Università degli Studi di Napoli "Parthenope"

Teaching schedule

Academic year: 
2018/2019
Belonging course: 
Course of Master's Degree Programme on APPLIED COMPUTER SCIENCE (MACHINE LEARNING AND BIG DATA)
Disciplinary sector: 
NUMERICAL ANALYSIS (MAT/08)
Language: 
Italian
Credits: 
12
Year of study: 
1
Cycle: 
Annualita' Singola
Hours of front activity: 
96

Language

Italian

Course description

The course analyzes methodologies, algorithms and software in the field of Scientific Computing, with a particular focus to Data Science applications and simulations by means of differential models. The course contains in-depth analysis of the MATLAB language, used for software development during the Laboratory activities that are an essential part of the course.
Knowledge and understanding: The student must demonstrate knowledge and understanding of advanced aspects of numerical analysis and scientific computing, with particular regard to linear algebra, approximation, systems of differential equations, conformal mappings and Fourier analysis, in a both theoretical and applicative context, and the MATLAB programming language and its development environment.
Applying knowledge and understanding: The student must demonstrate that he can use his acquired knowledge to solve computationally advanced problems of applied mathematics in Data Science and simulations fields, even when they appear in real world contexts and in different application disciplines, to develop numerical algorithms and analyze them from the point of view of accuracy and complexity, to use MATLAB language in a conscious way, in order to produce scientific software that can be used to effectively solve a variety of advanced problems. These skills are also expressed in a thorough and conscious use of calculation tools and advanced computer labs.
Making judgments: Students must be able to independently evaluate the results produced by scientific software. In addition, they must be able to identify, even using online repositories of scientific software, the most appropriate algorithms and software to solve a specific problem.
Communication skills: The student must be able to compile a presentation report of a numerical algorithm and to document his MATLAB implementation, also working in small team, using advanced scientific writing / documentation tools and using the applied/computational mathematics terminology , even in English.
Learning skills: The student must be able to update and independently analyze topics and specific applications of numerical analysis and scientific calculation, also by accessing databases, on-line repositories of scientific software and other methods provided by the network.

Prerequisites

For the students of the Applied Computer Science (Machine Learning and Big Data) MSc Program: it is necessary to have acquired the knowledge and skills transmitted by the following courses of the Computer Science BSc program: Mathematics I, Mathematics II, Numerical Methods, Algorithms and Data Structures and Lab ASD.

Syllabus

PART I - Data Science and Simulation (first semester)
Recall of Numerical Linear Algebra: interpretation of the matrix-vector and matrix-matrix multiplication - scalar product and orthogonal projection - projectors on subspaces.Use of MATLAB to solve problems in previous areas. Hours of lessons: 2, Laboratory hours: 4
Matrix factorizations: QR factorization - spectral decomposition - singular values ​​decomposition (SVD) - methods for the calculation of the factorizations. Applications to Data Science: the idea of ​​reducing the dimensionality of data, analysis of data in bioinformatics, analysis and compression of images, semantic indexing of texts, principal components analysis. Use of MATLAB to solve problems in previous areas. Hours of lessons: 6, Laboratory hours: 4
Google Pagerank algorithm: graphs and matrices - stochastic and positive matrices - power method - power method as smoother - Pagerank interpretation as random process - Markov chains - Markov chains applied to data analysis. Use of MATLAB to solve problems in previous areas. Hours of lessons: 4, Laboratory hours: 2
Resolution of non-linear equation systems: Newton's methods - fixed point method - computer graphic application - application to social network's reputation systems. Use of MATLAB to solve problems in previous areas. Hours of lessons: 2, Laboratory hours: 2
Calculation of maximums and minima of functions of several variables: steepest descent method - Newton's methods - Levenberg-Marquardt method - convergence, rate of convergence, stopping criteria - applications to computational modeling. Outline of the problem of global optimization: simulated annealing methods. Use of MATLAB to solve problems in previous areas. Hours of lessons: 4, Laboratory hours: 2
Numerical resolution of ordinary differential equations: finite differences - initial values problems​​- explicit and implicit methods - stability and convergence -boundary values problems ​​- applications to computational modeling. Use of MATLAB to solve problems in previous areas. Hours of lessons: 4, Laboratory hours: 4
Numerical resolution of differential equations to partial derivatives: stationary equations (Laplace and Poisson equations) - non-stationary equations (advection equation, diffusion equations) - finite difference methods - stability and convergence - applications to computational modeling. Use of MATLAB to solve problems in previous areas. Hours of lessons: 4, Laboratory hours: 4.

