Solving Multilinear Systems via Tensor Inversion
Solving Multilinear Systems via Tensor Inversion
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DOI：
10.1137/100804577
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发表时间：
2013-05
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通讯作者：
Michael J. Brazell;Na Li;C. Navasca;C. Tamon
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文献类型：
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作者：
Michael J. Brazell;Na Li;C. Navasca;C. Tamon
高阶张量反演对于偶数阶是可能的。这是由于这样一个事实，即一个张量群赋予的收缩产品是同构的一般线性群的程度$n$。有了这些同构群结构，我们推导出一个张量SVD，我们已经证明是等价于著名的规范多元分解和多线性SVD提供了一些约束条件得到满足。此外，在这个组结构的框架内，多线性系统推导和解决高维偏微分方程和大型离散量子模型的问题。我们还解决多线性系统，不适合在最小二乘意义上的框架。这些是当存在奇数个模式或当每个模式具有不同维度时的情况。在数值上，我们解决多线性系统使用迭代技术，即双共轭梯度和雅可比方法。
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.