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- Abella, Jaume, et al.
(författare)
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SAFEXPLAIN : Safe and Explainable Critical Embedded Systems Based on AI
- 2023
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Ingår i: DATE 23: Design, Automation And Test In Europe. - 9783981926378 ; , s. 1-6
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Konferensbidrag (refereegranskat)abstract
- Deep Learning (DL) techniques are at the heart of most future advanced software functions in Critical Autonomous AI-based Systems (CAIS), where they also represent a major competitive factor. Hence, the economic success of CAIS industries (e.g., automotive, space, railway) depends on their ability to design, implement, qualify, and certify DL-based software products under bounded effort/cost. However, there is a fundamental gap between Functional Safety (FUSA) requirements on CAIS and the nature of DL solutions. This gap stems from the development process of DL libraries and affects high-level safety concepts such as (1) explainability and traceability, (2) suitability for varying safety requirements, (3) FUSA-compliant implementations, and (4) real-time constraints. As a matter of fact, the data-dependent and stochastic nature of DL algorithms clashes with current FUSA practice, which instead builds on deterministic, verifiable, and pass/fail test-based software. The SAFEXPLAIN project tackles these challenges and targets by providing a flexible approach to allow the certification - hence adoption - of DL-based solutions in CAIS building on: (1) DL solutions that provide end-to-end traceability, with specific approaches to explain whether predictions can be trusted and strategies to reach (and prove) correct operation, in accordance to certification standards; (2) alternative and increasingly sophisticated design safety patterns for DL with varying criticality and fault tolerance requirements; (3) DL library implementations that adhere to safety requirements; and (4) computing platform configurations, to regain determinism, and probabilistic timing analyses, to handle the remaining non-determinism. © 2023 EDAA.
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| 4. |
- Barbarino, Giovanni, et al.
(författare)
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Block generalized locally Toeplitz sequences : theory and applications in the multidimensional case
- 2020
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Ingår i: Electronic Transactions on Numerical Analysis. - : Osterreichische Akademie der Wissenschaften. - 1068-9613. ; 53, s. 113-216
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Tidskriftsartikel (refereegranskat)abstract
- In computational mathematics, when dealing with a large linear discrete problem (e.g., a linear system) arising from the numerical discretization of a partial differential equation (PDE), knowledge of the spectral distribution of the associated matrix has proved to be useful information for designing/analyzing appropriate solvers-especially, preconditioned Krylov and multigrid solvers-for the considered problem. Actually, this spectral information is of interest also in itself as long as the eigenvalues of the aforementioned matrix represent physical quantities of interest, which is the case for several problems from engineering and applied sciences (e.g., the study of natural vibration frequencies in an elastic material). The theory of multilevel generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of matrices A(n) arising from virtually any kind of numerical discretization of PDEs. Indeed, when the mesh-fineness parameter n tends to infinity, these matrices A(n) give rise to a sequence {A(n)}(n), which often turns out to be a multilevel GLT sequence or one of its "relatives", i.e., a multilevel block GLT sequence or a (multilevel) reduced GLT sequence. In particular, multilevel block GLT sequences are encountered in the discretization of systems of PDEs as well as in the higher-order finite element or discontinuous Galerkin approximation of scalar/vectorial PDEs. In this work, we systematically develop the theory of multilevel block GLT sequences as an extension of the theories of (unilevel) GLT sequences [Garoni and Serra-Capizzano, Generalized Locally Toeplitz Sequences: Theory and Applications. Vol. I., Springer, Cham, 2017], multilevel GLT sequences [Garoni and Serra-Capizzano, Generalized Locally Toeplitz Sequences: Theory and Applications. Vol. II., Springer, Cham, 2018], and block GLT sequences [Barbarino, Garoni, and Serra-Capizzano, Electron. Trans. Numer. A(n)al., 53 (2020), pp. 28-112]. We also present several emblematic applications of this theory in the context of PDE discretizations.
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| 5. |
- Barbarino, Giovanni, et al.
(författare)
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Block generalized locally Toeplitz sequences : theory and applications in the unidimensional case
- 2020
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Ingår i: Electronic Transactions on Numerical Analysis. - : Osterreichische Akademie der Wissenschaften. - 1068-9613. ; 53, s. 28-112
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Tidskriftsartikel (refereegranskat)abstract
- In computational mathematics, when dealing with a large linear discrete problem (e.g., a linear system) arising from the numerical discretization of a differential equation (DE), knowledge of the spectral distribution of the associated matrix has proved to be useful information for designing/analyzing appropriate solvers-especially, preconditioned Krylov and multigrid solvers-for the considered problem. Actually, this spectral information is of interest also in itself as long as the eigenvalues of the aforementioned matrix represent physical quantities of interest, which is the case for several problems from engineering and applied sciences (e.g., the study of natural vibration frequencies in an elastic material). The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of matrices A(n) arising from virtually any kind of numerical discretization of DEs. Indeed, when the mesh-fineness parameter n tends to infinity, these matrices A(n) give rise to a sequence {A(n)}(n), which often turns out to be a GLT sequence or one of its "relatives", i.e., a block GLT sequence or a reduced GLT sequence. In particular, block GLT sequences are encountered in the discretization of systems of DEs as well as in the higher-order finite element or discontinuous Galerkin approximation of scalar/vectorial DEs. This work is a review, refinement, extension, and systematic exposition of the theory of block GLT sequences. It also includes several emblematic applications of this theory in the context of DE discretizations.
