Block Arnoldi Algorithm Research Articles

On Some Numerical Methods for Solving Large Differential Nonsymmetric Stein Matrix Equations

In this paper, we propose a new numerical method based on the extended block Arnoldi algorithm for solving large-scale differential nonsymmetric Stein matrix equations with low-rank right-hand sides. This algorithm is based on projecting the initial problem on the extended block Krylov subspace to obtain a low-dimensional differential Stein matrix equation. The obtained reduced-order problem is solved by the backward differentiation formula (BDF) method or the Rosenbrock (Ros) method, the obtained solution is used to build the low-rank approximate solution of the original problem. We give some theoretical results and report some numerical experiments.

A Block Arnoldi Algorithm Based Reduced-Order Model Applied to Large-Scale Algebraic Equations of a 3-D Field Problem

In simulations of three-dimensional transient physics filled through a numerical approach, the order of the equation set of high-fidelity models is extremely high. To eliminate the large dimension of equations, a model order reduction (MOR) technique is introduced. In the existing MOR methods, the block Arnoldi algorithm-based MOR method is numerically stable, achieving a passively reduced order model. Nevertheless, this method performs poorly when it is applied to very wide-frequency transients. To eliminate this deficiency, multipoint MOR methods are emerging. However, it is hard to directly apply an existing multipoint MOR method to a 3-D transient field equation set. The implementation issues in a reduction process (such as the selection of expansion points, the number of moments matched at a point and the error bound) have not been explored in detail. In this respect, an adaptive multipoint model reduction model based on the Arnoldi algorithm is proposed to obtain the reduced-order models of a 3-D temperature field. The originality of this study is the proposal of a novel adaptive algorithm for selecting expansion points, matching moments automatically, using a posterior-error estimator based on temperature response coupled with a network topological method (NTM). The computational efficiency and accuracy of the proposed method are evaluated by the numerical results from solving the temperature field of a prototype insulated-gate bipolar transistor (IGBT).

Approximate solutions to large nonsymmetric differential Riccati problems with applications to transport theory

SummaryIn this paper, we consider large‐scale nonsymmetric differential matrix Riccati equations with low‐rank right‐hand sides. These matrix equations appear in many applications such as control theory, transport theory, applied probability, and others. We show how to apply Krylov‐type methods such as the extended block Arnoldi algorithm to get low‐rank approximate solutions. The initial problem is projected onto small subspaces to get low dimensional nonsymmetric differential equations that are solved using the exponential approximation or via other integration schemes such as backward differentiation formula (BDF) or Rosenbrock method. We also show how these techniques can be easily used to solve some problems from the well‐known transport equation. Some numerical examples are given to illustrate the application of the proposed methods to large‐scale problems.

On applying weighted seed techniques to GMRES algorithm for solving multiple linear systems

In the present paper, we are concerned by weighted Arnoldi like methods for solving large and sparse linear systems that have different right-hand sides but have the same coefficient matrix. We first give detailed descriptions of the weighted Gram-Schmidt process and of a Ruhe variant of the weighted block Arnoldi algorithm. We also establish some theoretical results that links the iterates of the weighted block Arnoldi process to those of the non weighted one. Then, to accelerate the convergence of the classical restarted block and seed GMRES methods, we introduce the weighted restarted block and seed GMRES methods. Numerical experiments that are done with different matrices coming from the Matrix Market repository or from the university of Florida sparse matrix collection are reported at the end of this work in order to compare the performance and show the effectiveness of the proposed methods.

On some Krylov subspace based methods for large-scale nonsymmetric algebraic Riccati problems

In the present paper, we consider large scale nonsymmetric matrix Riccati equations with low rank right hand sides. These matrix equations appear in many applications such as transport theory, Wiener–Hopf factorization of Markov chains, applied probability and others. We show how to apply directly Krylov methods such as the extended block Arnoldi algorithm to get low rank approximate solutions. We also combine the Newton method and block Krylov subspace methods to get approximations of the desired minimal nonnegative solution. We give some theoretical results and report some numerical experiments for the well known transport equation.

A REDUCED ORDER MODEL BASED ON BLOCK ARNOLDI METHOD FOR AEROELASTIC SYSTEM

A reduced-order model (ROM) based on block Arnoldi algorithm to quickly predict flutter boundary of aeroelastic system is investigated. First, a mass–damper–spring dynamic system is tested, which shows that the low dimension system produced by the block Arnoldi method can keep a good dynamic property with the original system in low and high frequencies. Then a two-degree of freedom transonic nonlinear aerofoil aeroelastic system is used to validate the suitability of the block Arnoldi method in flutter prediction analysis. In the aerofoil case, the ROM based on a linearized model is obtained through a high-fidelity nonlinear computational fluid dynamics (CFD) calculation. The order of the reduced model is only 8 while it still has nearly the same accuracy as the full 9600-order model. Compared with the proper orthogonal decomposition (POD) method, the results show that, without snapshots the block Arnoldi/ROM has a unique superiority by maintaining the system stability aspect. The flutter boundary of the aeroelastic system predicted by the block Arnoldi/ROM agrees well with the CFD and reference results. The Arnoldi/ROM provides an efficient and convenient tool to quick analyze the system stability of nonlinear transonic aeroelastic systems.

