials of degree $m-1 \geq 0$. We show that on uniform meshes the iterated collocation solution possesses the global superconvergence order $m+1$, and that the solution's order of local superconvergence at the nodes of the underlying mesh cannot exceed $m+2$, in sharp contrast to problems with nonvanishing delays. Moreover, the optimal order $p^{\ast } = m+2$ can be attained only under suitable assumptions. This result also resolves a conjecture of 1997 regarding the attainable order of local superconvergence for Volterra integral equations containing a proportional delay $qt$ with $q \in (0,1)$.

Convergence analysis of Schwarz methods without overlap for the Helmholtz equation

In this paper, the continuous and discrete optimal transmission conditions for the Schwarz algorithm without overlap for the Helmholtz equation are studied. Since such transmission conditions lead to non-local operators, they are approximated through two different approaches. The first approach, called optimized, consists of an approximation of the optimal continuous transmission conditions with partial differential operators, which are then optimized for efficiency. The second approach, called approximated, is based on pure algebraic operations performed on the optimal discrete transmission conditions. After demonstrating the optimal convergence properties of the Schwarz algorithm new numerical investigations are performed on a wide range of unstructured meshes and arbitrary mesh partitioning with cross points. Numerical results illustrate for the first time the effectiveness, robustness and comparative performance of the optimized and approximated Schwarz methods on a model problem and on industrial problems.

Robust optimal design and convergence properties analysis of iterative learning control approaches

In this paper, we address four major issues in the field of iterative learning control (ILC) theory and design. The first issue is concerned with ILC design in the presence of system interval uncertainties. Targeting at time-optimal (fastest convergence) and robustness properties concurrently, we formulate the ILC design into a min–max optimization problem and provide a systematic solution for linear-type ILC consisting of the first-order and higher-order ILC schemes. Inherently relating to the first issue, the second issue is concerned with the performance evaluation of various ILC schemes. Convergence speed is one of the most important factors in ILC. A learning performance index— Q-factor—is introduced, which provides a rigorous and quantified evaluation criterion for comparing the convergence speed of various ILC schemes. We further explore a key issue: how does the system dynamics affect the learning performance. By associating the time weighted norm with the supreme norm, we disclose the dynamics impact in ILC, which can be assessed by global uniform bound and monotonicity in iteration domain. Finally we address a rather controversial issue in ILC: can the higher-order ILC outperform the lower-order ILC in terms of convergence speed and robustness? By applying the min–max design, which is robust and optimal, and conducting rigorous analysis, we reach the conclusion that the Q-factor of ILC sequences of lower-order ILC is lower than that of higher-order ILC in terms of the time-weighted norm.

A discrete Galerkin method for a catalytic combustion model

We apply a novel cost-effective spline method to a one-dimensional model of catalytic combustion in a monolith reactor. The model includes terms for catalytic reaction, heat and mass transfer between the channel wall and the gas, axial conduction in the solid wall, and heat exchange by radiative transfer. This leads to a nonlinear integrodifferential-algebraic system. The computational scheme is based on a discrete Petrov-Galerkin Method, discussed in detail in the recent work [1], and seeks spline approximations to the solutions. It is more cost-effective than the usual orthogonal collocation method and has been proved recently that it retains all stable and optimal convergence properties of the orthogonal collocation on finite elements. It also provides an approach which retains the coupling of the solution components which was not present in previous work on this problem. The numerical experiments obtained using the method are verified against solutions provided in the literature.

An implicit boundary element solution with consistent linearization for free surface flows and non‐linear fluid–structure interaction of floating bodies

AbstractIn this work, a new comprehensive method has been developed which enables the solution of large, non‐linear motions of rigid bodies in a fluid with a free surface. The application of the modern Eulerian–Lagrangian approach has been translated into an implicit time‐integration formulation, a development which enables the use of larger time steps (where accuracy requirements allow it). Novel features of this project include: (1) an implicit formulation of the rigid‐body motion in a fluid with a free surface valid for both two or three dimensions and several moving bodies; (2) a complete formulation and solution of the initial conditions; (3) a fully consistent (exact) linearization for free surface flows valid for any boundary elements such that optimal convergence properties are obtained when using a Newton–Raphson solver. The proposed framework has been completed with details on implementation issues referring mainly to the computation of the complete initial conditions and the consistent linearization of the formulation for free surface flows.The second part of the paper demonstrates the mathematical and numerical formulation through numerical results simulating large free surface flows and non‐linear fluid structure interaction. The implicit formulation using a fully consistent linearization based on the boundary element method and the generalized trapezoidal rule has been applied to the solution of free surface flows for the evolution of a triangular wave, the generation of tsunamis and the propagation of a wave up to overturning. Fluid–structure interaction examples include the free and forced motion of a circular cylinder and the sway, heave and roll motion of a U‐shaped body in a tank with a flap wave generator. The presented examples demonstrate the applicability and performance of the implicit scheme with consistent linearization. Copyright © 2001 John Wiley & Sons. Ltd.

A stabilized finite element method for generalized stationary incompressible flows

In this paper, we describe a finite element formulation for the numerical solution of the stationary incompressible Navier–Stokes equations including Coriolis forces and the permeability of the medium. The stabilized method is based on the algebraic version of the sub-grid scale approach. We first describe this technique for general systems of convection–diffusion–reaction equations and then we apply it to the linearized flow equations. The important point is the design of the matrix of stabilization parameters that the method has. This design is based on the identification of the stability problems of the Galerkin method and a scaling of variables argument to determine which coefficients must be included in the stabilization matrix. This, together with the convergence analysis of the linearized problem, leads to a simple expression for the stabilization parameters in the general situation considered in the paper. The numerical analysis of the linearized problem also shows that the method has optimal convergence properties.