PART II - Geometrical mappings and Transforms (second semester)
Spaces and transformations: Main Spaces (linear, affine, projective.) and related transformations - eigenspaces - conformal mappings. - applications to computational graphics - use of MATLAB to solve problems in previous areas. Hours of lessons: 10, Laboratory hours: 6
Approximation in the least squares sense: Linear least squares, best approximation - continuous least squares linear approximation - nonlinear squared minima - use of MATLAB in approximation. use of MATLAB to solve problems in previous areas. Hours of lessons: 10, Laboratory hours: 6
Insights on the Fourier and Laplace Transforms: 1D and 2D Fourier transforms - applications to the analysis and synthesis of sounds and images - Fourier discrete transformation - FFT algorithms - use of MATLAB in Fourier analysis. Application of the Laplace Transform to differential equations. Hours of lessons: 10, Laboratory hours: 6

PART I - Data Science and Simulation (first semester)
Recall of Numerical Linear Algebra: interpretation of the matrix-vector and matrix-matrix multiplication - scalar product and orthogonal projection - projectors on subspaces. 6h
Matrix factorizations: QR factorization - spectral decomposition - singular values ​​decomposition (SVD) - methods for the calculation of the factorizations. Applications to Data Science: the idea of ​​reducing the dimensionality of data, analysis of data in bioinformatics, analysis and compression of images, semantic indexing of texts, principal components analysis. 12h
Google Pagerank algorithm: graphs and matrices - stochastic and positive matrices - power method - power method as smoother - Pagerank interpretation as random process - Markov chains - Markov chains applied to data analysis. 6h
Resolution of non-linear equation systems: Newton's methods - fixed point method - computer graphic application - application to social network's reputation systems. 4h
Calculation of maximums and minima of functions of several variables: steepest descent method - Newton's methods - Levenberg-Marquardt method - convergence, rate of convergence, stopping criteria - applications to computational modeling. Outline of the problem of global optimization: simulated annealing methods. 4h
Numerical resolution of ordinary differential equations: finite differences - initial values problems​​- explicit and implicit methods - stability and convergence -boundary values problems ​​- applications to computational modeling. 8h
Numerical resolution of differential equations to partial derivatives: stationary equations (Laplace and Poisson equations) - non-stationary equations (advection equation, diffusion equations) - finite difference methods - stability and convergence - applications to computational modeling. 8h

PART II - Geometrical mappings and Transforms (second semester)
Spaces and Transformations.
Main Spaces (linear, affine, projective spaces) and related mappings.
Applications of Eigenvalues and Eigenspaces: PCA derivation and its geometrical meaning.
Conformal mappings. - Applications to computer graphics - Use of MATLAB to solve problems in previous areas.
The Best Approximation, with respect to the Euclidean norm, for finite dimensional subspaces and for Hilbert spaces - Least Squares Approximation - Linear least squares approximation: discrete and continuous case - Use of MATLAB in the least squares approximation.
Insights on the Fourier Transform.
1D and 2D Fourier Series and Transforms and their applications to the analysis and synthesis of sounds and images - Discrete Fourier Transform - FFT algorithms - Use of MATLAB in Fourier analysis.
Laplace Transform and its application to Differential Equations.

