Deflated BiCG with an Application to Model Reduction

Mathematical Problems in Engineering ◽

10.1155/2015/372109 ◽

2015 ◽

Vol 2015 ◽

pp. 1-8

Author(s):

Jing Meng ◽

Pei-Yong Zhu ◽

Hou-Biao Li

Keyword(s):

Model Reduction ◽

Linear Systems ◽

Invariant Subspaces ◽

Numerical Examples ◽

Right Hand ◽

Ritz Vectors ◽

Left And Right

Most calculations in model reduction involve the solutions of a sequence of dual linear systems with multiple right-hand sides. To solve such systems efficiently, a new deflated BiCG method is explored in this paper. The proposed algorithm uses harmonic Ritz vectors to approximate left and right invariant subspaces inexpensively via small descenting direction vectors found by subsequent runs of deflated BiCG and then derives the deflated subspaces for the next pair of dual linear systems. This process leads to faster convergence for the next pair of systems. Numerical examples illustrate the effectiveness of the proposed method.

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A Relaxed Kaczmarz Method for Fuzzy Linear Systems

10.21203/rs.3.rs-283515/v1 ◽

2021 ◽

Author(s):

Ke Wang ◽

Shijun Zhang ◽

Shiheng Wang

Keyword(s):

Linear Systems ◽

Convergence Theorem ◽

Iterative Scheme ◽

Coefficient Matrix ◽

Systems Of Equations ◽

Numerical Examples ◽

Right Hand ◽

Kaczmarz Method ◽

Fuzzy Linear Systems ◽

Linear Systems Of Equations

Abstract A relaxed Kaczmarz method is presented for solving a class of fuzzy linear systems of equations with crisp coefficient matrix and fuzzy right-hand side. The iterative scheme is established and the convergence theorem is provided. Numerical examples show that the method is effective.

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Recycling Krylov Subspaces for Solving Linear Systems with Successively Changing Right-hand Sides Arising in Model Reduction

Lecture Notes in Electrical Engineering - Model Reduction for Circuit Simulation ◽

10.1007/978-94-007-0089-5_6 ◽

2011 ◽

pp. 125-140 ◽

Cited By ~ 2

Author(s):

Peter Benner ◽

Lihong Feng

Keyword(s):

Model Reduction ◽

Linear Systems ◽

Krylov Subspaces ◽

Right Hand

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MINRES Seed Projection Methods for Solving Symmetric Linear Systems with Multiple Right-Hand Sides

Mathematical Problems in Engineering ◽

10.1155/2014/357874 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6

Author(s):

Xin Li ◽

Hao Liu ◽

Jingfu Zhu

Keyword(s):

Linear Systems ◽

Projection Method ◽

Symmetric Matrix ◽

Coefficient Matrix ◽

Projection Methods ◽

Numerical Examples ◽

Right Hand ◽

Symmetric Systems

We consider the MINRES seed projection method for solving multiple right-hand side linear systemsAX=B, whereA∈Rn×nis a nonsingular symmetric matrix,B∈Rn×p. In general, GMRES seed projection method is one of the effective methods for solving multiple right-hand side linear systems. However, when the coefficient matrix is symmetric, the efficiency of this method would be weak. MINRES seed projection method for solving symmetric systems with multiple right-hand sides is proposed in this paper, and the residual estimation is analyzed. The numerical examples show the efficiency of this method.

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Visual context differentially mediates left- and right-hand reaching movements

PsycEXTRA Dataset ◽

10.1037/e520562012-747 ◽

2009 ◽

Author(s):

Jos J. Adam ◽

Susan Hoonhorst ◽

Rick Muskens ◽

Jay Pratt ◽

Martin H. Fischer

Keyword(s):

Reaching Movements ◽

Visual Context ◽

Right Hand ◽

Left And Right

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Optimal Estimation of Unknown Right-Hand Sides of First Order Linear Systems of Periodic Ordinary Differential Equations from Indirect Noisy Observations of Their Solutions

2020 IEEE 2nd International Conference on System Analysis & Intelligent Computing (SAIC) ◽

10.1109/saic51296.2020.9239223 ◽

2020 ◽

Author(s):

Oleksandr Nakonechnyi ◽

Yuri Podlipenko

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Linear Systems ◽

Optimal Estimation ◽

Order Linear ◽

First Order ◽

Right Hand

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On Existence of Stabilizing Switching Laws within a Class of Unstable Linear Systems

Abstract and Applied Analysis ◽

10.1155/2013/681523 ◽

2013 ◽

Vol 2013 ◽

pp. 1-11 ◽

Cited By ~ 1

Author(s):

Sendren Sheng-Dong Xu ◽

Chih-Chiang Chen

Keyword(s):

Control Systems ◽

Linear Systems ◽

Switched Systems ◽

The Other ◽

Linear Control ◽

Linear Control Systems ◽

Problem Statement ◽

Theoretical Derivation ◽

Control Laws ◽

Numerical Examples

The equivalence of two conditions, condition (3) and condition (4) stated in Problem Statement section, regarding the existence of stabilizing switching laws between two unstable linear systems first appeared in (Feron 1996). Although Feron never published this result, it has been referenced in almost every survey on switched systems; see, for example, (Liberzon and Morse 1999). This paper proposes another way to prove the equivalence of two conditions regarding the existence of stabilizing switching laws between two unstable linear systems. One is effective for theoretical derivation, while the other is implementable, and a class of stabilizing switching laws have been explicitly constructed by Wicks et al. (1994). With the help of the equivalent relation, a condition for the existence of controllers and stabilizing switching laws between two unstabilizable linear control systems is then proposed. Then, the study is further extended to the issue concerning the construction of quadratically stabilizing switching laws among unstable linear systems and unstabilizable linear control systems. The obtained results are employed to study the existence of control laws and quadratically stabilizing switching laws within a class of unstabilizable linear control systems. The numerical examples are illustrated and simulated to show the feasibility and effectiveness of the proposed methods.

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