New substructuring domain decomposition methods for advection–diffusion equations

In this paper we consider a nonsymmetric elliptic problem and we use the techniques related to the Steklov–Poincaré operators to propose new substructuring iterative procedures. In particular, we propose two methods that generalize the well-known Neumann–Neumann and Dirichlet–Neumann iterative procedures. We prove that our methods, that use symmetric and positive-definite preconditioners, lead to the construction of iterative schemes with optimal convergence properties. Numerical results for the finite element discretization are given.

On the connection between some linear triangular Reissner-Mindlin plate bending elements

We consider three triangular plate bending elements for the Reissner-Mindlin model. The elements are the MIN3 element of Tessler and Hughes [19], the stabilized MITC3 element of Brezzi, Fortin and Stenberg [5] and the T3BL element of Xu, Auricchio and Taylor [2, 17, 20]. We show that the bilinear forms of the stabilized MITC3 and MIN3 elements are equivalent and that their implementation may be simplified by using numerical integration of reduced order. The T3BL element is shown to be essentially the same as the MIN3 and stabilized MITC3 elements with reduced integration. We finally introduce a general stabilized finite element formulation which covers all three methods. For this class of methods we prove the stability and optimal convergence properties.

Fourier frequency adaptive regularization for smoothing data

The problem of smoothing data through a transform in the Fourier domain is analyzed. It is well known that this problem has a very easy solution and optimal convergence properties; moreover, the GCV criterion is able to give an estimate of the regularization parameter that is asymptotically optimal in the average. The presence of just one regularization parameter in the problem means that all Fourier coefficients are smoothed with the same law, regardless of the function. Here we introduce a frequency adaptive regularization method where a regularization parameter is introduced for each coefficient, able to smooth different frequencies taking into account both function and noise. We give convergence results for the method; moreover an ideal choice of the regularization parameters is provided basing on the minimization of the L 2 risk. Numerical experiments are worked out on some significant test functions in order to show performance of the method. Comparison with results achievable with the wavelet regularization and the wavelet adaptive regularization methods is finally performed.

Flexible Conjugate Gradients

We analyze the conjugate gradient (CG) method with preconditioning slightly variable from one iteration to the next. To maintain the optimal convergence properties, we consider a variant proposed by Axelsson that performs an explicit orthogonalization of the search directions vectors. For this method, which we refer to as flexible CG, we develop a theoretical analysis that shows that the convergence rate is essentially independent of the variations in the preconditioner as long as the latter are kept sufficiently small. We further discuss the real convergence rate on the basis of some heuristic arguments supported by numerical experiments. Depending on the eigenvalue distribution corresponding to the fixed reference preconditioner, several situations have to be distinguished. In some cases, the convergence is as fast with truncated versions of the algorithm or even with the standard CG method, whereas quite large variations are allowed without too much penalty. In other cases, the flexible variant effectively outperforms the standard method, while the need for truncation limits the size of the variations that can be reasonably allowed.

A Scalable Substructuring Method by Lagrange Multipliers for Plate Bending Problems

We present a new Lagrange multiplier-based domain decomposition method for solving iteratively systems of equations arising from the finite element discretization of plate bending problems. The proposed method is essentially an extension of the finite element tearing and interconnecting substructuring algorithm to the biharmonic equation. The main idea is to enforce continuity of the transversal displacement field at the subdomain crosspoints throughout the preconditioned conjugate gradient iterations. The resulting method is proved to have a condition number that does not grow with the number of subdomains but rather grows at most polylogarithmically with the number of elements per subdomain. These optimal properties hold for numerous plate bending elements that are used in practice including the Hsieh--Clough--Tocher element, the discrete Kirchhoff triangle, and a class of nonlocking elements for the Reissner--Mindlin plate models. Computational experiments are reported and shown to confirm the theoretical optimal convergence properties of the new domain decomposition method. Computational efficiency is also demonstrated with the numerical solution in 45 iterations and 105 seconds on a 64-processor IBM SP2 of a plate bending problem with almost one million degrees of freedom.

Optimal v-cycle algebraic multilevel preconditioning

We consider algebraic multilevel preconditioning methods based on the recursive use of a 2 × 2 block incomplete factorization procedure in which the Schur complement is approximated by a coarse grid matrix. As is well known, for discrete second-order elliptic PDEs, optimal convergence properties are proved for such basic two-level schemes under mild assumptions on the PDE coefficients, but their recursive use in a simple V-cycle algorithm does not generally lead to optimal order convergence. In the present paper, we analyse the combination of these techniques with a smoothing procedure much the same as the one used in standard multigrid algorithms, except that smoothing is not required on the finest grid. Theoretical results prove optimal convergence properties for the V-cycle under an assumption similar to the ‘approximation property’ of the classical multigrid convergence theory. On the other hand, numerical experiments made on both 2D and 3D problems show that the condition number is close to that of the two-level method. Further, the method appears robust in the presence of discontinuity and anisotropy, even when the material interfaces are not aligned with the coarse grid. Copyright © 1999 John Wiley & Sons, Ltd.