ECTS Credits: 9   |   Semester: 1   |   Year: 1   |   Campus: University of L'Aquila   |   Language: English   |   Code: DT0626

Unit Coordinator: Marco Di Francesco, Michele Palladino

Aims:

Introducing basic tools of advanced real analysis such as metric spaces, Banach spaces, Hilbert spaces, bounded operators, weak convergences, compact operators, weak and strong compactness in metric spaces, spectral theory, in order to allow the student to formulate and solve linear ordinary differential equations partial differential equations, classical variational problems, and numerical approximation problems in an "abstract" form. Provide a primer of abstract measure and integration to be used in advanced probability and analysis courses.

Content:

• Metric spaces, normed linear spaces. Topology in metric spaces. Compactness.
• Spaces of continuous functions. Convergence of function sequences. Approximation by polynomials. Compactness in spaces of continuous functions. Arzelà's theorem. Contraction mapping theorem.
• Crash course on abstract measure and integration. Measurable spaces and measurable functions. Borel and Lebesgue measures. Integrals on measure spaces. Limit exchange convergence theorems. Lp spaces. Product measures. Signed measures and Radon-Nicodym Theorem. Riesz representation theorem for measures.
• Introduction to the theory of linear bounded operators on Banach spaces. Bounded operators. Dual norm. Examples. Riesz' lemma. Norm convergence for bounded operators.
• Hilbert spaces. Elementary properties. Orthogonality. Orthogonal projections. Bessel's inequality. Orthonormal bases. Examples.
• Bounded operators on Hilbert spaces. Dual of a Hilbert space. Adjoin operator, self-adjoint operators, unitary operators. Applications. Weak convergence on Hilbert spaces. Banach-Alaoglu's theorem.
• Introduction to spectral theory. Compact operators. Spectral theorem for self-adjoint compact operators on Hilbert spaces. Hilbert-Schmidt operators. Functions of operators.

Pre-requisites:

Basic calculus and analysis in several variables, linear algebra.

Reading list:

• John K. Hunter, Bruno Nachtergaele, Applied Analysis. World Scientific.
• H. Brezis, Funtional Analysis, Sobolev Spaces, and partial differential equations. Springer.
• Piermarco Cannarsa, Teresa D’Aprile, Introduction to Measure Theory and Functional Analysis, Springer.

ECTS Credits: 6   |   Semester: 1   |   Year: 1   |   Campus: University of L'Aquila   |   Language: English   |   Code: I0183

Unit Coordinator: Amadori Debora

Aims:

LEARNING OBJECTIVES.
The course aims at providing basic properties and main techniques to solve basic partial differential equations.
Those objectives contribute to the learning goals of the entire course of studies, as the inner coherence of the master degree in Mathematical Modelling was verified at the time of the planning of the master program.

LEARNING OUTCOMES.
At the end of the course, the student should:

1. know basic properties (existence, uniqueness, etc.) and main techniques (characteristics, separation of variables, Fourier methods, Green's functions, similarity solutions, etc.) to solve basic partial differential equations (semilinear first order PDEs, heat, Laplace, wave equations);
2. understand and be able to explain thesis and proofs in the field of basic partial differential equations;
3. have strengthened the logic and computational skills;
4. be able to read and understand other mathematical texts on related topics.

Content:

First order partial differential equations. Definition of characteristic vectors and characteristic surfaces. Characteristics for (semi)linear partial differential equations of first order in two independent variables. Existence and uniqueness to initial value problems for first order semilinear partial differential equations in two independent variables Duhamel’s principle for non homogeneous first order partial differential equations.

Second order partial differential equations. Classification of second order semilinear partial differential equations in two independent variables. Canonical form for second order semilinear partial differential equations in two independent variables. Classification for second order semilinear partial differential equations in many independent variables.

Heat equation. Derivation of heat equation and well–posed problems in one space dimension. Solution of Cauchy–Dirichlet problem for one dimensional heat equation by means of Fourier method of separation of variables. Energy method and uniqueness. Maximum principle. Fundamental solution. Solution of global Cauchy problem. Non homogeneous problem: Duhamel’s principle.

