講師：Biswa Nath Datta (Northern Illinois University)
題目：Computational and optimization methods for quadratic inverse
eigenvalue problems arising in mechanical vibration and structural
dynamics : Linking mathematics to industry
日時：7月10日(火)10:30-12:00
場所：京都大学工学部総合校舎213講義室

概要：
The Quadratic Eigenvalue Problem is to find eigenvalues and
eigenvectors of a quadratic matrix pencil of the form P(\lambda) =
M\lambda^2+C\lambda+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.