Teaching Methods

Traditional teaching in the presence, held in a suitable computing lab. Each class also presents the resolution of an advanced problem and the analysis of the related algorithms and software. Moreover, each class requires students to deepen some aspects of the lecture, both through theoretical analysis and through the development of algorithms or algorithm variants (in MATLAB).

Textbooks

C. Moler - Numerical Computing with MATLAB, SIAM, 2005. Downloadable from www.mathworks.com
M. Rizzardi - Experimenting mathematics with MATLAB: elements of complex analysis (in Italian), Liguori Ed. 2008.
G. Giunta - ACS notes Part I (in Italian), 2014. Downloadable from the e-learning platform of the Department of Science and Technology.
All the lessons may be watched in streaming as animated presentations in Flash format with the audio commentary by the Teacher through the e-learning platform of the Department of Sciences and Technologies; the slides (format .pdf and .pps) of all the lessons are available on the same platform, together with exercises, in-depth projects, notes for the laboratory, and an operational introduction to MATLAB.

Learning assessment

The objective of the verification procedure is to quantify, for each student, the level of achievement of the previously indicated training objectives.
The verification procedure consists mainly of an mid-term exam and a final exam, both oral. The first exam is focused on the theoretical, applicative and
implementative aspects of the methods and algorithms of linear Algebra, nonlinear systems, unconstrained optimization, ordinary and partial differential
equations; the final examn deepens the theoretical, applicative and implementative aspects related to Fourier analysis and geometric transformations.
The two tests are also aimed at evaluating the ability to programming in Matlab, access to the main repositories of software for scientific
applications and in general to the conscious use of the basic tools in Scientific Computing. The ability ito work in team and in writing
accompanying reports to application software is assessed throughout the course, held in a computer lab and involving students in a continuous activity
of team-working.

More information

All detailed information on the course can be found on the page of the Course on the e-learning platform of the Department of Science and Technology: http: //e-scienzeetecnologie.uniparthenope.it/
in particular:
http://e-scienzeetecnologie.uniparthenope.it/course/category.php?id=13
Lectures are in Italian. The professors are fluent in English and available to interact with students in English, also during the examination.

TraduttoreDisattiva traduzione istantanea

Inglese

Italiano

Francese

Rileva lingua

Italiano

Inglese

Spagnolo

Tutte le informazioni di dettaglio sul corso sono reperibili sulla pagina del Corso in piattaforma di e-learning del Dipartimento di Scienze e Tecnologie:http://e-scienzeetecnologie.uniparthenope.it/
in particolare:
http://e-scienzeetecnologie.uniparthenope.it/course/category.php?id=13

287/5000

All detailed information on the course can be found on the page of the Course in e-learning platform of the Department of Science and Technology: http: //e-scienzeetecnologie.uniparthenope.it/
in particular:
http://e-scienzeetecnologie.uniparthenope.it/course/category.php?id=13

TraduttoreDisattiva traduzione istantanea

Inglese

Italiano

Francese

Rileva lingua

Italiano

Inglese

Spagnolo

Tutte le informazioni di dettaglio sul corso sono reperibili sulla pagina del Corso in piattaforma di e-learning del Dipartimento di Scienze e Tecnologie:http://e-scienzeetecnologie.uniparthenope.it/
in particolare:
http://e-scienzeetecnologie.uniparthenope.it/course/category.php?id=13

287/5000

All detailed information on the course can be found on the page of the Course on e-learning platform of the Department of Science and Technology: http: //e-scienzeetecnologie.uniparthenope.it/
in particular:
http://e-scienzeetecnologie.uniparthenope.it/course/category.php?id=13

TraduttoreDisattiva traduzione istantanea

Inglese

Italiano

Francese

Rileva lingua

Italiano

Inglese

Spagnolo

Tutte le informazioni di dettaglio sul corso sono reperibili sulla pagina del Corso in piattaforma di e-learning del Dipartimento di Scienze e Tecnologie:http://e-scienzeetecnologie.uniparthenope.it/
in particolare:
http://e-scienzeetecnologie.uniparthenope.it/course/category.php?id=13

287/5000

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