Solving Multilinear Systems via Tensor Inversion

Higher order tensor inversion is possible for even order. This is due to the fact that a tensor group endowed with the contracted product is isomorphic to the general linear group of degree $n$. With these isomorphic group structures, we derive a tensor SVD which we have shown to be equivalent to we...

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Vydané v:SIAM journal on matrix analysis and applications Ročník 34; číslo 2; s. 542 - 570
Hlavní autori: Brazell, M., Li, N., Navasca, C., Tamon, C.
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: Philadelphia Society for Industrial and Applied Mathematics 01.01.2013
Predmet:
Applied mathematics
Decomposition
Eigenvalues
Engineering
Equivalence
Inversions
Iterative methods
Least squares method
Linear algebra
Mathematical analysis
Mathematical models
Numerical analysis
Partial differential equations
Signal processing
Tensors
ISSN:0895-4798, 1095-7162
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Shrnutí:Higher order tensor inversion is possible for even order. This is due to the fact that a tensor group endowed with the contracted product is isomorphic to the general linear group of degree $n$. With these isomorphic group structures, we derive a tensor SVD which we have shown to be equivalent to well-known canonical polyadic decomposition and multilinear SVD provided that some constraints are satisfied. Moreover, within this group structure framework, multilinear systems are derived and solved for problems of high-dimensional PDEs and large discrete quantum models. We also address multilinear systems which do not fit the framework in the least-squares sense. These are cases when there is an odd number of modes or when each mode has distinct dimension. Numerically we solve multilinear systems using iterative techniques, namely, biconjugate gradient and Jacobi methods.
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ISSN:0895-4798
1095-7162
DOI:10.1137/100804577