Ritz Vectors Research Articles

Numerical experiments on a flexible variant of GMRES‐DR

AbstractThe Flexible GMRES (FGMRES [1]) and the GMRES with deflated restarting (GMRES‐DR [2]) methods are two algorithms derived from GMRES [3], that are considered as powerful when solving large non hermitian systems of linear equations. GMRES‐DR is a variant of GMRES with an improved restarting technique that maintains in the Krylov subspace harmonic Ritz vector from the previous restart. In situations where the convergence of restarted GMRES is slow and where the matrix has few eigenvalues close to the origin, this technique has proved very efficient. The new method that we propose is the Flexible GMRES with deflated restarting (FGMRES‐DR [6]), which combines the two above mentioned algorithms in order to yield better performance. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Abstract perturbed Krylov methods

We introduce the framework of “abstract perturbed Krylov methods”. This is a new and unifying point of view on Krylov subspace methods based solely on the matrix equation AQ k + F k = Q k + 1 C ̲ k = Q k C k + q k + 1 c k + 1 , k e k T and the assumption that the matrix C k is unreduced Hessenberg. We give polynomial expressions relating the Ritz vectors, quasi-orthogonal residual iterates and quasi-minimal residual iterates to the starting vector q 1 and the perturbation term F k . The properties of these polynomials and similarities between them are analyzed in some detail. The results suggest the interpretation of abstract perturbed Krylov methods as additive overlay of several abstract exact Krylov methods.

Restarted block Lanczos bidiagonalization methods

The problem of computing a few of the largest or smallest singular values and associated singular vectors of a large matrix arises in many applications. This paper describes restarted block Lanczos bidiagonalization methods based on augmentation of Ritz vectors or harmonic Ritz vectors by block Krylov subspaces.

A modified harmonic block Arnoldi algorithm with adaptive shifts for large interior eigenproblems

The harmonic block Arnoldi method can be used to find interior eigenpairs of large matrices. Given a target point or shift τ to which the needed interior eigenvalues are close, the desired interior eigenpairs are the eigenvalues nearest τ and the associated eigenvectors. However, it has been shown that the harmonic Ritz vectors may converge erratically and even may fail to do so. To do a better job, a modified harmonic block Arnoldi method is coined that replaces the harmonic Ritz vectors by some modified harmonic Ritz vectors. The relationships between the modified harmonic block Arnoldi method and the original one are analyzed. Moreover, how to adaptively adjust shifts during iterations so as to improve convergence is also discussed. Numerical results on the efficiency of the new algorithm are reported.

A New Method for Wind-Induced Response Analysis of Long Span Roofs

Long span roofs are usually characterised by their dense distribution of natural frequencies and remarkable coupling effects among different natural modes. Those traditional methods, such as modal superposition method, are not suitable for analyzing wind-induced response of these structures. In this paper, a new method named as Ritz-POD method is proposed to analyse wind-induced response of long span roofs in frequency domain, which synthesises proper orthogonal decomposition (POD) technique and direct superposition method with Ritz vectors. The eigenvectors of fluctuating wind load with POD technique are regarded as spatial distributions of this load to generate Ritz vectors, and wind-induced response is analysed by direct superposition method with Ritz vectors. Therefore, it can take into account multimode buffeting response automatically, even the case that structures vibrate in high order modes. According to this method, the wind-induced response of a single layer reticular shell is analysed, and compared to the results achieved with modal superposition method. It shows that Ritz-POD method is more accurate and efficient with fewer Ritz vectors.

Using cross-product matrices to compute the SVD

This paper concerns accurate computation of the singular value decomposition (SVD) of an $m\times n$ matrix $A$ . As is well known, cross-product matrix based SVD algorithms compute large singular values accurately but generally deliver poor small singular values. A new novel cross-product matrix based SVD method is proposed: (a) Use a backward stable algorithm to compute the eigenpairs of $A^{\rm T}A$ and take the square roots of the large eigenvalues of it as the large singular values of $A$ ; (b) form the Rayleigh quotient of $A^{\rm T}A$ with respect to the matrix consisting of the computed eigenvectors associated with the computed small eigenvalues of $A^{\rm T}A$ ; (c) compute the eigenvalues of the Rayleigh quotient and take the square roots of them as the small singular values of $A$ . A detailed quantitative error analysis is conducted on the method. It is proved that if small singular values are well separated from the large ones then the method can compute the small ones accurately up to the order of the unit roundoff $\epsilon$ . An algorithm is developed that is not only cheaper than the standard Golub–Reinsch and Chan SVD algorithms but also can update or downdate a new SVD by adding or deleting a row and compute certain refined Ritz vectors for large matrix eigenproblems at very low cost. Several variants of the algorithm are proposed that compute some or all parts of the SVD. Typical numerical examples confirm the high accuracy of our algorithm.

