Shift-Invert Arnoldi's Method with Preconditioned Iterative Solves

A domain decomposition method with Lagrange multipliers for the Stokes problem is developed and analysed. A common approach to solve the Stokes problem, termed the Uzawa algorithm, is to decouple the velocity and the pressure. This approach yields the Schur complement system for the pressure Lagrange multiplier which is solved with an iterative solver. Each outer iteration of the Uzawa procedure involves the inversion of a Laplacian in each spatial direction. The objective of this paper is to effectively solve this inner system (the vector Laplacian system) by applying the finite‐element tearing and interconnecting (FETI) method. Previously calculated search directions for the FETI solver are reused in subsequent outer Uzawa iterations. The advantage of the approach proposed in this paper is that pressure is continuous across the entire computational domain. Numerical tests are performed by solving the driven cavity problem. An analysis of the number of outer Uzawa iterations and inner FETI iterations is reported. Results show that the total number of inner iterations is almost numerically scalable since it grows asymptotically with the mesh size and the number of subdomains. Copyright © 2003 John Wiley & Sons, Ltd.

Computing the matrix exponential for large sparse matrices is essential for solving evolution equations numerically. Conventionally, the shift-invert Arnoldi method is used to compute a matrix exponential and a vector product. This method transforms the original matrix using a shift, and generates the Krylov subspace of the transformed matrix for approximation. The transformation makes the approximation converge faster. In this paper, a new method called the double-shift-invert Arnoldi method is explored. This method uses two shifts for transforming the matrix, and this transformation results in a faster convergence than the shift-invert Arnoldi method.

In the hydrodynamic stability analysis of fluid flow, an eigenproblem with a large asymmetric matrix can be produced by the linearized Navier-Stokes equations. To find the leading eigenvalues and eigenvectors is one of the key issues in the stability analysis. In this study, the Krylov subspace method has been used to develop a general algorithm and then solve the eigenproblem with a large asymmetric matrix in the linearized Navier-Stokes equations. The Krylov subspace method is to construct a set of linearly independent vectors from the linearized equations in which the leading eigenvalues with the largest real parts are included, and orthonormalize the vectors. Then, the Arnoldi's method is used for extracting the desirable eigenvalues with the largest real parts. The advantage of the present method is its ability to calculate a given number of leading eigenvalues including the largest real parts, and corresponding eigenvectors in a large asymmetric matrix. Therefore, this algorithm is general for the hydrodynamic stability analysis with asymmetric matrices resulting from the linearized Navier-Stokes equations. This method has been verified and applied to detect the critical Reynolds number of fluid flow past a circular cylinder. The finite element method with equal-order interpolation was used in this application for discretization of the linearized Navier-Stokes equations.

The two-stage iterative methods for symmetric positive definite matrices

There have been some developments in the area of large and sparse matrix computations. A class of classical methods known as the Krylov subspace methods that include the Lanczos and Arnoldi methods, have been found to be suitable for sparse matrix computations. We give a brief overview of some of the recently developed Arnoldi and Lanczos based methods that seem to be suitable for large and sparse control problems. The research in this area is still in its infancy.

ADAPTIVE MODELING FOR BIT RATE CONTROL

This work presents an efficient technique for simultaneously reconstructing an image of a moving object from X-ray projections along with estimating the motion parameters. Current state-of-the-art iterative methods rely on objective functions that contain the solution of an iterative procedure as one of its terms. This complicates the reconstruction process as it leads to nested iterations and makes analytic differentiation impractical. The presented technique relies on an objective function that simultaneously depends on the image and the motion parameters. The derivatives of this objective function towards the image and the motion parameters are known exactly and implemented matrix free and in parallel. Moreover, the stepsizes of both the iterative reconstruction and motion parameter estimation schemes are chosen following a mathematical formulation that guarantees a fast convergence speed. The result is a method that not only needs less iterations, but also removes the need to tune the finite differences stepsize and the number of inner iterations, which allows for efficient, scalable reconstruction using modern optimizers, without nested iterations.

