In this paper we study the convergence of the well-known Greedy Rank-One Update Algorithm. It is used to construct the rank-one series solution for full-rank linear systems. The existence of the rank one approximations is also not new, but surprisingly the focus there has been more on the applications side more that in the convergence analysis. Our main contribution is to prove the convergence of the algorithm and also we study the required rank one approximation in each step. We also give some numerical examples and describe its relationship with the Finite Element Method for High-Dimensional Partial Differential Equations based on the tensorial product of one-dimensional bases. We illustrate this situation taking as a model problem the multidimensional Poisson equation with homogeneous Dirichlet boundary condition.

Tensor-based methods are receiving a growing interest in scientific computing for the numerical solution of problems defined in high dimensional tensor product spaces. A family of methods called proper generalized decompositions (PGD) methods have been recently introduced for the a priori construction of tensor approximations of the solution of such problems. In this paper, we give a mathematical analysis of a family of progressive and updated PGDs for a particular class of problems associated with the minimization of a convex functional over a reflexive tensor Banach space.

In this paper we introduce and develop the notion of minimal subspaces in the framework of algebraic and topological tensor product spaces. This mathematical structure arises in a natural way in the study of tensor representations. We use minimal subspaces to prove the existence of a best approximation, for any element in a Banach tensor space, by means of a tensor given in a typical representation format (Tucker, hierarchical, or tensor train). We show that this result holds in a tensor Banach space with a norm stronger than the injective norm and in an intersection of finitely many Banach tensor spaces satisfying some additional conditions. Examples using topological tensor products of standard Sobolev spaces are given.

Traditionally, drug development involved the individual synthesis and biological evaluation of hundreds to thousands of compounds with the intention of highlighting their biological activity, selectivity, and bioavailability, as well as their low toxicity. On average, this process of new drug development involved, in addition to high economic costs, a period of several years before hopefully finding a drug with suitable characteristics to drive its commercialization. Therefore, the chemical synthesis of new compounds became the limiting step in the process of searching for or optimizing leads for new drug development. This need for large chemical libraries led to the birth of high-throughput synthesis methods and combinatorial chemistry. Virtual combinatorial chemistry is based on the same principle as real chemistry—many different compounds can be generated from a few building blocks at once. The difference lies in its speed, as millions of compounds can be produced in a few seconds. On the other hand, many virtual screening methods, such as QSAR (Quantitative Sturcture-Activity Relationship), pharmacophore models, and molecular docking, have been developed to study these libraries. These models allow for the selection of molecules to be synthesized and tested with a high probability of success. The virtual combinatorial chemistry–virtual screening tandem has become a fundamental tool in the process of searching for and developing a drug, as it allows the process to be accelerated with extraordinary economic savings.