Structural Health Monitoring via Measured Ritz Vectors Utilizing Artificial Neural Networks

Abstract: A pattern recognition approach for structural health monitoring (SHM) is presented that uses damage-induced changes in Ritz vectors as the features to characterize the damage patterns defined by the corresponding locations and severity of damage. Unlike most other pattern recognition methods, an artificial neural network (ANN) technique is employed as a tool for systematically identifying the damage pattern corresponding to an observed feature. An important aspect of using an ANN is its design but this is usually skipped in the literature on ANN-based SHM. The design of an ANN has significant effects on both the training and performance of the ANN. As the multi-layer perceptron ANN model is adopted in this work, ANN design refers to the selection of the number of hidden layers and the number of neurons in each hidden layer. A design method based on a Bayesian probabilistic approach for model selection is proposed. The combination of the pattern recognition method and the Bayesian ANN design method forms a practical SHM methodology. A truss model is employed to demonstrate the proposed methodology.

A dynamic thick restarted semi-refined ABLE algorithm for computing a few selected eigentriplets of large nonsymmetric matrices

ABLE method can be used to compute eigentriplets of large scale nonsymmetric matrices. However, there is no guarantee for the Ritz vectors obtained by this method to converge even if the subspace is good enough. Furthermore, for the sake of storage limitation and amount of computations, restarting techniques are often needed. In order to deal with these problems, we propose a dynamic thick restarted semi-refined ABLE algorithm (SABLE) in which we use semi-refined Ritz vectors to approximate the desired eigenvectors. The relationship between the novel method and the classical one is given. Theoretical results show that the new method can circumvent the possible danger that exists in the standard one in some degree. Numerical experiments are made on real world problems, and comparisons are drown on the thick restarted ABLE algorithm (TABLE) and the dynamic thick restarted semi-refined ABLE algorithm (SABLE). They show that the latter is often more powerful and attractive than its standard counterpart.

Design-Oriented Structural Model Order Reduction for Strain-Actuated Flight Vehicle Structures

This paper assesses the accuracy of a number of structural-model order-reduction techniques as applied to strain-actuated (smart) flight-vehicle structures modeled with finite elements (FE). Model order reduction is an essential facet when working in a design-oriented optimization environment. Therefore, to eventually build a design-oriented multidisciplinary optimization (MDO) capability for large-scale strain-actuated structures (e.g., aircraft, spacecraft), insight into the affects of model order reduction on the accuracy of design constraints and sensitivities must first be gained. In this paper reduced-order models are solved using mode displacement (MD) and mode acceleration (MA) methods. Furthermore, the effect of the chosen reduction bases on model accuracy is also investigated. Specifically, the total number of basis modes used is varied. Fictitious masses (FMs) may be added to areas of the structure where local effects predominate, and mass-normalized static solution vectors (Ritz vectors) of important actuation modes may be appended to the bases. By examining the convergence of design constraints and their sensitivities for a variety of model order-reduction approaches (solution methods and reduction bases), valuable insight is gained into the technical challenges that must be addressed when creating a robust MDO capability for strain-actuated structures.

Influence of Seed Layer Moduli on Finite Element Method–Based Modulus Backcalculation Results

The determination of pavement layer moduli from falling weight deflectometer test data is known as backcalculation analysis. Generally, backcalculation analysis is unstable–-greatly influenced by several causes of error. They may be categorized as modeling error in the forward analysis, deflection measurement error, or numerical computation error due to instability in the backcalculation procedure, for example. Because of these problems, the seed values selected for layer moduli greatly influence backcalculation results. To reduce the effects of measurement error, truncated singular-value decomposition is used for regularization. Variable scaling, often used in optimization algorithms, is implemented to improve numerical accuracy. A Ritz vector reduction method is used to solve a large system of dynamic equations in dynamic backcalculation efficiently, and various other means are introduced to decrease computational time. Recent updates of Dynamic Back Analysis for Layer Moduli software, first developed in 1993 and whose solver is based on axi-symmetric finite element method, are presented, as are examples of airfield pavement applications. Results are compared with results from Back Analysis for Layer Moduli static backcalculation software, whose solver was developed using multilayered linear elastic theory. Experience indicates that a dynamic backcalculation is superior to static back-calculation. The results from the two methods are presented and compared.