This paper considers an implicitly restarted Krylov subspace method that approximates a stable, linear transfer function $f(s)$ of order n by one of order m, where $n\gg m$. It is well known that oblique projections onto a Krylov subspace may generate unstable partial realizations. To remedy this situation, the oblique projectors obtained via classical Krylov subspace methods are supplemented with further projectors which enable the formation of stable partial realizations directly from $f(s)$. A key feature of this process is that it may be incorporated into an implicit restart scheme. A second difficulty arises from the fact that Krylov subspace methods often generate partial realizations that contain nonessential modes. To this end, balanced truncation may be employed to discard the unwanted part of the reduced-order model. This paper proposes oblique projection methods for large-scale model reduction that simultaneously compute stable reduced-order models while discarding all nonessential modes. It is shown that both of these tasks may be effected by a single oblique projection process. Furthermore, the process is shown to naturally fit into an implicit restart framework. The theoretical properties of these methods are thoroughly investigated, and exact low-dimensional expressions for the $\L^{\infty}$-norm of the residual errors are derived. Finally, the behavior of the algorithm is illustrated on two large-scale examples.

1 - Implicitly restarted arnoldi/lanczos methods and large scale svd applications

Effect of time-step size on flow around generic car models

In this paper, we apply the projective integration method to moment models of the Boltzmann-BGK equation and investigate the numerical properties of the resulting scheme. Projective integration is an explicit, asymptotic-preserving scheme that is tailored to problems with a large spectral gap between fast and slow eigenvalues of the model. A spectral analysis of the moment model shows a clear spectral gap and reveals the multi-scale nature of the model. The new scheme overcomes the severe time step constraint of standard explicit schemes like the forward Euler scheme by performing a number of inner iterations and then extrapolating the solution forward in time. The projective integration scheme is non-intrusive and yields fast and accurate solutions, as demonstrated using a 1D shock tube test case. These observations open up many possibilities for further use of the scheme for high-resolution discretizations and different collision models.

The monotonicity of two-stage iterative methods

Convergence study of DGFEM SN based 2D/1D coupling method for solving neutron transport k-eigenvalue problems with Fourier analysis

A performance of high-order Cellwise Relaxation Implicit Discontinuous Galerkin (DGCRI) scheme on unsteady flow computation is investigated. The inviscid Taylor-Green vortex is first computed to demonstrate the capability of the high-order DG-CRI scheme on the homogeneous turbulence computation. The higher-order DG-CRI scheme gives better resolved small vortex structures, which is well indicated in the later decay of the kinetic energy and the higher enstrophy. In addition, the effects of the CFL number and the number of inner iterations are discussed by comparing with the results by explicit time integration. Then, the high-order DG method is applied for the LES computation of the turbulent channel flow, and the advantages of the high-order DG method in predicting accurate turbulent statistics are shown.

Based on the transient planar pitching method, the dynamic derivative calculation function is implemented on an unstructured computational fluid (CFD) software NNW-FlowStar. Then the pitch-damping dynamic derivatives of the standard model of the ANF missile at zero angles of attack and different Mach numbers (0.5∼4.5) are calculated. The influence of various numerical and physical parameters on the calculation results of dynamic derivatives is studied in detail. It is found that the dynamic derivatives of pitch damping are independent of the values of amplitude (0.125° ≤ A ≤ 1°) and reduction frequency (0.025 ≤ k ≤ 0.2). When the number of calculation steps of the sine oscillation period is 200 and the number of inner iterations is 40, the prediction accuracy and computing efficiency can meet the engineering requirements simultaneously in this case. The influence of different grid resolutions on the dynamic derivative calculation results is studied. Results show that the grid requirements of dynamic derivative calculation are consistent with those of steady-state calculations. Finally, the calculation results are compared with the free flight test results, which verifies the reliability of the methods and parameters suggested in this paper.