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On a general solution of partially described inverse quadratic eigenvalue problems and its applications
"... In this paper, we consider to solve a general form of real and symmetric n × n matrices M, C, K with M being positive definite for an inverse quadratic eigenvalue problem (IQEP): Q(λ)x ≡ (λ 2 M + λC + K)x = 0, so that Q(λ) has a partially prescribed subset of k eigenvalues and eigenvectors (k ≤ n). ..."
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In this paper, we consider to solve a general form of real and symmetric n × n matrices M, C, K with M being positive definite for an inverse quadratic eigenvalue problem (IQEP): Q(λ)x ≡ (λ 2 M + λC + K)x = 0, so that Q(λ) has a partially prescribed subset of k eigenvalues and eigenvectors (k ≤ n). Via appropriate choice of free variables in the general form of IQEP, for k = n: we solve (i) an IQEP with K semipositive definite, (ii) an IQEP having additionally assigned n eigenvalues, (iii) an IQEP having additionally assigned r eigenpairs (r ≤ √ n) under closed complex conjugation; for k < n: we solve (i) a unique monic IQEP with k = n−1 which has an additionally assigned complex conjugate eigenpair, (ii) an IQEP having additionally assigned 2(n−k) complex eigenvalues with nonzero imaginary parts. Some numerical results are given to show the solvability of the above described IQEPs.
Spillover Phenomenon in Quadratic Model Updating
 AIAA JOURNAL
, 2007
"... Model updating concerns the modification of an existing but inaccurate model with measured data. For models characterized by quadratic pencils, the measured data usually involve incomplete knowledge of natural frequencies, mode shapes, or other spectral information. In conducting the updating, it is ..."
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Cited by 12 (5 self)
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Model updating concerns the modification of an existing but inaccurate model with measured data. For models characterized by quadratic pencils, the measured data usually involve incomplete knowledge of natural frequencies, mode shapes, or other spectral information. In conducting the updating, it is often desirable to match only the part of observed data without tampering with the other part of unmeasured or unknown eigenstructure inherent in the original model. Such an updating, if possible, is said to have no spillover. This paper studies the spillover phenomenon in the updating of quadratic pencils. In particular, it is shown that an updating with no spillover is always possible for undamped quadratic pencils, whereas spillover for damped quadratic pencils is generally unpreventable.
A dual optimization approach to inverse quadratic eigenvalue problems with partial eigenstructure
 SIAM J. Sci. Comput
"... The inverse quadratic eigenvalue problem (IQEP) arises in the field of structural dynamics. It aims to find three symmetric matrices, known as the mass, the damping, and the stiffness matrices, respectively, such that they are closest to the given analytical matrices and satisfy the measured data. T ..."
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Cited by 8 (6 self)
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The inverse quadratic eigenvalue problem (IQEP) arises in the field of structural dynamics. It aims to find three symmetric matrices, known as the mass, the damping, and the stiffness matrices, respectively, such that they are closest to the given analytical matrices and satisfy the measured data. The difficulty of this problem lies in the fact that in applications the mass matrix should be positive definite and the stiffness matrix positive semidefinite. Based on an equivalent dual optimization version of the IQEP, we present a quadratically convergent Newtontype method. Our numerical experiments confirm the high efficiency of the proposed method.
Structured quadratic inverse eigenvalue problem, I. Serially linked systems
 SIAM J. Sci. Comput
"... Abstract. Quadratic pencils arising from applications are often inherently structured. Factors contributing to the structure include the connectivity of elements within the underlying physical system and the mandatory nonnegativity of physical parameters. For physical feasibility, structural constra ..."