Influence of Seed Layer Moduli on Finite Element Method-Based Modulus Backcalculation Results

The determination of pavement layer moduli from falling weight deflec-tometer test data is known as backcalculation analysis. Generally, back-calculation analysis is unstable—greatly influenced by several causes of error. They may be categorized as modeling error in the forward analysis, deflection measurement error, or numerical computation error due to instability in the backcalculation procedure, for example. Because of these problems, the seed values selected for layer moduli greatly influence backcalculation results. To reduce the effects of measurement error, truncated singular-value decomposition is used for regularization. Variable scaling, often used in optimization algorithms, is implemented to improve numerical accuracy. A Ritz vector reduction method is used to solve a large system of dynamic equations in dynamic backcalculation efficiently, and various other means are introduced to decrease computational time. Recent updates of Dynamic Back Analysis for Layer Moduli software, first developed ...

Comparison of Different Methods of Ritz Vectors Extraction

Ritz vectors are parameters, which describe object dynamics in similar way as modal vectors. Their advantage with respect to modal vectors is a fact, that less number of them is required to estimate object response. It is also proofed, that Ritz vectors are more sensitive for damage detection then modal vectors. The problem which arises, when one wants to use Ritz vectors for damage assessment is that there are no many methods to extract Ritz vectors from operational data. In addition, the methods require knowledge of analytical mass matrix or state a space model matrix, which makes them rather complicated. In the paper two Ritz vectors extraction methods are taken into account. First bases on flexibility matrix and second is related with state space model. Both of them are evaluated with respect to analytical procedure of Ritz vectors calculation. For the vectors evaluation, a simulation data from theoretical model were used. In next step Ritz vectors were extracted from data measured on a real structure. On the structure damage was simulated. The Ritz vectors extracted with use of both tested methods were used to detect damage.

Model order reduction of linear and nonlinear 3D thermal finite-element description of microwave devices for circuit analysis: Research Articles

Electrothermal models of power devices are necessary for the accurate analysis of their performances. For this reason, this article deals with a methodology to obtain an electrothermal model based on a reduced model of a 3D thermal finite-element (FE) description for its thermal part and on pulsed electrical measurements for its electrical part. The reduced thermal model is based on the Ritz vector approach, which ensures a steady-state solution in every case. An equivalent SPICE subcircuit implementation for circuit simulation is proposed and discussed. An extension of the method to a nonlinear reduced model based on the Kirchoff transformation is also proposed. The complete models have been successfully implemented in circuit simulators for several HBT or PHEMT device structures. Many results concerning devices and circuits are presented, including simulation of both the static and dynamic collector-current collapse in HBTs due to the thermal phenomenon. Moreover, the results in terms of the circuit for an X-band high-power amplifier are also presented. As for the nonlinear approach, results concerning an homogeneous structure is given. © 2005 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2005.

Restarted GMRES augmented with harmonic Ritz vectors for shifted linear systems

Frommer and Glassner [Frommer, A. and Glassner, U., 1998, Restarted GMRES for shifted linear systems, SIAM Journal on Scientific Computing, 19, 15–26.] develop a variant of the restarted GMRES method for shifted linear systems at the expense of only one matrix–vector multiplication per iteration. However, restarting slows down the convergence, even stagnation. We present a variant of the restarted GMRES augmented with some approximate eigenvectors for the shifted systems. The convergence can be much faster at little extra expense. Numerical experiments show its efficiency.

A refined harmonic Rayleigh-Ritz procedure and an explicitly restarted refined harmonic Arnoldi algorithm

The work is fourfold. First, a refined harmonic Rayleigh-Ritz procedure is proposed, some relationships are established between the refined harmonic Ritz vector and the harmonic Ritz vector, an a priori error bound is derived for the refined harmonic Ritz vector, and some properties are established on Rayleigh quotients and residual norms. Second, a resulting refined harmonic Arnoldi method is discussed, and how to compute the refined harmonic Ritz vectors cheaply and accurately is considered. Third, an explicitly restarted refined harmonic Arnoldi algorithm is developed over an augmented Krylov subspace. Finally, numerical examples are reported that compare the new algorithm with the implicitly restarted harmonic Arnoldi algorithm (IRHA) and the implicitly restarted refined harmonic Arnoldi algorithm (IRRHA). Numerical results confirm efficiency of the new algorithm.

