250041 VO Low-rank tensors and the data-driven solution of PDEs (2021S)

This course focuses on the MPS/TT low-rank decomposition of multidimensional arrays, which appears naturally as a representation of functions from low-rank refinement in the construction of finite-element approximations. The course opens with an introduction to low-rank tensor decompositions from a linear-algebraic perspective and then proceeds to low-rank tensor methods for second-order linear elliptic problems. The low-rank approximation of functions, depending on their regularity, is rigorously analyzed and then placed in the context of PDE analysis. The course also treats state-of-the-art methods for solving optimality equations (ALS, AME): the construction, implementation and numerical analysis of such methods is presented.

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This course spotlights the intersection of two areas of modern applied mathematics:
* low-rank approximation and analysis of abstract data represented by multi-dimensional arrays
and
* adaptive numerical methods for solving PDE problems.
For the mentioned two areas, however seemingly disjoint, the idea of exactly representing or approximating «data» in a suitable low-dimensional subspace of a large (possibly infinite-dimensional) space is equally natural. The notions of matrix rank and of low-rank matrix approximation, presented in basic courses of linear algebra, are central to one of many possible expressions of this idea.

In psychometrics, signal processing, image processing and (vaguely defined) data mining, low-rank tensor decompositions have been studied as a way of formally generalizing the notion of rank from matrices to higher-dimensional arrays (tensors). Several such generalizations have been proposed, including the canonical polyadic (CP) and Tucker decompositions and the tensor-SVD, with the primary motivation of analyzing, interpreting and compressing datasets. In this context, data are often thought of as parametrizations of images, video, social networks or collections of texts, whereas data representing functions are mostly considered as convenient test examples.

On the other hand, the «tensor-train» (TT) and the more general «hierarchical Tucker» decompositions were developed in the community of numerical mathematics, more recently and with particular attention to PDE problems. In fact, exactly the same and very similar representations had long been used for the numerical simulation of many-body quantum systems by computational chemists and physicists under the names of «matrix-product states» (MPS) and «multilayer multi-configuration time-dependent Hartree». These low-rank tensor decompositions are based on subspace approximation, which can be performed adaptively and iteratively, in a multilevel fashion. In a broader context of PDE problems, this leads to numerical methods that are formally based on generic discretizations but effectively operate on adaptive, data-driven discretizations constructed «online», in the course of computation. In several settings of practical importance, such methods achieve the accuracy of problem-specific, sophisticated methods.

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The goal of the course is to introduce students to the foundations of modern low-rank methods for the numerical solution of PDE and to the state-of-the-art research in this area.
In particular, the course is to provide students with ample opportunity for starting own research.
In addition, through numerous connections with basic courses (linear algebra, numerical mathematics and numerical analysis, real and complex analysis, analysis of PDE), this course serves to reinforce a broader perspective of mathematics as an integrated field with its distinctive methods of inquiry that span across its pure and applied branches.