Towards a Vector Field Based Approach to the Proper Generalized Decomposition (PGD)

Towards a Vector Field Based Approach to the Proper Generalized Decomposition (PGD)

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A novel algorithm called the Proper Generalized Decomposition (PGD) is widely used by the engineering community to compute the solution of high dimensional problems.However, it is well-known that the bottleneck of its practical implementation oswald the octopus stuffed animal focuses on the computation of the so-called best rank-one approximation.Motivated by this fact, we are going to discuss some of the geometrical aspects of the best rank-one approximation procedure.More precisely, our main result is to construct explicitly a vector field over a low-dimensional vector space and to prove that we can identify its stationary points with the critical points of the best rank-one optimization problem.

To obtain this result, we endow the set of tensors with fixed probiowrap rank-one with an explicit geometric structure.