Laplace equation. Laplace and Poisson equation: well-posed problems; uniqueness by means of energy method. Mean value property and maximum principles. Laplace equation in a disk by means of separation of variables. Poisson’s formula. Harnack’s inequality and Liouville’s Theorem. Fundamental solution of Laplace operator. Solution of Poisson’s equation in the whole space. Green’s functions and Green’s representation formula.

Wave equation. Transversal vibrations of a string. Well–posed problems in one space dimension. D’Alembert formula. Characteristic parallelogram. Domain of dependence and range of influence. Fundamental solution for one dimensional wave equation. Duhamel’s principle for non homogeneous one dimensional wave equations. Special solutions of multi–d wave equation: planar and spherical waves. Well–posedness for initial, boundary value problems: uniqueness by means of energy estimates. Separation of variables. Domain of dependence and range of influence in several space variables. Fundamental solution for multi–dimensional wave equation. Solution of 3–d wave equation: Kirchhoff’s formula and strong Huygens’ principle. Wave equation in two dimensions: method of descent. Fundamental solution in 2–d. Duhamel’s principle for non homogeneous wave equation in 3–d: delayed potentials.

Pre-requisites:

Students must know the basic notions of mathematical analysis, including Fourier series and ordinary differential equations, and the basic notions of continuum mechanics.

Reading list:

- L.C. Evans. Partial Differential Equations. Graduate Studies in Mathematics, Vol. 19, AMS, 2010.

- S. Salsa. Partial Differential Equations in Actions: from Modelling to Theory. Springer–Verlag Italia, 2008.

- S. Salsa, G. Verzini. Equazioni a derivate parziali: complementi ed esercizi. Springer–Verlag Italia. 2005.

- W.A. Strauss. Partial Differential Equations: An Introduction. John Wiley & Sons Inc., 2008.

- E.C. Zachmanoglou, D.W. Thoe. lntroduction to Partial Differential Equations with Applications. Dover Publications, Inc., 1986.

ECTS Credits: 6   |   Semester: 1   |   Year: 1   |   Campus: University of L'Aquila   |   Language: English   |   Code: DT0821

Unit Coordinator: Cristina Pignotti, Michele Palladino

Aims:

To get the mathematical basics of control theory and optimal control theory.
To know classical problems of control theory governed by ordinary differential equations and to deal with them by using the concepts learned.
To know and construct significant applications of optimal control theory in life sciences, physics, and economics.
To know some examples of control problems for models governed by partial differential equations.

Content:

Controllability of linear systems and bang-bang principle. Controllability of nonlinear systems. Stabilizability of linear and nonlinear systems.

Basic optimal control problems. Necessary conditions. Adjoint equation. Pontryagin’s Maximum Principle. Existence and uniqueness results for minimizers. Hamiltonian and autonomous problems. Optimality conditions. State conditions at the final time. Payoff terms. States with fixed endpoints. 

Dynamic Programming and Hamilton-Jacobi Equations.
Optimal control problems in biology, physics, and economics. Control problems for multiagent systems.

An introduction to controllability and stabilization of partial differential equations: the wave equation and the heat equation.

Pre-requisites:

Basic calculus and analysis (differential and integral calculus with functions of many variables). Ordinary differential equations. 

Reading list:

-L.C. Evans, An introduction to mathematical control theory, Berkley, Lecture notes.

-A. Bressan, B. Piccoli, Introduction to Mathematical Theory of Control, AIMS Book Series, 2007

-A. Isidori, Nonlinear Control Systems: An Introduction, Springer

-S. Lenhart and J.T. Workman, Optimal Control Applied to Biological Models, Chapman & Hall/CRC.
-S. Anita, V. Arnautu and V. Capasso, An Introduction to Optimal Control Problems in Life Sciences and Economics, Birkhauser.

