5,333 research outputs found
Tensor and Matrix Inversions with Applications
Higher order tensor inversion is possible for even order. We have shown that
a tensor group endowed with the Einstein (contracted) product is isomorphic to
the general linear group of degree . With the isomorphic group structures,
we derived new tensor decompositions which we have shown to be related to the
well-known canonical polyadic decomposition and multilinear SVD. Moreover,
within this group structure framework, multilinear systems are derived,
specifically, for solving high dimensional PDEs and large discrete quantum
models. We also address multilinear systems which do not fit the framework in
the least-squares sense, that is, when the tensor has an odd number of modes or
when the tensor has distinct dimensions in each modes. With the notion of
tensor inversion, multilinear systems are solvable. Numerically we solve
multilinear systems using iterative techniques, namely biconjugate gradient and
Jacobi methods in tensor format
Lyapunov Exponents without Rescaling and Reorthogonalization
We present a new method for the computation of Lyapunov exponents utilizing
representations of orthogonal matrices applied to decompositions of M or
MM_trans where M is the tangent map. This method uses a minimal set of
variables, does not require renormalization or reorthogonalization, can be used
to efficiently compute partial Lyapunov spectra, and does not break down when
the Lyapunov spectrum is degenerate.Comment: 4 pages, no figures, uses RevTeX plus macro (included). Phys. Rev.
Lett. (in press
Tensor Network alternating linear scheme for MIMO Volterra system identification
This article introduces two Tensor Network-based iterative algorithms for the
identification of high-order discrete-time nonlinear multiple-input
multiple-output (MIMO) Volterra systems. The system identification problem is
rewritten in terms of a Volterra tensor, which is never explicitly constructed,
thus avoiding the curse of dimensionality. It is shown how each iteration of
the two identification algorithms involves solving a linear system of low
computational complexity. The proposed algorithms are guaranteed to
monotonically converge and numerical stability is ensured through the use of
orthogonal matrix factorizations. The performance and accuracy of the two
identification algorithms are illustrated by numerical experiments, where
accurate degree-10 MIMO Volterra models are identified in about 1 second in
Matlab on a standard desktop pc
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