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Cited by 6 (1 self)
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Abstract. Quadratic pencils arising from applications are often inherently structured. Factors contributing to the structure include the connectivity of elements within the underlying physical system and the mandatory nonnegativity of physical parameters. For physical feasibility, structural constraints must be respected. Consequently, they impose additional challenges on the inverse eigenvalue problems which intend to construct a structured quadratic pencil from prescribed eigeninformation. Knowledge of whether a structured quadratic inverse eigenvalue problem is solvable is interesting in both theory and applications. However, the issue of solvability is problem dependent and has to be addressed structure by structure. This paper considers one particular structure where the elements of the physical system, if modeled as a massspring system, are serially linked. The discussion recasts both undamped or damped problems in a framework of inequality systems that can be adapted for numerical computation. Some open questions are described. AMS subject classifications. 65F18, 15A22, 93B55
Semidefinite programming techniques for structured quadratic inverse eigenvalue problems
, 2009
"... Abstract. In the past decade or so, semidefinite programming (SDP) has emerged as a powerful tool capable of handling a remarkably wide range of problems. This article describes an innovative application of SDP techniques to quadratic inverse eigenvalue problems (QIEPs). The notion of QIEPs is of f ..."
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Cited by 4 (2 self)
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Abstract. In the past decade or so, semidefinite programming (SDP) has emerged as a powerful tool capable of handling a remarkably wide range of problems. This article describes an innovative application of SDP techniques to quadratic inverse eigenvalue problems (QIEPs). The notion of QIEPs is of fundamental importance because its ultimate goal of constructing or updating a vibration system from some observed or desirable dynamical behaviors while respecting some inherent feasibility constraints well suits many engineering applications. Thus far, however, QIEPs have remained challenging both theoretically and computationally due to the great variations of structural constraints that must be addressed. Of notable interest and significance are the uniformity and the simplicity in the SDP formulation that solves effectively many otherwise very difficult QIEPs. AMS subject classifications. 65F18, 15A22, 93B55 Key words. semidefinite programming, quadratic pencil, inverse eigenvalue problem, structural constraint,
STRUCTURED QUADRATIC INVERSE EIGENVALUE PROBLEM, II. GENERALLY LINKED SYSTEMS
"... Abstract. Quadratic pencils arise in many areas of important applications. The underlying physical systems often impose inherent structures, which include the predetermined innerconnectivity among elements within the physical system and the mandatory nonnegativity of physical parameters, on the pen ..."
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Abstract. Quadratic pencils arise in many areas of important applications. The underlying physical systems often impose inherent structures, which include the predetermined innerconnectivity among elements within the physical system and the mandatory nonnegativity of physical parameters, on the pencils. In the inverse problem of reconstructing a quadratic pencil from prescribed eigeninformation, respecting the desirable structure becomes important but more challenging both theoretically and practically. The issue of whether a structured inverse eigenvalue problem is solvable or not is problem dependent and has to be addressed structure by structure. In an earlier work, physical systems that can be modeled under the paradigm of a serially linked massspring system have been considered via specifically formulated inequality systems. In this paper, the framework is generalized to arbitrary generally linked systems. In particular, given any configuration of innerconnectivity in a massspring system, this paper presents a mechanism that systematically and automatically generates a corresponding inequality system. A numerical approach is proposed to determine whether the inverse problem is solvable and, if yes, computes the coefficient matrices while providing an estimate of the residual error. The most important feature of this approach is that it is problem independent, that is, the approach is general and robust for any kind of physical configuration. The ideas discussed in this paper have been implemented into a software package by which some numerical experiments are reported.
INVERSE MODE PROBLEM FOR REAL AND SYMMETRIC QUADRATIC MODELS
, 2009
"... Abstract. Many natural phenomena can be modeled by a secondorder dynamical system M ¨y+C ˙y+Ky = f(t), where y(t) stands for an appropriate state variable and M, C, K are timeinvariant, real and symmetric matrices. In contrast to the classical inverse vibration problem where a model is to be deter ..."
