Page 442 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 442
MATRIX COMPUTATIONS AND NUMERICAL (MS-47)
On the convergence of the Jacobi-type method for computing orthogonal
tensor decomposition
Erna Begovic´, ebegovic@fkit.hr
University of Zagreb, Croatia
Tensor decompositions are a central problem of numerical multilinear algebra. For a general
third-order tensors A ∈ Rn×n×n we are looking for its SVD-like decomposition
A = S ×1 U ×2 V ×3 W,
where U, V, W are orthogonal n × n matrices and S is an n × n × n tensor such that
n
diag(S) 2 = Si2ii → max .
i=1
To obtain this decomposition we are using the alternating least squares approach and a Jacobi-
type method. The algorithm works on 2 × 2 × 2 subtensors. In each iteration the sum of
the squares of two diagonal entries is maximized using Jacobi rotations. We show how the
rotation angles are calculated and prove the convergence of the algorithm. Moreover, we discuss
different initializations of the algorithm.
Inverses of k-Toeplitz matrices for resonator arrays with multiple
receivers
Jose Brox, josebrox@mat.uc.pt
Centre for Mathematics of the University of Coimbra, Portugal
Coauthor: José Alberto
Wireless power transfer systems allow to avoid electrical contact and transfer power in rough
environments with water, dust or dirt. They are used in electrical vehicle and mobile devices
charging, biomedical devices powering, etc. But they have a drawback: in case of misalignment
or distance from the transmitter to the receiver, the efficiency and power transmitted can drop
abruptly. To overcome this inconvenience, arrays of resonators arranged in a plane are used to
transfer power over longer distances through magnetic coupling, with receivers placed over the
array to absorb the power transmitted. In the literature, these arrays have been examined using
magnetoinductive wave theory or through the circuit analysis of the array, however considering
only arrays with one receiver placed over them. Here we present the study of arrays with
multiple receivers for which an arbitrary pattern of receivers is repeated over every k resonators.
In this case, the impedance matrix representing the circuit is tridiagonal with equal upper and
lower diagonals and periodic main diagonal of period k. We show how to invert those matrices,
by computing their determinants through linear recurrence relations and then using them to
compute the minors appearing in the cofactor matrix. In this way we are able to provide rational
formulas for the currents, power transmission and efficiency of the system.
Work published in Applied Mathematics and Computation 377 (2020).
440
On the convergence of the Jacobi-type method for computing orthogonal
tensor decomposition
Erna Begovic´, ebegovic@fkit.hr
University of Zagreb, Croatia
Tensor decompositions are a central problem of numerical multilinear algebra. For a general
third-order tensors A ∈ Rn×n×n we are looking for its SVD-like decomposition
A = S ×1 U ×2 V ×3 W,
where U, V, W are orthogonal n × n matrices and S is an n × n × n tensor such that
n
diag(S) 2 = Si2ii → max .
i=1
To obtain this decomposition we are using the alternating least squares approach and a Jacobi-
type method. The algorithm works on 2 × 2 × 2 subtensors. In each iteration the sum of
the squares of two diagonal entries is maximized using Jacobi rotations. We show how the
rotation angles are calculated and prove the convergence of the algorithm. Moreover, we discuss
different initializations of the algorithm.
Inverses of k-Toeplitz matrices for resonator arrays with multiple
receivers
Jose Brox, josebrox@mat.uc.pt
Centre for Mathematics of the University of Coimbra, Portugal
Coauthor: José Alberto
Wireless power transfer systems allow to avoid electrical contact and transfer power in rough
environments with water, dust or dirt. They are used in electrical vehicle and mobile devices
charging, biomedical devices powering, etc. But they have a drawback: in case of misalignment
or distance from the transmitter to the receiver, the efficiency and power transmitted can drop
abruptly. To overcome this inconvenience, arrays of resonators arranged in a plane are used to
transfer power over longer distances through magnetic coupling, with receivers placed over the
array to absorb the power transmitted. In the literature, these arrays have been examined using
magnetoinductive wave theory or through the circuit analysis of the array, however considering
only arrays with one receiver placed over them. Here we present the study of arrays with
multiple receivers for which an arbitrary pattern of receivers is repeated over every k resonators.
In this case, the impedance matrix representing the circuit is tridiagonal with equal upper and
lower diagonals and periodic main diagonal of period k. We show how to invert those matrices,
by computing their determinants through linear recurrence relations and then using them to
compute the minors appearing in the cofactor matrix. In this way we are able to provide rational
formulas for the currents, power transmission and efficiency of the system.
Work published in Applied Mathematics and Computation 377 (2020).
440