ECTS Credits: 6   |   Semester: 1   |   Year: 1   |   Campus: University of L'Aquila   |   Language: English   |   Code: I0459

Unit Coordinator: Bruno Rubino

Aims:

The course is intended to introduce and develop an understanding of the concepts in nonlinear dynamical systems and bifurcation theory, and an ability to analyze nonlinear dynamic models of physical systems. The emphasis is to be on understanding the underlying basis of local bifurcation analysis techniques and their applications to structural and mechanical systems.

Content:

Review of: first-order nonlinear ODE, first-order linear systems of autonomous ODE. Local theory for nonlinear dynamical systems: linearization, stable manifold theorem, stability and Liapunov functions, planar non-hyperbolic critical points, center manifold theory, normal form theory. Global theory for nonlinear systems: limit sets and attractors, limit cycles and separatrix cycles, Poincaré map. Hamiltonian systems. Poincaré-Bendixson theory. Bifurcation theory for nonlinear systems: structural stability, bifurcation at non-hyperbolic equilibrium points, Hopf bifurcations, bifurcation at non hyperbolic periodic orbits. Applications.

Pre-requisites:

Ordinary differential equations

Reading list:

Lawrence Perko, Differential equations and dynamical systems, Springer-Verlag, 2001

ECTS Credits: 3   |   Semester: 1   |   Year: 1   |   Campus: University of L'Aquila   |      |   Code: DT0807

Unit Coordinator: Roberta Antonetti

Aims:

The aim of this course is to provide students with knowledge of Italian language at level A1 (beginner). The course will concentrate on four aspects of communication: speaking, writing, listening and reading on a basic level. Attention is given to correct pronunciation of Italian.
Furthermore, the goals of the course are to enable students to:
- develop the language proficiency required to communicate effectively in Italian at level A1;
- develop awareness of the nature of language and language learning;
- develop transferable skills;
- form a sound base of the skills, language and attitudes required for progression to work or further study, either in Italian or another subject area.
On successful completion of this module, a student should be able to:
- understand familiar words and very basic phrases concerning himself/herself, his/her family and immediate concrete surroundings when people speak slowly and clearly;
- read familiar names, words and very simple sentences, for example on notices and posters or in catalogues;
- interact in a simple way provided the other person is prepared to repeat or rephrase things at a slower rate of speech;
- ask and answer simple questions in areas of immediate need or on very familiar topics;
- use simple phrases and sentences to describe where he/she lives and people he/she knows;
- write short messages, postcards, fill in forms with personal details or a hotel registration form.

Content:

Students will be required to show knowledge and understanding of the broad topic areas listed below. These provide contexts for the acquisition of vocabulary and the study of grammar and structures. 
VOCABULARY AND TOPIC AREAS 
Clothes and accessories; colors; countries, nationalities and languages; culture, customs and celebrations; education e.g. learning institutions, education and training, learning tools, subjects; family and friends; feelings and emotions; food and drink; interests, sports and activities; jobs; measurements e.g. size, shape, weight; numbers (cardinal, ordinal) and money; rooms and furniture; shops and places; street directions; the human body and health e.g. parts of the body, health and illness; time expressions e.g. telling the time, dates, days of the week, months, seasons; travel and transport; weather.

FUNCTIONAL SYLLABUS
Accepting/ thanking; applying for a job; asking about personal information; asking permission; buying and asking prices; describing people and objects; giving opinions; giving instructions; greeting and introducing; inviting, refusing; making suggestions; requesting, offering; talking about likes and dislikes; talking about routines; telling the time.

GRAMMAR SYLLABUS
Sounds and spelling; nouns: gender, number; irregular plural nouns; definite article; indefinite article; adjectives, first group adjectives and second group adjectives, position of adjectives; formal address; pronouns, subject pronouns and direct and indirect object pronouns; possessive adjectives; prepositions simple and prepositions with articles; interrogatives; adverbs; present tense of regular and irregular verbs; modal verbs and the verb sapere; present perfect and some verbs with an irregular past participle.