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Abstract. Many natural phenomena can be modeled by a secondorder dynamical system M ¨y+C ˙y+Ky = f(t), where y(t) stands for an appropriate state variable and M, C, K are timeinvariant, real and symmetric matrices. In contrast to the classical inverse vibration problem where a model is to be determined from natural frequencies corresponding to various boundary conditions, the inverse mode problem concerns the reconstruction of the coefficient matrices (M, C, K) from a prescribed or observed subset of natural modes. This paper set forth a mathematical framework for the inverse mode problem and resolves some open questions raised in the literature. In particular, it shows that, given merely the desirable structure of the spectrum, namely, given the cardinalities of real or complex eigenvalues but not the actual eigenvalues, the set of eigenvectors can be completed via solving an underdetermined nonlinear system of equations. This completion suffices to construct symmetric coefficient matrices (M, C, K) whereas the underlying system can have arbitrary eigenvalues. Generic conditions under which the real symmetric quadratic inverse mode problem is solvable are discussed. Applications to important tasks such as updating models without spillover or constructing models with positive semidefinite coefficient matrices are discussed. AMS subject classifications. 65F18, 15A22, 93B55
A Smoothing Newton’s Method for the Construction of a Damped Vibrating System From Noisy Test Eigendata
, 2008
"... In this paper we consider an inverse problem for a damped vibration system from the noisy measured eigendata, where the mass, damping, and stiffness matrices are all symmetric positive definite matrices with the mass matrix being diagonal and the damping and stiffness matrices tridiagonal. To take i ..."
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In this paper we consider an inverse problem for a damped vibration system from the noisy measured eigendata, where the mass, damping, and stiffness matrices are all symmetric positive definite matrices with the mass matrix being diagonal and the damping and stiffness matrices tridiagonal. To take into consideration the noise in the data, the problem is formulated as a convex optimization problem involving quadratic constraints on the unknown mass, damping, and stiffness parameters. Then we propose a smoothing Newtontype algorithm for the optimization problem, which improves a preexisting estimate of a solution to the inverse problem. We show that the proposed method converges both globally and quadratically. Numerical examples are also given to demonstrate the efficiency of our method.
Symmetric Tridiagonal Inverse Quadratic Eigenvalue Problems with Partial Eigendata
, 2007
"... In this paper we concern the inverse problem of constructing the nbyn real symmetric tridiagonal matrices C and K so that the monic quadratic pencil Q(λ): = λ2I + λC + K (where I is the identity matrix) possesses the given partial eigendata. We first provide the sufficient and necessary conditions ..."
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In this paper we concern the inverse problem of constructing the nbyn real symmetric tridiagonal matrices C and K so that the monic quadratic pencil Q(λ): = λ2I + λC + K (where I is the identity matrix) possesses the given partial eigendata. We first provide the sufficient and necessary conditions for the existence of an exact solution to the inverse problem from the selfconjugate set of prescribed four eigenpairs. To find a physical solution for the inverse problem where the matrices C and K are weakly diagonally dominant and have positive diagonal elements and negative offdiagonal elements, we consider the inverse problem from the partial measured noisy eigendata. We propose a regularized smoothing Newton method for solving the inverse problem. The global and quadratic convergence of our approach is established under some mild assumptions. Some numerical examples and a practical engineering application in vibrations show the efficiency of our method.
over04.tex” THE SPILLOVER PHENOMENON IN QUADRATIC MODEL UPDATING
, 2007
"... Abstract. Model updating concerns the modification of an existing but inaccurate model with measured data. For models characterized by quadratic pencils, the measured data usually involve incomplete knowledge of natural frequencies, mode shapes, or other spectral information. In conducting the updat ..."
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Abstract. Model updating concerns the modification of an existing but inaccurate model with measured data. For models characterized by quadratic pencils, the measured data usually involve incomplete knowledge of natural frequencies, mode shapes, or other spectral information. In conducting the updating, it is often desirable to match only the part of observed data without tampering with the other part of unmeasured or unknown eigenstructure inherent in the original model. Such an updating, if possible, is said to have no spillover. This paper studies the spillover phenomenon in the updating of quadratic pencils. In particular, it is shown that an updating with no spillover is always possible for undamped quadratic pencils whereas spillover for damped quadratic pencils is generally unpreventable. AMS subject classifications. 65F18, 15A22, 93B55