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| 6. |
- Barbarino, Giovanni, et al.
(författare)
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Rectangular GLT sequences
- 2022
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Ingår i: Electronic Transactions on Numerical Analysis. - : Osterreichische Akademie der Wissenschaften, Verlag. - 1068-9613. ; 55, s. 585-617
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Tidskriftsartikel (refereegranskat)abstract
- The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computingthe asymptotic spectral distribution of square matrices An arising from the discretization of differential problems.Indeed, as the mesh fineness parameter n increases to ∞, the sequence {An}n often turns out to be a GLT sequence.In this paper, motivated by recent applications, we further enhance the GLT apparatus by developing a full theory ofrectangular GLT sequences as an extension of the theory of classical square GLT sequences. We also provide twoexamples of application as an illustration of the potential of the theory presented herein
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| 8. |
- Benedusi, Pietro, et al.
(författare)
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Fast Parallel Solver for the Space-time IgA-DG Discretization of the Diffusion Equation
- 2021
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Ingår i: Journal of Scientific Computing. - : Springer. - 0885-7474 .- 1573-7691. ; 89:1
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Tidskriftsartikel (refereegranskat)abstract
- We consider the space-time discretization of the diffusion equation, using an isogeometric analysis (IgA) approximation in space and a discontinuous Galerkin (DG) approximation in time. Drawing inspiration from a former spectral analysis, we propose for the resulting space-time linear system a multigrid preconditioned GMRES method, which combines a preconditioned GMRES with a standard multigrid acting only in space. The performance of the proposed solver is illustrated through numerical experiments, which show its competitiveness in terms of iteration count, run-time and parallel scaling.
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| 14. |
- Di Benedetto, M. D., et al.
(författare)
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Wireless ventilation control for large-scale systems : The mining industrial case
- 2009
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Ingår i: 2009 6th IEEE Annual Communications Society Conference on Sensor, Mesh and Ad Hoc Communications and Networks Workshops. - 9781424439386
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Konferensbidrag (refereegranskat)abstract
- Mining ventilation is an interesting example of a large scale system with high environmental impact where advanced control strategies can bring major improvements. Indeed, one of the first objectives of modern mining industry is to fulfill environmental specifications [1] during the ore extraction and crushing, by optimizing the energy consumption or the production of polluting agents. The mine electric consumption was 4 % of total industrial electric demand in the US in 1994 (6 % in 2007 in South Africa) and 90 % of it was related to motor system energy [2]. Another interesting figure is given in [3] where it is estimated that the savings associated with global control strategies for fluid systems (pumps, fans and compressors) represent approximately 20 % of the total manufacturing motor system energy savings. This motivates the development of new control strategies for large scale aerodynamic processes based on appropriate automation and a global consideration of the system. More specifically, the challenge in this work is focused on the mining ventilation since as much as 50 % or more of the energy consumed by the mining process may go into the ventilation (including heating the air). It is clear that investigating automatic control solutions and minimizing the amount of pumped air to save energy consumption (proportional to the cube of airflow quantity [4]) is of great environmental and industrial interest.
