The aim of this set of lectures is the presentation of some state of the art techniques for the computation of incomplete factorizations of given large and sparse matrices. We will recall the general setting in which this kind of strategy have been devised and some applications. We will focus our attention mainly on two particular class of algorithms, the one that descends from the modification of the classical LU factorization, namely the Incomplete LU factorization with its variants, and the use of incomplete biconjugation technique for the direct computation of an approximate inverse. Both algorithmical and mathematical issues will be addressed. At the end of this overview we will focus on some techniques for computing a cost-effective update of the incomplete factorization.
Lecture Notes P1
Lecture Notes P2
Lecture Notes P3
Exercises

2 lectures (Moscow) - 3 lectures (Rome)

References:
[1] Bertaccini, Daniele, and Filippone Salvatore. "Sparse approximate inverse preconditioners on high performance GPU platforms." Computers & Mathematics with Applications 71.3 (2016): 693-711.
[2] Van Duin, Arno CN. "Scalable parallel preconditioning with the sparse approximate inverse of triangular matrices." SIAM journal on matrix analysis and applications 20.4 (1999): 987-1006.
[3] Benzi, Michele. "Preconditioning techniques for large linear systems: a survey."Journal of computational Physics 182.2 (2002): 418-477.

3-4 lectures (Moscow)
3-4 lectures (Rome)

Graphs are ubiquitous in the real world; social networks, transport networks, computer networks are objects that can be satisfactorily modeled as graphs: the so-called complex networks. In various contexts, matrix theory and techniques borrowed from numerical linear algebra play a major role in their analysis and treatment. For example, matrix functions and eigenvector computations are required by certain centrality indices to understand roles or find out rankings of nodes and edges; and spectral techniques are efficient, widespread tools to solve community detection and network partitioning problems. The aim of this mini-course is to introduce students to the basic tools in linear algebra and matrix theory that underpin the modern complex network science.
Short outline:
Perron-Frobenius theory, nodal domains, spectral properties of graph matrices (adjacency, laplacian, modularity matrices) and their applications in complex network analysis: eigenvector-based centrality indices, graph partitioning and community detection problems.
10 lectures

This course will not be an advanced study of convex analysis, nor a survey of all possible algorithms for convex optimization. Instead, we will concentrate only on few good algorithms (including barrier and interior-point methods) and few interesting applications (including pass-band filter design and wireless channel estimation). Our main goal will be to develop a working and concrete knowledge of convex optimization: an ability to recognize and solve convex optimization problems.
Short outline:
Convex sets. Separating hyperplanes. Convex functions. Conjugate function. Duality. Optimality conditions. Theorems on alternatives. Best approximation in C-norm. Optimal detector design and hypothesis testing. Distance between sets. Extremal volume ellipsoids. Primal-dual interior-point methods.
8 lectures

Brief introduction in the state-of-the-art problems of matrix calculations in wireless communication should allow participants to understand importance of matrix methods for systems, which become part of our life.
The course is presented by 3 parts:

Part 1. September 9. Introduction in wireless communication and its relation to linear algebra problems (90 minutes)

• System model of wireless communication
• Capacity of the system and linear space extension
• Definition of system matrix and its properties discussion
• Outlook of matrix methods that are playing important role in telecom.
• Some tasks for the weekend.

Part 2. September 12. State-of-the-Art solutions and possible optimization. Matrix methods and requirements to them from the industry (90 minutes)

• Not well-posed problems in the sense of Hadamard (ill-posed problems)
• \(QR\), \(LU\), iterative methods and requirements to them from wireless systems
• Important role of \(SVD\) in ultra-high rate communications
• Belief propagation approach in detection problems
• Topic for next time discussion (task for home preparation)

Part 3. September 13. Implementation issues and future Outlook (90 minutes)

• The simplest one doesn't mean the fastest one
• Robustness & accuracy
• Task analysis and discussion

3 lectures
Summary
Lecture 1 Lecture 2 Lecture 3

Isogeometric Analysis (IgA) is a paradigm for the analysis of problems governed by partial differential equations. Its goal is to improve the connection between numerical simulation and Computer Aided Design (CAD) systems. In its original formulation, the main idea in IgA is to use directly the geometry provided by CAD systems and to approximate the unknown solutions of differential equations by the same type of functions. Tensor-product B-splines and their rational extension, the so-called NURBS, are the dominant technology in CAD systems used in engineering and thus also in IgA. Galerkin discretizations have been intensively employed in the IgA context. This talk is devoted to the study of the spectrum of the resulting stiffness matrices, when the fineness parameters tend to zero so that the related matrix-size tends to infinity. We consider elliptic problems with constant coefficients.
1 seminar
Reference
Excercises

3-4 lectures (Moscow)

3-4 lectures (Rome)

Definition and construction of self-similar fractals. Self-similar energies on finitely ramified self-similar fractals. Eigenforms. Existence and specially uniqueness of eigenforms. The uniqueness problem is treated by means of the Perron-Frobenius theory.

Recently, the class of Generalized Locally Toeplitz (GLT) sequences has been introduced as a generalization both of classical Toeplitz sequences and of variable coefficient differential operators and, for every sequence of the class, it has been demonstrated that it is possible to give a rigorous description of the asymptotic spectrum in terms of a function (the symbol) that can be easily identified.
This generalizes the notion of a symbol for differential operators (discrete and continuous) or for Toeplitz sequences for which it is identified through the Fourier coefficients and is related to the classical Fourier Analysis.
The GLT class has nice algebraic properties and indeed it has been proven that it is stable under linear combinations, products, and inversion when the sequence which is inverted shows a sparsely vanishing symbol (sparsely vanishing symbol means a symbol which vanishes at most in a set of zero Lebesgue measure). Furthermore, the GLT class virtually includes any approximation by local methods (Finite Difference, Finite Element, Isogeometric Analysis etc) of partial differential equations (PDEs) and, based on this, we demonstrate that our results on GLT sequences can be used in a PDE setting in various directions:

1. as a generalized Fourier Analysis for the design and for the study of preconditioned iterative and semi-iterative methods, when dealing with variable coefficients, non rectangular domains, non uniform gridding or triangulations,
2. for a multigrid analysis of convergence and for providing spectral information on large preconditioned systems in the variable coefficient case,
3. in order to provide a tool for the stability analysis of PDE numerical schemes (e.g. a necessary von Neumann criterium for variable coefficient systems of PDEs is obtained, uniformly with respect to the boundary conditions), etc.

We will discuss specifically problems (1) and (2) and other possible directions in which the GLT analysis can be conveniently employed.

Keywords: Toeplitz and Generalized Locally Toeplitz matrices, approximations of PDE and related methods, such as Finite Differences, Finite Volumes, Finite Elements. Isogeometric Analysis: collocations, Galerkin and discontinuous Galerkin
,br> Selected Bibliography
2 lectures (Rome)

Norms, convexity, separation. Majorization and doubly stochastic matrices. Interlacing properties for the eigenvalues of Hermitian matrices. Weyl inequalities and Hoffman-Wieland theorem. Lidskii theorem. Unitarily invariant norms. Ky Fan norms. John von Neumann and Mirski theorems.
Excercises

3+3 lectures

The idea of the course is to introduce the theory of semidefinite programming (optimization). SDP is not only a powerful numerical approach but also a very useful analytical tool for finding approximate solutions of several NP hard problems. We will consider some basic facts about semidefinite programs and some nontrivial examples of so called semidefinite relaxations of NP hard problems.
Lecture Notes P1 Exercises

4 lectures + 4 seminars