Pre-requisites:

There are no prerequisites for attending the course.

Reading list:

"Dieci A1" - Alma edizioni, Firenze 2019.
“New Italian grammar in practice”, Alma edizioni, Firenze 2015.

ECTS Credits: 6   |   Semester: 2   |   Year: 1   |   Campus: Hamburg University of Technology   |   Language: English   |   Code: DT0651

Unit Coordinator: Daniel Ruprecht

Aims:

Students are able to list numerical methods for the solution of ordinary differential equations and explain their core ideas, repeat convergence statements for the treated numerical methods (including the prerequisites tied to the underlying problem), explain aspects regarding the practical execution of a method, select the appropriate numerical method for concrete problems, implement the numerical algorithms efficiently and interpret the numerical results.

Content:

• Numerical methods for Initial Value Problems: single step methods, multistep methods, stiff problems, differential algebraic equations (DAE) of index 1;
• Numerical methods for Boundary Value Problems: multiple shooting method, difference methods

Pre-requisites:

Analysis, Linear Algebra, Basic MATLAB knowledge

Reading list:

• E. Hairer, S. Noersett, G. Wanner: Solving Ordinary Differential Equations I: Nonstiff Problems
• E. Hairer, G. Wanner: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems

ECTS Credits: 9   |   Semester: 2   |   Year: 1   |   Campus: Hamburg University of Technology   |   Language: English   |   Code: DT0652

Unit Coordinator: Sabine Le Borne, Daniel Ruprecht

Aims:

Students can list classical and modern iteration methods and their interrelationships, repeat convergence statements for iterative methods and explain aspects regarding the efficient implementation of iteration methods.

They will learn the fundamental concepts of parallel programming and how to translate them into efficient, parallel code.

They will learn how to compile parallel code and how to model and measure performance of parallelized software. 

Content:

• Sparse systems: orderings and storage formats, direct solvers;
• Classical methods: basic notions, convergence;
• Projection methods;
• Rylov space methods;
• Preconditioning (e.g. ILU);
• Multigrid methods;
• Domain decomposition methods, shared versus distributed memory parallelization;
• Message Passing Interface;
• OpenMP;
• Threads;
• Processes;
• HPC architecture;
• Performance models;
• Speedup and parallel efficiency 

Pre-requisites:

• Analysis,
• Linear Algebra,
• Programming experience in C and C++, FORTRAN, Python or a similar programming language.

Reading list:

• Y. Saad. Iterative methods for sparse linear systems
• M. Olshanskii, E. Tyrtyshnikov. Iterative methods for linear systems: theory and applications
• Thomas Rauber, Parallel Programming : for Multicore and Cluster Systems, Berlin [u.a.] Springer 2010
• Bertil Schmidt, Parallel programming : concepts and practice, Amsterdam Morgan Kaufmann 201

Additional info:

Parallel programming for interdisciplinary mathematics (DT0857)

Scientific programming for interdisciplinary mathematics (DT0858)

ECTS Credits: 6   |   Semester: 2   |   Year: 1   |   Campus: Hamburg University of Technology   |   Language: English   |   Code: DT0653

Unit Coordinator: Thomas Schmidt

Aims:

The module introduces to variational minimization problems and/or variational methods for PDEs.

It may cover problems in a classical smooth setting as well as theory in Sobolev spaces.

Content:

A selection out of the following:

• Model problems and examples (Dirichlet energy, isoperimetric and brachistochrone problems, minimal surfaces, Bolza and Weierstrass examples, …),
• Existence and uniqueness of minimizers by direct methods,
• Weak lower semicontinuity of (quasi)convex variational integrals,
• Necessary and sufficient (PDE) conditions for minimizers,
• Problems with constraints (obstacles, capacities, manifold and volume constraints, ...),
• Generalized minimizers (relaxation, Young measures, ...),
• Variational principles and applications,
• Duality theory,
• Outlook on regularity.

Pre-requisites:

A solid background in analysis and linear algebra is necessary.

