@article{lin_dong_chu_2010, title={Inverse mode problems for real and symmetric quadratic models}, volume={26}, ISSN={["1361-6420"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-77952566534&partnerID=MN8TOARS}, DOI={10.1088/0266-5611/26/6/065003}, abstractNote={Many natural phenomena can be modeled by a second-order dynamical system , where stands for an appropriate state variable and M, C, K are time-invariant, 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 of the actual eigenvalues, the set of eigenvectors can be completed via solving an under-determined 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 spill-over or constructing models with positive semi-definite coefficient matrices are discussed.}, number={6}, journal={INVERSE PROBLEMS}, author={Lin, Matthew M. and Dong, Bo and Chu, Moody T.}, year={2010}, month={Jun} }
 @article{lin_chu_2010, title={On the nonnegative rank of Euclidean distance matrices}, volume={433}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-77953135527&partnerID=MN8TOARS}, DOI={10.1016/j.laa.2010.03.038}, abstractNote={The Euclidean distance matrix for n distinct points in Rr is generically of rank r + 2. It is shown in this paper via a geometric argument that its nonnegative rank for the case r = 1 is generically n.}, number={3}, journal={Linear Algebra and Its Applications}, author={Lin, M.M. and Chu, Moody}, year={2010}, pages={681–689} }
 @article{lin_dong_chu_2010, title={Semi-definite programming techniques for structured quadratic inverse eigenvalue problems}, volume={53}, ISSN={["1572-9265"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-77950457631&partnerID=MN8TOARS}, DOI={10.1007/s11075-009-9309-9}, abstractNote={In the past decade or so, semi-definite 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.}, number={4}, journal={NUMERICAL ALGORITHMS}, author={Lin, Matthew M. and Dong, Bo and Chu, Moody T.}, year={2010}, month={Apr}, pages={419–437} }
 @article{dong_lin_chu_2009, title={Parameter reconstruction of vibration systems from partial eigeninformation}, volume={327}, ISSN={["1095-8568"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-69049084391&partnerID=MN8TOARS}, DOI={10.1016/j.jsv.2009.06.026}, abstractNote={Quadratic matrix polynomials are fundamental to vibration analysis. Because of the predetermined interconnectivity among the constituent elements and the mandatory nonnegativity of the physical parameters, most given vibration systems will impose some inherent structure on the coefficients of the corresponding quadratic matrix polynomials. In the inverse problem of reconstructing a vibration system from its observed or desirable dynamical behavior, respecting the intrinsic structure becomes important and challenging both theoretically and practically. The issue of whether a structured inverse eigenvalue problem is solvable 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 mass–spring 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 interconnectivity in a mass–spring 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 it is so, 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.}, number={3-5}, journal={JOURNAL OF SOUND AND VIBRATION}, author={Dong, Bo and Lin, Matthew M. and Chu, Moody T.}, year={2009}, month={Nov}, pages={391–401} }