The first aim of this proposal is to develop theoretical and computational methods for common problems in tensors: small rank approximations, best rank approximations, uniqueness of best rank approximations, approximations of tensors preserving symmetries, rank of tensors, norm computations, finding the number of singular values tuples, computing singular value tuples, computing nonlinear eigenvalues. The second aim of this proposal is the use of theoretical and computational methods in tensors for central problems in Quantum Information Sciences: separability, the capacity of quantum channels and the computational complexity of these quantities.
High dimensional problems arise in many areas of applications such as mathematical biology, chemistry, physics, image processing, finance or engineering. The typical mathematical model uses tensors, i.e., multidimensional arrays, to represent the physical model and the computational methods for simulation and optimization can usually be based on operations with these tensors. For higher order tensors the theory and the development of numerical methods is much more complicated than for matrices, i.e. two dimensional arrays. The aim of this proposal to study, find and apply new techniques in tensors to advance the current machinery for applications in biology, engineering and science.