Familiarity with functional analysis, Sobolev spaces, and PDEs can be advantageous.

Reading list:

• H. Attouch, G. Buttazzo, G. Michaille, Variational Analysis in Sobolev and BV Spaces, Applications to PDEs and Optimization, MOS-SIAM Series on Optimization 17, Philadelphia, 2014.
• G. Buttazzo, M. Giaquinta, S. Hildebrandt, One-Dimensional Variational Problems, An Introduction, Oxford Lecture Series in Mathematics and its Applications 15, Clarendon Press, Oxford, 1998.
• B. Dacorogna, Introduction to the Calculus of Variations, Imperial College Press, London, 2014.
• B. Dacorogna, Direct Methods in the Calculus of Variations, Applied Mathematical Sciences, Springer, Berlin, 2008.
• I. Ekeland, R. Témam, Convex Analysis and Variational Problems, Classics in Applied Mathematics 28, SIAM, Philadelphia, 1999.
• M. Giaquinta, S. Hildbrandt, Calculus of Variations 1, The Lagrangian Formalism, Grundlehren der Mathematischen Wissenschaften 310, Springer, Berlin, 1996.
• E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, Singapore, 2003.
• F. Rindler, Calculus of Variations, Universitext, Springer, Cham, 2018.
• F. Santambrogio, Optimal Transport for Applied Mathematicians, Calculus of Variations, PDEs, and Modeling, Progress in Nonlinear Differential Equations and Their Applications 87, Birkhäuser/Springer, Cham, 2015.
• M. Struwe, Variational Methods, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 34, Springer, Berlin, 2008.

ECTS Credits: 3   |   Semester: 2   |   Year: 1   |   Campus: Hamburg University of Technology   |      |   Code: DT0668

Unit Coordinator: Kathrin Heuking

Aims:

• To understand sentences and frequently used expressions related to familiar everyday expressions, university life and its requirements.
• To describe your place of residence and the city of Hamburg.
• To complete simple forms.
• To impart basic grammar structures and vocabulary.
• To enable to read and understand.

Content:

• To introduce somebody to somebody,
• Talking about yourself;
• Alphabet,
• Spelling,
• So called "w-questions",
• Numbers (1-1 Mio),
• Time,
• Days of the week,
• Pronunciation.
• Simple role plays.
• Grammar: word order, construction of German sentences, affirmative sentence, questions, regular verbs → present tense, auxiliary verbs "haben" and "sein", pronouns (Nominativ), negative answer to a question "nicht" and "kein". 

Reading list:

Will be announced in lectures.

ECTS Credits: 6   |   Semester: 2   |   Year: 1   |   Campus: Hamburg University of Technology   |   Language: English   |   Code: DT0654

Unit Coordinator: Matthias Schulte

Aims:

This course provides an introduction to probability theory and stochastic processes with special emphasis on applications and examples.

The first part covers some important concepts from measure theory, stochastic convergence and conditional expectation, while the second part deals with some important classes of stochastic processes. 

Content:

• Measure and probability spaces
• Integration and expectation
• Types of stochastic convergence
• Law of large numbers
• Central limit theorem
• Radon-Nikodym theorem
• Conditional expectation
• Martingales
• Markov chains
• Poisson processes 

Pre-requisites:

Familiarity with the basic concepts of probability

Reading list:

• H. Bauer, Probability theory and elements of measure theory, second edition, Academic Press, 1981.
• A. Klenke, Probability Theory: A Comprehensive Course, second edition, Springer, 2014.
• G. F. Lawler, Introduction to Stochastic Processes, second edition, Chapman & Hall/CRC, 2006.
• A. N. Shiryaev, Probability, second edition, Springer, 1996. 