Block Krylov subspace methods for large-scale matrix computations in control

In this paper we show how to improve the approximate solution of the large Sylvester equation obtained by an arbitrary method. Such problems appear in many areas of control theory such as the computation of Hankel singular values, model reduction algorithms and others. Moreover, we propose a new method based on refinement process and weighted block Arnoldi algorithm for solving large Sylvester matrix equation. The numerical tests report the effectiveness of these methods.

Mathematical Modeling of Transient Responses in a Large Scale Multiconductor System

This paper proposes a reduced-order model for the large scale circuits representing the multiconductor systems. It is based on the block Arnoldi algorithm for calculating congruence transformation matrix. By setting up the state equations in frequency domain for the multiconductor systems, the transfer functions are calculated and so the frequency response curves are obtained. A comparison is made between those curves obtained from the circuits with and without reduced-order treatment and a better agreement appears between them. By using inverse Laplace transform, the lightning transient responses in the multiconductor systems are given in time-domain. A better agreement is shown between calculated and experimental results, which confirm the validity of the proposed model.

A reduced-order model for whole-chip thermal analysis of microfluidic lab-on-a-chip systems

This paper presents a Krylov subspace projection-based Reduced Order Model (ROM) for whole microfluidic chip thermal analysis, including conjugate heat transfer. Two key steps in the reduced order modeling procedure are described in detail, including (1) the acquisition of a 3D full-scale computational model in the state-space form to capture the dynamic thermal behavior of the entire microfluidic chip; and (2) the model order reduction using the Block Arnoldi algorithm to markedly lower the dimension of the full-scale model. Case studies using practically relevant thermal microfluidic chip are undertaken to establish the capability and to evaluate the computational performance of the reduced order modeling technique. The ROM is compared against the full-scale model and exhibits good agreement in spatiotemporal thermal profiles (<0.5% relative error in pertinent time scales) and over three orders-of-magnitude acceleration in computational speed. The salient model reusability and real-time simulation capability renders it amenable for operational optimization and in-line thermal control and management of microfluidic systems and devices.

Block Arnoldi-based methods for large scale discrete-time algebraic Riccati equations

In the present paper, we present block Arnoldi-based methods for the computation of low rank approximate solutions of large discrete-time algebraic Riccati equations (DARE). The proposed methods are projection methods onto block or extended block Krylov subspaces. We give new upper bounds for the norm of the error obtained by applying these block Arnoldi-based processes. We also introduce the Newton method combined with the block Arnoldi algorithm and present some numerical experiments with comparisons between these methods.

Model Order Reduction for Pre-Stressed Harmonic Analysis of Micromechanical Beam Resonators

A second-order system model order reduction method for pre-stressed harmonic analysis of electrostatically actuated microbeams is demonstrated, which produces a low dimensional approximation of the original system and enables a substantial reduction of simulation time. The moment matching property for second-order dynamic systems is studied and the block Arnoldi algorithm is adopted for the generation of the Krylov subspace, which extracts the low order model from the discretized system assembled through finite element analysis. The difference between two successive reduced models suggests the choice of the order for the reduce model. A detailed comparison research among the full model and the reduced models is performed. The research results confirm the effectiveness of the presented method.

An Arnoldi Algorithm for Power-Delivery Networks With Variable Dielectric Constant and Loss Tangent

The tracking-sensitivity algorithm is an efficient method to calculate the changes in the network matrix of a circuit, when a global parameter, such as temperature, is continuously varied. In this paper, we apply the concept of tracking sensitivity on the frequency-domain simulation of power-delivery networks (PDNs) in chip packages and printed circuit boards. The global variable we consider is the complex permittivity of the dielectric. We present a methodology to calculate the change in the PDN impedance due to a variation in the dielectric constant, loss tangent, or both. This information is useful not only to understand any deviations between simulations and measurements, but also to understand the impact of the variability of these dielectric properties on the performance of the PDN. Using the tracking-sensitivity algorithm, we can efficiently recalculate the impedance matrix after changing a global parameter. The classical tracking-sensitivity algorithm was based on a power series expansion. This paper presents a more accurate approach based on the block Arnoldi algorithm.

A thick-restarted block Arnoldi algorithm with modified Ritz vectors for large eigenproblems

The block Arnoldi method is one of the most commonly used techniques for large eigenproblems. In this paper, we exploit certain modified Ritz vectors to take the place of Ritz vectors in the thick-restarted block Arnoldi algorithm, and propose a modified thick-restarted block Arnoldi algorithm for large eigenproblems. We then consider how to periodically combine the refined subspace iterative method with the modified thick-restarting block Arnoldi algorithm for computing a few dominant eigenpairs of a large matrix. The resulting algorithm is called a Subspace-Block Arnoldi algorithm. Numerical experiments show the efficiency of our new algorithms.

Use of near-breakdowns in the block Arnoldi method for solving large Sylvester equations

In this paper we construct an approximate solution to large Sylvester equations of the form AX+XB=CD^T. The construction uses a new variant of the block Arnoldi algorithm which exploits the near-breakdowns, that is, the near singularities in the generated basis. As a consequence, the algorithm eliminates the directions which do not contribute to the approximate solution by keeping in the generated basis only the ''active'' vectors detected by a criterion based on the residual associated with the approximate solution. The effectiveness of the proposed algorithm is demonstrated on several examples, including the case where the matrix B has a small or a large size.

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.

Connecting heat transfer macromodels for array MEMS structures

Different methodologies to extract a dynamic compact thermal model of a microelectronic or MEMS device have been developed in recent years. They include strategies based on data fitting, a time-constant spectrum, modal analysis and finally formal model reduction. Researchers seek compact thermal multiport representation for system level simulation. However, thermal flux is not lumped by nature as electrical flow and, as a matter of fact, there appears to be very few works on how to couple dynamic compact thermal models with each other. In the present work, we take a finite element model of a MOS-transistor-based microhotplate array made in ANSYS as a case study. We consider two available techniques to make the model reduction. First, we employ the block Arnoldi algorithm that makes model reduction of the whole array at once. Second, we use the modified Guyan algorithm for a single hotplate and couple reduced models via substructuring. We compare both techniques with each other and discuss the possibility of combining the best parts of the two approaches.

Efficient optimization of transient dynamic problems in MEMS devices using model order reduction

One of the main obstacles to including transient dynamic effects into the performance functions of a structural optimization for microelectromechanical systems (MEMS) is the high computational cost of each time-dependent response simulation. This paper focuses on the application of model order reduction techniques to optimal design so as to reduce the transient analysis time for the optimization process. To do this, our open-source software mor4ansys performs model order reductions via the block Arnoldi algorithm directly to ANSYS finite element models. We adopt a micro accelerometer as an example to demonstrate the advantages of this approach. The harmonic and transient results of a reduced-order model of the accelerometer yield very good agreement with that from the original high-dimensional ANSYS model. The use of reduced-order models within the optimization iterations produces almost the same results as those without model order reduction, and speeds up the total computation by at least an order of magnitude.

Block Krylov Subspace Methods for Large Algebraic Riccati Equations

In the present paper, we present numerical methods for the computation of approximate solutions to large continuous-time and discrete-time algebraic Riccati equations. The proposed methods are projection methods onto block Krylov subspaces. We use the block Arnoldi process to construct an orthonormal basis of the corresponding block Krylov subspace and then extract low rank approximate solutions. We consider the sequential version of the block Arnoldi algorithm by incorporating a deflation technique which allows us to delete linearly and almost linearly dependent vectors in the block Krylov subspace sequences. We give some theoretical results and present numerical experiments for large problems.

An iterative block Arnoldi algorithm with modified approximate eigenvectors for large unsymmetric eigenvalue problems

A modified m-step Arnoldi algorithm proposed by Jia and Elsner for computing a few selected eigenpairs of large unsymmetric matrices is generalized to its block version. In the new method, we use new modified approximate eigenvectors obtained from a linear combination of Ritz vectors and the wasted ( m+1)th block basis vector, whose residual norms are satisfied with some ( p+1)-dimensional minimization problems. The resulting block Arnoldi algorithm is not only better than the standard m-step one both in theory and in practice but also cheaper than the standard ( m+1)-step one. The relationships among the residual norms of Ritz pairs and those of new approximate eigenpairs are analyzed. Theoretical results show that the new method can overcome the drawback of nonconvergence, which exists in the conventional one in some extent. We carry out some numerical experiments, which exhibit that the new algorithm is more efficient and works better than its counterparts.

A convergence analysis of GMRES and FOM methods for Sylvester equations

We discuss convergence properties of the GMRES and FOM methods for solving large Sylvester equations of the form AX-XB=C. In particular we show the importance of the separation between the fields of values of A and B on the convergence behavior of GMRES. We also discuss the stagnation phenomenon in GMRES and its consequence on FOM. We generalize the issue of breakdown in the block-Arnoldi algorithm and explain its consequence on FOM and GMRES methods. Several numerical tests illustrate the theoretical results.