We present a novel formulation, based on the latest advancement in polynomial system solving via linear algebra, for identifying limit cycles in general n-dimensional autonomous nonlinear polynomial systems. The condi...We present a novel formulation, based on the latest advancement in polynomial system solving via linear algebra, for identifying limit cycles in general n-dimensional autonomous nonlinear polynomial systems. The condition for the existence of an algebraic limit cycle is first set up and cast into a Macaulay matrix format whereby polynomials are regarded as coefficient vectors of monomials. This results in a system of polynomial equations whose roots are solved through the null space of another Macaulay matrix. This two-level Macaulay matrix approach relies solely on linear algebra and eigenvalue computation with robust numerical implementation. Furthermore, a state immersion technique further enlarges the scope to cover also non-polynomial (including exponential and logarithmic) limit cycles. Application examples are given to demonstrate the efficacy of the proposed framework.展开更多
In this article,two new algorithms are presented that convert a given data tensor train into either a Tucker decomposition with orthogonal matrix factors or a multi-scale entanglement renormalization ansatz(MERA).The ...In this article,two new algorithms are presented that convert a given data tensor train into either a Tucker decomposition with orthogonal matrix factors or a multi-scale entanglement renormalization ansatz(MERA).The Tucker core tensor is never explicitly computed but stored as a tensor train instead,resulting in both computationally and storage efficient algorithms.Both the multilinear Tucker-ranks as well as the MERA-ranks are automatically determined by the algorithm for a given upper bound on the relative approximation error.In addition,an iterative algorithm with low computational complexity based on solving an orthogonal Procrustes problem is proposed for the first time to retrieve optimal rank-lowering disentangler tensors,which are a crucial component in the construction of a low-rank MERA.Numerical experiments demonstrate the effectiveness of the proposed algorithms together with the potential storage benefit of a low-rank MERA over a tensor train.展开更多
This paper considers the optimal model reduction problem of matrix second-order linear systems in the sense of Hilbert-Schmidt-Hankel norm, with the reduced order systems preserving the structure of the original syste...This paper considers the optimal model reduction problem of matrix second-order linear systems in the sense of Hilbert-Schmidt-Hankel norm, with the reduced order systems preserving the structure of the original systems. The expressions of the error function and its gradient are derived. Two numerical examples are given to illustrate the presented model reduction technique.展开更多
文摘We present a novel formulation, based on the latest advancement in polynomial system solving via linear algebra, for identifying limit cycles in general n-dimensional autonomous nonlinear polynomial systems. The condition for the existence of an algebraic limit cycle is first set up and cast into a Macaulay matrix format whereby polynomials are regarded as coefficient vectors of monomials. This results in a system of polynomial equations whose roots are solved through the null space of another Macaulay matrix. This two-level Macaulay matrix approach relies solely on linear algebra and eigenvalue computation with robust numerical implementation. Furthermore, a state immersion technique further enlarges the scope to cover also non-polynomial (including exponential and logarithmic) limit cycles. Application examples are given to demonstrate the efficacy of the proposed framework.
基金the Ministry of Education and Science of the Russian Federation(grant 14.756.31.0001).
文摘In this article,two new algorithms are presented that convert a given data tensor train into either a Tucker decomposition with orthogonal matrix factors or a multi-scale entanglement renormalization ansatz(MERA).The Tucker core tensor is never explicitly computed but stored as a tensor train instead,resulting in both computationally and storage efficient algorithms.Both the multilinear Tucker-ranks as well as the MERA-ranks are automatically determined by the algorithm for a given upper bound on the relative approximation error.In addition,an iterative algorithm with low computational complexity based on solving an orthogonal Procrustes problem is proposed for the first time to retrieve optimal rank-lowering disentangler tensors,which are a crucial component in the construction of a low-rank MERA.Numerical experiments demonstrate the effectiveness of the proposed algorithms together with the potential storage benefit of a low-rank MERA over a tensor train.
基金supported by the National Nature Science Foundation of China (No. 60804032)the Central University Basic Research Foundation of South China University of Technology (No. 2009zm0178)the Small Project Funding of HKU from HKU SPACE Research Fund (No.201007176165)
文摘This paper considers the optimal model reduction problem of matrix second-order linear systems in the sense of Hilbert-Schmidt-Hankel norm, with the reduced order systems preserving the structure of the original systems. The expressions of the error function and its gradient are derived. Two numerical examples are given to illustrate the presented model reduction technique.