COMPARATIVE STUDY OF MEASUREMENT WITH PREDICTION ON DYNAMIC RESPONSE OF PAVEMENT

The pavement design using analytically or mechanistic-empirical based methodologies is now replacing the empirical one. To make the analytically based design advance, comparative study of measurement with prediction on dynamic response of pavement has been done. To effectively backcalculate the material properties of pavement, the Ritz vector is introduced, and the requisite number of Ritz vector needed in the forward and sensitivity analyses is adaptively determined in the developed program. The material properties of constructed layers for pavement can be backcalculated from the FWD test data. Moreover, the predicted surface deflections, soil pressure, and strains are in good agreement with the measured ones for the prepared pavements. It is confirmed that the developed FEM program has adaptability to the tested pavements.

Model order reduction of linear and nonlinear 3D thermal finite-element description of microwave devices for circuit analysis

Electrothermal models of power devices are necessary for the accurate analysis of their performances. For this reason, this article deals with a methodology to obtain an electrothermal model based on a reduced model of a 3D thermal finite-element (FE) description for its thermal part and on pulsed electrical measurements for its electrical part. The reduced thermal model is based on the Ritz vector approach, which ensures a steady-state solution in every case. An equivalent SPICE subcircuit implementation for circuit simulation is proposed and discussed. An extension of the method to a nonlinear reduced model based on the Kirchoff transformation is also proposed. The complete models have been successfully implemented in circuit simulators for several HBT or PHEMT device structures. Many results concerning devices and circuits are presented, including simulation of both the static and dynamic collector-current collapse in HBTs due to the thermal phenomenon. Moreover, the results in terms of the circuit for an X-band high-power amplifier are also presented. As for the nonlinear approach, results concerning an homogeneous structure is given. © 2005 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2005.

Augmented Implicitly Restarted Lanczos Bidiagonalization Methods

New restarted Lanczos bidiagonalization methods for the computation of a few of the largest or smallest singular values of a large matrix are presented. Restarting is carried out by augmentation of Krylov subspaces that arise naturally in the standard Lanczos bidiagonalization method. The augmenting vectors are associated with certain Ritz or harmonic Ritz vectors. Computed examples show the new methods to be competitive with available schemes.

A refined jacobi-davidson method and its correction equation

A central problem in the Jacobi-Davidson method is to expand a projection subspace by solving a certain correction equation. It has been commonly accepted that the correction equation always has a solution. However, it is proved in this paper that this is not true. Conditions are given to decide when it has a unique solution or many solutions or no solution. A refined Jacobi-Davidson method is proposed to overcome the possible nonconvergence of Ritz vectors by computing certain refined approximation eigenvectors from the subspace. A corresponding correction equation is derived for the refined method. Numerical experiments are conducted and efficiency of the refined method is confirmed.

Non‐linear dynamic analysis using one‐dimensional updated subspaces

The purpose of this paper is to improve the effectiveness of ordinary reduction methods performance, in nonlinear dynamic analysis. In this paper, the error vector due to linear and nonlinear dynamic analysis in generalized subspaces is extracted, and is decomposed into two independent components, namely outside and inside components. Based on the inside error component, a new iterative reduction method, one‐dimensional generalized subspace procedure (ODGS), is proposed where an innovative criterion is defined for updating the base vectors necessary for stiffness changes in nonlinear dynamic analysis. In this study, the performance of ODGS for linear and nonlinear analysis of elastodynamic systems including non‐proportional damping based on the Ritz generalized subspace has been proposed. Numerical examples show the competency of the proposed method in both economy and exactness. Time saving gained from the ODGS method could be recompensed to get much more accurate results consuming the same CPU time. This iterative method is more effective than the ordinary reduction methods. Since the method is directly derived from the discrete model based on the finite element method (FEM), the complexity of the structure does not affect directly the effectiveness of ODGS. Therefore, whenever the FEM is effectively capable to represent the topology of the structure, the ODGS results will also represent the system response properly. Same as any other reduction methods, accuracy of this iterative reduction method is directly related to the number of selected Ritz vectors, according to convergence criterion.