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| 23. |
- Ekström, Sven-Erik, 1977-
(författare)
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Matrix-Less Methods for Computing Eigenvalues of Large Structured Matrices
- 2018
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- When modeling natural phenomena with linear partial differential equations, the discretized system of equations is in general represented by a matrix. To solve or analyze these systems, we are often interested in the spectral behavior of these matrices. Whenever the matrices of interest are Toeplitz, or Toeplitz-like, we can use the theory of Generalized Locally Toeplitz (GLT) sequences to study the spectrum (eigenvalues). A central concept in the theory of GLT sequences is the so-called symbol, that is, a function associated with a sequence of matrices of increasing size. When sampling the symbol and when the related matrix sequence is Hermitian (or quasi-Hermitian), we obtain an approximation of the spectrum of a matrix of a fixed size and we can therefore see its general behavior. However, the so-computed approximations of the eigenvalues are often affected by errors having magnitude of the reciprocal of the matrix size.In this thesis we develop novel methods, which we call "matrix-less" since they neither store the matrices of interest nor depend on matrix-vector products, to estimate these errors. Moreover, we exploit the structures of the considered matrices to efficiently and accurately compute the spectrum.We begin by considering the errors of the approximate eigenvalues computed by sampling the symbol on a uniform grid, and we conjecture the existence of an asymptotic expansion for these errors. We devise an algorithm to approximate the expansion by using a small number of moderately sized matrices, and we show through numerical experiments the effectiveness of the algorithm. We also show that the same algorithm works for preconditioned matrices, a result which is important in practical applications. Then, we explain how to use the approximated expansion on the whole spectrum for large matrices, whereas in earlier works its applicability was restricted only to certain matrix sizes and to a subset of the spectrum. Next, we demonstrate how to use the so-developed techniques to investigate, solve, and improve the accuracy in the eigenvalue computations for various differential problems discretized by the isogeometric analysis (IgA) method. Lastly, we discuss a class of non-monotone symbols for which we construct the sampling grid yielding exact eigenvalues and eigenvectors.To summarize, we show, both theoretically and numerically, the applicability of the presented matrix-less methods for a wide variety of problems.
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| 26. |
- Garoni, Carlo, et al.
(författare)
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Block generalized locally Toeplitz sequences : From the theory to the applications
- 2018
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Ingår i: Axioms. - : MDPI AG. - 2075-1680. ; 7, s. 49:1-29
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Tidskriftsartikel (refereegranskat)abstract
- The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of matrices An arising from virtually any kind of numerical discretization of differential equations (DEs). Indeed, when the mesh fineness parameter n tends to infinity, these matrices An give rise to a sequence {An}n, which often turns out to be a GLT sequence or one of its “relatives”, i.e., a block GLT sequence or a reduced GLT sequence. In particular, block GLT sequences are encountered in the discretization of systems of DEs as well as in the higher-order finite element or discontinuous Galerkin approximation of scalar DEs. Despite the applicative interest, a solid theory of block GLT sequences has been developed only recently, in 2018. The purpose of the present paper is to illustrate the potential of this theory by presenting a few noteworthy examples of applications in the context of DE discretizations.
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| 28. |
- Garoni, Carlo, et al.
(författare)
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Block GLT sequences : Matrix functions and engineering application
- 2019
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Ingår i: The Electronic Journal of Linear Algebra. - : University of Wyoming Libraries. - 1537-9582 .- 1081-3810. ; 35, s. 204-222
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Tidskriftsartikel (refereegranskat)abstract
- The theory of block generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the spectral distribution of block-structured matrices arising from the discretization of differential problems, with a special reference to systems of differential equations (DEs) and to the higher-order finite element or discontinuous Galerkin approximation of both scalar and vectorial DEs. In the present paper, the theory of block GLT sequences is extended by proving that $\{f(A_n)\}_n$ is a block GLT sequence as long as $f$ is continuous and $\{A_n\}_n$ is a block GLT sequence formed by Hermitian matrices. It is also provided a relevant application of this result to the computation of the distribution of the numerical eigenvalues obtained from the higher-order isogeometric Galerkin discretization of second-order variable-coefficient differential eigenvalue problems (a topic of interest not only in numerical analysis but also in engineering).
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| 36. |
- Garoni, Carlo, et al.
(författare)
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Multilevel Generalized Locally Toeplitz Sequences : An Overview and an Example of Application
- 2019
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Ingår i: International Conference on Numerical Analysis and Applied Mathematics (ICNAAM-2018). - : AIP Publishing. - 9780735418547
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Konferensbidrag (refereegranskat)abstract
- The theory of multilevel generalized locally Toeplitz sequences is a powerful apparatus for computing/analyzing the spectral distribution of matrices arising from the numerical discretization of partial differential equations. In this note, we present an overview of this theory along with an example of application.
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| 37. |
- Garoni, Carlo, et al.
(författare)
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NURBS in isogeometric discretization methods : A spectral analysis
- 2020
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Ingår i: Numerical Linear Algebra with Applications. - : Wiley. - 1070-5325 .- 1099-1506. ; 27:6
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Tidskriftsartikel (refereegranskat)abstract
- Nonuniform rational B-splines (NURBS) are the most common representation form in isogeometric analysis. In this article, we study the spectral behavior of discretization matrices arising from isogeometric Galerkin and collocation methods based ond-variate NURBS of degrees(p(1), horizontal ellipsis ,p(d)), and applied to general second-order partial differential equations defined on ad-dimensional domain. The spectrum of these matrices can be compactly and accurately described by means of a so-called symbol. We compute this symbol and show that it is the same as in the case of isogeometric discretization matrices based ond-variate polynomial B-splines of degrees(p(1), horizontal ellipsis ,p(d)). The theoretical results are confirmed with a selection of numerical examples.
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