LINEAR ALGEBRA AND ITS APPLICATIONS

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Special Issue on Tensors and Multilinear Algebra

Tensors are increasingly ubiquitous in various areas of applied, computational, and industrial mathematics because multilinear algebra arises naturally in many contexts. For example, graphical models often involve nonnegative tensors in the form of multiway probability tables; symmetric tensors appear in the guise of cumulants in data analysis while more general tensors appear as matchgates in holographic algorithms and as computational networks in quantum chemistry; the notion of tensor rank is a natural measure for the degree of entanglement of quantum states. Numerous others applications exist in the fields of algebraic statistics, approximation theory, bioinformatics, chemometrics, computational complexity, machine learning, neuroscience, pattern recognition, quantum chemistry, and quantum computing. Consequently, there is a huge demand for analysis and computational methods for multilinear objects.

In recent years there have been substantial efforts to further develop the mathematical and computational aspects of tensor methods, with a goal of making them as easy to use as their matrix counterparts. Although it may seem that reformulating a tensor problem in terms of matrices would be beneficial, ignoring the multilinear structure of an object typically leads to inefficient algorithms and fails to adequately exploit the underlying structure. Furthermore, there are some applications that are only only tractable when we employ data approximations based on structured tensors. This pertains to many numerical problems for partial differential equations and integro-differential equations, in particular to quantum chemistry computations. On the other hand, there are some situations where linear algebraic techniques are key to advancing multilinear algebra. Conversely, certain linear algebra problems, e.g., with Kronecker structure, can benefit from multilinear algebra techniques. Thus, the purpose of this special issue is to bring together the aforementioned three lines of research — multilinear algebraic techniques in multilinear algebra, linear algebraic techniques in multilinear algebra, and multilinear algebraic techniques in linear algebra. The issue will attract the attention of mathematicians to challenges and recent findings in the rapidly developing field of applied and numerical multilinear algebra.

This special issue will be open to all papers with significant new results where either multilinear algebraic methods play an important role or new tools and problems of multilinear algebraic nature are presented. Survey papers that illustrate common themes across disciplines and application areas, and especially where multilinear algebraic or tensor techniques play a central role are also welcomed. Papers must meet the publication standards of Linear Algebra and Its Applications and will be refereed in the usual way.

Areas and topics of interest for this special issue include, but are not limited to: