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Spring Semester Seminars in Numerical Linear Algebra and beyond Department of Mathematics, University of Bologna

This webpage lists some of the seminars organized at the Department of Mathematics, University of Bologna. The presentations mainly cover Numerical Linear Algebra problems and related topics but broader contributions are welcomed as well.

Forthcoming events:


Time&Location: 15:45 - Aula B. Levi, Dipartimento di Matematica, Università di Bologna

Title: Image segmentation via an adaptive “filtered” scheme based on a modified level-set method 

Abstract: This seminar addresses the problem of image segmentation through an accurate high-order scheme based on the level-set method. In this approach, the curve evolution is described as the 0-level set of a representation function, but the velocity that drives the curve to the boundary of the object has been modified in order to obtain a new velocity with additional properties that are extremely useful to develop a more stable high-order approximation with a small additional cost. The approximation scheme proposed here is the 2D version of an adaptive “filtered” scheme, which combines two building blocks (a monotone scheme and a high-order scheme) via a filter function and smoothness indicators that allow one to detect the regularity of the approximate solution adapting the scheme in an automatic way. Some numerical tests on synthetic and real images confirm the accuracy of the proposed method and the advantages given by the new velocity. 


Organizers: Davide Palitta, Valeria Simoncini


Past events:

Time&Location: 14:30 - Aula Levi (Laboratorio Primo Piano), Dipartimento di Matematica, Università di Bologna

Title: Some recent developments of deterministic CholeskyQR

Abstract: CholeskyQR is a popular algorithm for QR factorization in both academia and industry. In order to have good orthogonality, CholeskyQR2 is developed by repeating CholeskyQR twice. Shifted CholeskyQR3 introduces a shifted item in order to deal with ill-conditioned matrices with good orthogonality. This talk primarily focuses on deterministric methods. We define a new matrix norm and make improvements to the shifted item and error estimations in CholeskyQR algorithms. We use such a technique and provide an analysis to some sparse matrices in the industry for CholeskyQR. Moreover, we combine CholeskyQR and our new matrix norm with randomized models for probabilistic error analysis and make amelioration to CholeskyQR. A new 3-step algorithm without CholeskyQR2 is also developed with good orthogonality.


Time&Location: 14:30 - Seminario I, Dipartimento di Matematica, Università di Bologna

Title: Spatiotemporal variability of solar radiations within an urban context: a characterisation by means of Principal Component Analysis

Abtract: Sunlight constitutes an abundant and endless natural fuel, available worldwide. In a society where a substantial part of the global energy yield is being directly expended at the city scale, urban areas appear as serious candidates for the production of solar energy. Their intrinsic complexity yet makes it challenging. The morphological heterogeneity between urban geometries and intricacy of their materials optical properties especially contribute together to causing important spatiotemporal variations in the distribution of incident solar radiations. The field of irradiance received by a specific urban region (e.g. façade, building, district) may thus rapidely become the result of complex miscellaneous interactions between many degrees of freedom. Besides, Principal Component Analysis (PCA) has been widely validated as an efficient algorithm to identify the principal behavioural features, or modes of variability, of a high-dimensional phenomenon. An approach is proposed here for analysing the variations in space and time of the solar resource within an urban context by means of PCA. A parametric investigation is conducted on a set of theoretical 100×100 m² urban districts, defined as arrangements of cuboid-like buildings, with various typological indicators (Total Site Coverage, Average Building Height) and surface materials (Lambertian, highly-specular) at three different latitudes. For each configuration, the distribution of irradiance incident on the facets of the central building is modelled via backwards Monte-Carlo ray tracing over a full year and under clear sky conditions, with a 15 min timestep and 1 m spatial resolution. PCA is subsequently applied to the simulated radiative fields to extract dominant modes of variation. First results validate energy-based orthogonal decompositions like PCA as efficient tools for characterising the variability distribution of multivariate phenomena in this context, allowing for the identification of district areas subjected to important spatial and temporal variations of the solar resource. Characteristic time scales are clearly represented across successive orders of decomposition. Information about the district morphology is also obtained, with the contribution of surrounding geometries being portrayed by specific spatial modes. Similar prevalent variables are further repetitively encountered across multiple evaluated surfaces, but at different modal ranks.


Time&Location: 14:30 - Seminario II, Dipartimento di Matematica, Università di Bologna

Title: Potential and applications of tensor-based algorithms

Abstract: Tensors have become widely used in various domains due to their practicality. Tensor factorization techniques are used to solve computationally demanding problems, analyze large datasets, and refine descriptions of complex phenomena. This presentation outlines the development of my research on tensors, including an overview of commonly used tensor methods and their applications in various fields such as remote sensing, multilinear algebra, numerical simulation, and signal processing. Criteria for selecting the most appropriate tensor technique depending on the problem under consideration will be emphasized. The presentation aims to outline the advantages and limitations inherent in these techniques. It explores the challenges and offers insights into current research directions driven by real-world, computational, and applied problems.


