The Quadratic Eigenvalue Problem is to find eigenvalues and eigenvectors a quadratic matrix pencil of the form P(L) = ML^2+CL+K , where the matrices M, C, and K are square matrices. Unfortunately, the problem has not been widely studied because of the intrinsic difficulties with solving the problem in a numerically effective way. Indeed, the state-of-the-art computational techniques are capable of computing only a few extremal eigenvalues and eigenvectors, especially if the matrices are large and sparse, which is often the case in practical applications. The inverse quadratic eigenvalue problem, on the other hand, refers to constructing the matrices M, C, and K, given the complete or partial spectrum and the associated eigenvectors. The inverse quadratic eigenvalue problem is equally important and arises in a wide variety of engineering applications, including mechanical vibrations, aerospace engineering, design of space structures, structural dynamics, etc.
Of special practical importance is to construct the coefficient matrices from the knowledge of only partial spectrum and the associated eigenvectors. The greatest computational challenge is to solve the partial quadratic inverse eigenvalue problem using the small number of eigenvalues and eigenvectors which are all that are computable using the state-of-the-art techniques. Furthermore, computational techniques must be able to take advantage of the exploitable physical properties, such as the symmetry, positive definiteness, sparsity, etc., which are computational assets for solution of large and sparse problems.
This talk will deal with two special quadratic inverse eigenvalue problems that arise in mechanical vibration and structural dynamics. The first one, Quadratic Partial Eigenvalue Assignment Problem(QPEVAP), arises in controlling dangerous vibrations in mechanical structures. Mathematically, the problem is to find two control feedback matrices such that a small amount of the eigenvalues of the associated quadratic eigenvalue problem, which are responsible for dangerous vibrations, are reassigned to suitably chosen ones while keeping the remaining large number of eigenvalues and eigenvectors unchanged. Additionally, for robust and economic control design, these feedback matrices must be found in such a way that they have the norms as small as possible and the condition number of the modified quadratic inverse problem is minimized. These considerations give rise to two nonlinear unconstrained optimization problems, known respectively, as Robust Quadratic Partial Eigenvalue Assignment Problem (RQPEVAP) and Minimum Norm Quadratic Partial Eigenvalue Assignment Problem (MNQPEVAP). The other one, the Finite Element Model Updating Problem (FEMUP), arising in the design and analysis of structural dynamics, refers to updating an analytical finite element model so that a set of measured eigenvalues and eigenvectors from a real-life structure are reproduced and the physical and structural properties of the original model are preserved. A properly updated model can be used in confidence for future designs and constructions. Another major application of FEMUP is the damage detections in structures. Solution of FEMUP also give rises to several constrained nonlinear optimization problems. I will give an overview of the recent developments on computational methods for these difficult nonlinear optimization problems and discuss directions of future research with some open problems for future research. The talk is interdisciplinary in nature and will be of interests to computational and applied mathematicians, and control and vibration engineers and optimization experts.
Bio:
Biswa Nath Datta is a Distinguished Research Professor at Northern Illinois University. He is a Professor in the Mathematical Sciences Department and an adjunct professor in the Electrical and Mechanical Engineering Department at NIU. He has served as the Director of the Applications Involvement Component (AIC) of the Doctoral Program at NIU. He developed the current “Computational Mathematics” program and took a major active role in the development of the Mathematical Sciences Ph.D program at NIU. Datta also held visiting professorship at the University of Illinois at Urbana-Champaign, Pennsylvania State University, Southern Illinois University, University of California at San Diego, State University of Campinas, Campinas, Brazil, as well as many other universities, research organizations, and industries around the world, including the Boeing Company and Wolfram Research Incorporation.
His research interests are interdisciplinary that blend numerical linear algebra and scientific computing (including large-scale and high performance computing) with control and vibration engineering. He research has produced computationally viable algorithms and high-quality engineering and scientific software packages scientific computing and electrical and vibration control systems design and analysis. He also pioneered the research in on large-scale and parallel computations in control. His current research involves development of computationally viable and mathematically sound algorithms for “Active Vibration Control” and “Model Updating” of large vibration systems modeled by finite element techniques. The thrusts of this research are in its applications to real-life problems arising in vibration industries, including automobiles, buildings, bridges, highways, air and space crafts, and in mathematical justifications of many ad hoc industrial techniques which lack solid mathematical foundations. Datta has authored more than 115 research papers, two books, Numerical Methods for Linear Control Systems-Design and Analysis, and Numerical Linear Algebra and Applications, and several software packages, including MATLAB-based toolboxes, MATCONTROL, MATCOM, and MATHEMATICA-based Advanced Numerical Methods. These packages are routinely used for classroom instructions, and academic and industrial research and developments.
His research has been supported by NSF, the Airforce Office of Scientific Research, US Department of Education, the Office of Naval Research (Japan), the Boeing Company, Wolfram Research Incorporation and numerous overseas funding agencies. In recognition of his research contributions, he has received several prestigious honors and awards. These included, election to IEEE Fellow in 2000, induction as an Academician of the Academy of Nonlinear Sciences (Russia) in 2002, Senior Fulbright Specialist Award in 2006 and 2009, and several IEEE Plaques and Medals of Honor. He is an IEEE Distinguished Lecturer and also has been honored by several IEEE sponsored conferences. Datta has served on the editorial board of premier mathematics journals in his areas of expertise, such as SIAM J. Matrix Analysis and Applications and Linear Algebra and its Applications (Special Editor) and is currently serving on the editorial board of about a dozen mathematics and engineering journals, including Numerical Linear Algebra with Applications, Mechanical Systems and Signal Processing, and Dynamical Systems. Datta has served as the vice-chair of the SIAM Linear Algebra Activity Group and has organized several successful interdisciplinary conferences sponsored by the American Mathematical Society and SIAM, and MTNS (Mathematical Theory of Networks and Systems), IEEE. He also co-edited several Proceedings books of these conferences.
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