671 research outputs found
Shifted Power Method for Computing Tensor Eigenpairs
Recent work on eigenvalues and eigenvectors for tensors of order m >= 3 has
been motivated by applications in blind source separation, magnetic resonance
imaging, molecular conformation, and more. In this paper, we consider methods
for computing real symmetric-tensor eigenpairs of the form Ax^{m-1} = \lambda x
subject to ||x||=1, which is closely related to optimal rank-1 approximation of
a symmetric tensor. Our contribution is a shifted symmetric higher-order power
method (SS-HOPM), which we show is guaranteed to converge to a tensor
eigenpair. SS-HOPM can be viewed as a generalization of the power iteration
method for matrices or of the symmetric higher-order power method.
Additionally, using fixed point analysis, we can characterize exactly which
eigenpairs can and cannot be found by the method. Numerical examples are
presented, including examples from an extension of the method to finding
complex eigenpairs
Perturbation bounds for the largest C-eigenvalue of piezoelectric-type tensors
In this paper, we focus on the perturbation analysis of the largest
C-eigenvalue of the piezoelectric-type tensor which has concrete physical
meaning which determines the highest piezoelectric coupling constant. Three
perturbation bounds are presented, theoretical analysis and numerical examples
show that the third perturbation bound has high accuracy when the norm of the
perturbation tensor is small
Reduced basis isogeometric mortar approximations for eigenvalue problems in vibroacoustics
We simulate the vibration of a violin bridge in a multi-query context using
reduced basis techniques. The mathematical model is based on an eigenvalue
problem for the orthotropic linear elasticity equation. In addition to the nine
material parameters, a geometrical thickness parameter is considered. This
parameter enters as a 10th material parameter into the system by a mapping onto
a parameter independent reference domain. The detailed simulation is carried
out by isogeometric mortar methods. Weakly coupled patch-wise tensorial
structured isogeometric elements are of special interest for complex geometries
with piecewise smooth but curvilinear boundaries. To obtain locality in the
detailed system, we use the saddle point approach and do not apply static
condensation techniques. However within the reduced basis context, it is
natural to eliminate the Lagrange multiplier and formulate a reduced eigenvalue
problem for a symmetric positive definite matrix. The selection of the
snapshots is controlled by a multi-query greedy strategy taking into account an
error indicator allowing for multiple eigenvalues
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