Robust Iterative Scheme Research Articles

A New Robust Iterative Scheme Applied in Solving a Fractional Diffusion Model for Oxygen Delivery via a Capillary of Tissues

In this paper, we constructed a new and robust fixed point iterative scheme called the UO iterative scheme for the approximation of a contraction mapping. The scheme converges strongly to the fixed point of a contraction mapping. A rate of convergence result is shown with an example, and our scheme, when compared, converges faster than some existing iterative schemes in the literature. Furthermore, the stability and data dependence results are shown. Our new scheme is applied in the approximation of the solution to the oxygen diffusion model. Finally, our results are applied in the approximation of the solution to the boundary value problems using Green’s functions with an example.

Comparative studies of iterative methods for solving the “optimally diffusive” coarse mesh finite difference accelerated transport equation

The emergence of advanced nuclear reactor systems with increasing complexity and heterogeneity has necessitated detailed high fidelity neutronic analysis. This requires exhaustive solution of the neutron transport equation which has the disadvantage of long computation times, especially for scattering dominant problems. In this paper, the performance behavior and stability analysis of several iterative methods for solving the optimally diffusive coarse mesh finite difference (odCMFD) accelerated transport equation has been presented. Efficacy of the acceleration iterations with varying coarse mesh size and convergence criteria of the low order problem, and their overall performance has been investigated for the 2-group LRA-BWR problem and the 7-group C5G7 problem. Prescriptions for different parameters and the most robust iterative scheme have been proposed based on these studies. The theoretical background of odCMFD based acceleration schemes and the numerical results of their application to different problems using the code DIAMOND have been presented in this paper.

Ionized, electron-attached, and excited states of molecular systems with spin-orbit coupling: Two-component GW and Bethe-Salpeter implementations.

We have implemented and applied the GW method and the static screened Bethe-Salpeter equation (BSE) for calculating linear-response properties for quasirelativistic molecular systems. Our ansatz is based on a two-component (2c) scheme that includes spin-orbit coupling as well as scalar relativistic effects. Efficient, state-of-the-art approaches including the analytic continuation (employing Padé approximants, scaling as N 4 with system size N ) and contour deformation schemes are presented to obtain the required 2c quasirelativistic GW quasiparticle energies. Screened exchange contributions are computed within the resolution-of-the-identity approximation, and working equations for the 2c GW/BSE method are given. The performance of the 2c GW/BSE method is assessed, and results are compared to other methods and experimental data. A robust iterative scheme for solving the eigenvalue problems occurring in the 2c GW/BSE and hybrid time-dependent density functional theories is presented.

An Optimal Eighth-Order Scheme for Multiple Zeros of Univariate Functions

We construct an optimal eighth-order scheme which will work for multiple zeros with multiplicity [Formula: see text], for the first time. Earlier, the maximum convergence order of multi-point iterative schemes was six for multiple zeros in the available literature. So, the main contribution of this study is to present a new higher-order and as well as optimal scheme for multiple zeros for the first time. In addition, we present an extensive convergence analysis with the main theorem which confirms theoretically eighth-order convergence of the proposed scheme. Moreover, we consider several real life problems which contain simple as well as multiple zeros in order to compare with the existing robust iterative schemes. Finally, we conclude on the basis of obtained numerical results that our iterative methods perform far better than the existing methods in terms of residual error, computational order of convergence and difference between the two consecutive iterations.

Analysis and Design of Short FRP-Confined Concrete-Encased Arbitrarily Shaped Steel Columns under Biaxial Loading

The FRP-confined concrete-encased steel column is a new form of hybrid column, which integrates advantages of all the constituent materials. Its structural performance, including load carrying capacity, ductility, and corrosion resistance, has been demonstrated to be excellent by limited experimental investigation. Currently, no systematic procedure, particularly for that with reinforced structural steel of arbitrary shapes, has been proposed for the sectional analysis and design for such novel hybrid columns under biaxial loading. The present paper aims at filling this research gap by proposing an approach for the rapid section analysis and providing rationale basis for FRP-confined concrete-encased arbitrarily shaped steel columns. A robust iterative scheme has been used with a traditional so-called fiber element method. The presented numerical examples demonstrated the validity and accuracy of the proposed approach.

Absolute Value Preconditioning for Symmetric Indefinite Linear Systems

We introduce a novel strategy for constructing symmetric positive definite (SPD) preconditioners for linear systems with symmetric indefinite matrices. The strategy, called absolute value preconditioning, is motivated by the observation that the preconditioned minimal residual method with the inverse of the absolute value of the matrix as a preconditioner converges to the exact solution of the system in at most two steps. Neither the exact absolute value of the matrix nor its exact inverse are computationally feasible to construct in general. However, we provide a practical example of an SPD preconditioner that is based on the suggested approach. In this example we consider a model problem with a shifted discrete negative Laplacian, and suggest a geometric multigrid (MG) preconditioner, where the inverse of the matrix absolute value appears only on the coarse grid, while operations on finer grids are based on the Laplacian. Our numerical tests demonstrate practical effectiveness of the new MG preconditioner, which leads to a robust iterative scheme with minimalist memory requirements.