Time&Location: 14:30 - Aula Arzelà, Dipartimento di Matematica, Università di Bologna

Title: Truncated LSQR for Matrix Least Squares Problems and Application to Dictionary Learning

Abstract: We are interested in the numerical solution of the matrix least squares problem min_X ∥AXB + CXD-F ∥_F , with A and C full column rank, B, D full row rank, F an n×n matrix of low rank, and ∥•∥_F the Frobenius norm. We derive a matrix-oriented implementation of LSQR, and devise an implementation of the truncation step that exploits the properties of the method. Experimental comparisons with the Conjugate Gradient method applied to the normal matrix equation and with a (new) sketched implementation of matrix LSQR illustrate the competitiveness  of the proposed algorithm. We also explore the applicability of our method in the context of Kronecker-based Dictionary Learning, and devise a representation of the data that seems to be promising for classification purposes.


Time&Location: 16:00 - Seminario II, Dipartimento di Matematica, Università di Bologna

Title: Learned regularization for linear inverse problems

Abstract: An inverse problem is the task of retrieving an unknown quantity from indirect observations. When the model describing the measurement acquisition is linear, this results in the inversion of a linear operator (a matrix, in a discrete formulation) which is usually ill-posed or ill-conditioned. A common strategy to tackle ill-posedness in inverse problems is to use regularizers, which are (families of) operators providing a stable approximation of the inverse map. Model-based regularization techniques often leverage prior knowledge of the exact solution, such as smoothness or sparsity with respect to a suitable representation; on the other side, in recent years many data-driven methods have been developed in the context of machine learning. Those techniques tackle the approximation of the inverse operator in suitable spaces of parametric functions (i.e., neural networks) and rely on large datasets of paired measurements and ground-truth objects. In this talk, I will focus on hybrid strategies, which aim at blending model-based and data-driven approaches, providing both satisfying numerical results and sound theoretical guarantees. I will describe a general framework to comprise many existing techniques in the theory of statistical learning, also reporting some recent theoretical advances (in the direction of generalization guarantees). I will help the discussion by presenting some relevant examples in the context of medical imaging and, specifically, in computed tomography.


Time&Location: 14:30 - Seminario I, Dipartimento di Matematica, Università di Bologna

Title: Row-aware Randomized SVD with applications

Abstract: We introduce a novel procedure for computing an SVD-type approximation of a tall matrix A. Specifically, we propose a randomization-based algorithm that improves the standard Randomized Singular Value Decomposition (RSVD). Most significantly, our approach, the Row-aware RSVD (R-RSVD), explicitly constructs information from the row space of A. This leads to better approximations to Range(A) while maintaining the same computational cost. The efficacy of the R-RSVD is supported by both robust theoretical results and extensive numerical experiments. Furthermore, we present an alternative algorithm inspired by the R-RSVD, capable of achieving comparable accuracy despite utilizing only a subsample of the rows of A, resulting in a significantly reduced computational cost. This method, that we name the Subsample Row-aware RSVD (Rsub-RSVD), is supported by a weaker error bound compared to the ones we derived for the R-RSVD, but still meaningful as it ensures that the error remains under control. Additionally, numerous experiments demonstrate that the Rsub-RSVD trend is akin to the one attained by the R-RSVD when the subsampling parameter is on the order of n, for a m×n A, with m >> n. Finally, we consider the application of our schemes in two very diverse settings which share the need for the computation of singular vectors as an intermediate step: the computation of CUR decompositions by the discrete empirical interpolation method (DEIM) and the construction of reduced-order models in the Loewner framework, a data-driven technique for model reduction of dynamical systems.


Time&Location: 11:00 - Aula B. Levi, Dipartimento di Matematica, Università di Bologna

Title: Computation  of centralities in temporal multilayer networks

Abstract: Multilayer networks are a type of complex network that consist of multiple layers, where each layer represents a different type of connection or interaction between the same set of nodes. These networks are used to model systems where entities are connected in multiple ways simultaneously, capturing the complexity of real-world relationships better than traditional single-layer networks. Through a particular interlayer structure, the dynamical evolution of a complex system over time can be represented. Computing the centrality of a temporal network can improve our understanding of how the most important nodes in a network change over time. Our focus is centered on the computation of the centralities of a multilayer temporal network whose modifications over time consist of low-rank updates of the edge adjacency matrix of a transport network. Using  Krylov subspace methods for matrix function approximations, we will exploit the particular structure of the problem to gain some computational advantages and modeling insights.