A robust iterative scheme for finite volume discretization of diffusive flux on highly skewed meshes

A new approach for diffusive flux discretization on a nonorthogonal mesh for finite volume method is proposed. This approach is based on an iterative method, Deferred correction introduced by M. Peric [J.H. Fergizer, M. Peric, Computational Methods for Fluid Dynamics, Springer, 2002]. It converges on highly skewed meshes where the former approach diverges. A convergence proof of our method is given on arbitrary quadrilateral control volumes. This proof is founded on the analysis of the spectral radius of the iteration matrix. This new approach is applied successfully to the solution of a Poisson equation in quadrangular domains, meshed with highly skewed control volumes. The precision order of used schemes is not affected by increasing skewness of the grid. Some numerical tests are performed to show the accuracy of the new approach.

Fast Passivity Enforcement for Pole-Residue Models by Perturbation of Residue Matrix Eigenvalues

Rational models must be passive in order to avoid unstable time domain simulations. This paper introduces a fast approach for passivity enforcement of pole-residue models. This is achieved by perturbing the eigenvalues of the residue matrices, as opposed to the existing approach of perturbing matrix elements. This leads to large savings in computation time with only a small increase of the modeling error. This fast residue perturbation (FRP) approach is merged with the modal perturbation technique, leading to fast modal perturbation (FMP). Usage of FMP over FRP achieves to retain the relative accuracy of the admittance matrix eigenvalues. A complete approach is obtained by combining the passivity enforcement step with passivity assessment via the Hamiltonian matrix eigenvalues and a robust iteration scheme, giving a guaranteed passive model. Application of FMP to a six-port power transformer shows that the approach is able to remove large out-of band passivity violations without corrupting the in-band behavior. This is shown to mitigate an unstable simulation. The approach is also demonstrated for a high-speed interconnect and a transmission line.

Robust norm equivalencies for diffusion problems

Additive multilevel methods offer an efficient way for the fast solution of large sparse linear systems which arise from a finite element discretization of an elliptic boundary value problem. These solution methods are based on multilevel norm equivalencies for the associated bilinear form using a suitable subspace decomposition. To obtain a robust iterative scheme, it is crucial that the constants in the norm equivalence do not depend or depend only weakly on the ellipticity constants of the problem. In this paper we present such a robust norm equivalence for the model problem $- \nabla \omega \nabla u=f$ with a scalar diffusion coefficient $\omega$ in $\Omega \subset \mathbb {R}^2$. Our estimates involve only very weak information about $\omega$, and the results are applicable for a large class of diffusion coefficients. Namely, we require $\omega$ to be in the Muckenhoupt class $A_{1}(\Omega )$, a function class well-studied in harmonic analysis. The presented multilevel norm equivalencies are a main step towards the realization of an optimal and robust multilevel preconditioner for scalar diffusion problems.

A comprehensive study of a multiplicative elastoplasticity model coupled to damage including parameter identification

In this contribution various aspects for a plasticity model coupled to damage are considered. The formulation of the model is performed in the intermediate configuration which occurs as a consequence of the multiplicative decomposition of the deformation gradient. We will resort to thermodynamic consistency, continuous tangent operator, algorithmic tangent operator and sensitivity analysis for parameter identification. Furthermore, for the discretized constitutive problem a robust iteration scheme with a two-level algorithm is proposed. In the numerical example material parameters are determined by least-squares minimization based on experimental data obtained with an optical method.

A Fast and Accurate Method for Numerical Analysis of Elastic Layered Contacts

Solution of contact problems for layered elastic solids generally requires the use of numerical methods. Recently, the fast Fourier transform (FFT) technique has been applied to such contacts. While very fast, FFT is strictly applicable only to periodic contact problems. When it is applied to essentially non-periodic contacts, an error is introduced in the numerical solution. A new method that overcomes the limitation of the “straightforward” FFT approach for solving non-periodic layered contact problems is introduced in the present article. A special correction procedure based on the multi-level multi-summation technique is used to compensate the FFT results for the periodicity error. The use of a robust iteration scheme based on the conjugate gradient method ensures that the new method is applicable to contact problems involving real rough surfaces. Numerical examples demonstrate that the new method is both accurate and fast. [S0742-4787(00)00501-4]

Iterative schemes for nonsymmetric and indefinite elliptic boundary value problems

The purpose of this paper is twofold. The first is to describe some simple and robust iterative schemes for nonsymmetric and indefinite elliptic boundary value problems. The schemes are based in the Sobolev space H 1 ( Ω ) {H^1}(\Omega ) and require minimal hypotheses. The second is to develop algorithms utilizing a coarse-grid approximation. This leads to iteration matrices whose eigenvalues lie in the right half of the complex plane. In fact, for symmetric indefinite problems, the iteration is reduced to a well-conditioned symmetric positive definite system which can be solved by conjugate gradient iteration. Applications of the general theory as well as numerical examples are given.

Parametric Methods for Source Location Estimation

Abstract Approximate maximum-likelihood estimates for locating of wide band sources in the presence of partly unknown noise fields are developed. Alternatively, two least squares methods fitting a parametric model of the spectral density matrix to a corresponding nonparametric estimate are investigated. Furthermore, applying the expectation maximization (EM) algorithm, a computationally robust iteration scheme for maximizing the log-likelihood function is derived. Using wide band data from a North Sea experiment, we compare the performance of the maximum likelihood method via the EM algorithm with the MUSIC algorithm combined with the rotational signal-subspace (RSS) focussing technique.