ECTS Credits: 6   |   Semester: 2   |   Year: 1   |   Campus: Hamburg University of Technology   |      |   

Unit Coordinator: Martin Burger

Aims:

This course will provide an introduction to some basic mathematical problems in image formation and image reconstruction. In addition to modelling forward problems, we consider classical regularization strategies, ensuring well-posedness of the image reconstruction problems. Beyond this classical setting, we dive into modern deep-learning methods, which allow solving inverse problems in a data-dependent approach. Finally, we also consider uncertainty quantification, where we employ the Bayesian view point of inverse problems.

Content:

• Modelling forward problems
• Regularization strategies
• Convex optimization algorithms
• Deep-learning methods: post-processing strategies, learned regularization.
• Uncertainty quantification

Pre-requisites:

Analysis, Linear Algebra, Basic Numerical Analysis and some programming skills

Reading list:

• Mueller, J. L., & Siltanen, S. (Eds.). (2012). Linear and nonlinear inverse problems with practical applications. Society for Industrial and Applied Mathematics.

• Benning, M., & Burger, M. (2018). Modern regularization methods for inverse problems. Acta numerica, 27, 1-111.

• Natterer, F., & Wübbeling, F. (2001). Mathematical methods in image reconstruction. Society for Industrial and Applied Mathematics.

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of L'Aquila   |   Language: English   |   Code: DT0113

Unit Coordinator: Corrado Lattanzio

Aims:

LEARNING OBJECTIVES.
The course aims at providing advanced mathematical notions used in the field of (applied) mathematical analysis and their applications to a variety of topics, including the basic equations of mathematical physics and some current research topics about linear and nonlinear partial differential equations.
Those objectives contribute to the learning goals of the entire course of studies, as the inner coherence of the master degree in Mathematics was verified at the time of the planning of the master program.
LEARNING OUTCOMES.
At the end of the course, the student should:
1. know the advanced mathematical notions used in the field of (applied) mathematical analysis, as measure theory, Sobolev Spaces, distributions, and their applications to the theory of linear and non-linear partial differential equations;
2. understand and be able to explain thesis and proofs in advanced mathematical analysis;
3. have strengthened the logic and computational skills;
4. be able to read and understand other mathematical texts on related topics.

Content:

Distributions. Locally integrable functions. The space of test function D(Ω). Distributions associated to locally integrable functions. Singular distributions. Examples. Operations on distributions: sum, products times functions, change of variables, restrictions, tensor product. Differentiation and his properties; comparison with classical derivatives. Differentiation of jump functions. Partition of unity. Support of a distribution; compactly supported distributions. Fourier transform and tempered distributions. Convolution between distributions and regularization of distributions. Denseness of D(Ω) in D′(Ω).

Sobolev spaces. Definition of weak derivatives and his motivation. Sobolev spaces Wk,p(Ω) and their properties. Interior and global approximation by smooth functions. Extensions. Traces. Embeddings theorems: Gagliardo–Nirenberg–Sobolev inequality and embedding theorem for p < n. Hölder spaces. Morrey inequality. Embedding theorem for p > n. Sobolev inequalities in the general case. Compact embeddings: Rellich–Kondrachov theorem, Poincaré inequalities. Embedding theorem for p = n. Characterization of the dual space H-1.

Second order parabolic equations. Definition of parabolic operator. Weak solutions for linear parabolic equations. Existence of weak solutions: Galerkin approximation, construction of approximating solutions, energy estimates, existence and uniqueness of solutions.

First order nonlinear hyperbolic equations. Scalar conservation laws: derivation, examples. Weak solutions, Rankine-Hugoniot conditions, entropy conditions. L1 stability, uniqueness and comparison for weak entropy solutions. Convergence of the vanishing viscosity and existence of the weak, entropy solution. Riemann problem.

Measures. System of sets, Positive Measures, Outer Measures, Construction of Measures, Signed Measures, Borel and Radon Measures
Integration. Measurable Functions, Simple Functions, Convergence Almost Everywhere, Integral of Measurable Functions, Convergences of Integrals, Fubini-Tonelli Theorems.
Differentiation. The Radon-Nikodym Theorem, Differentiation on Euclidean space, Differentiation of the real line.
Radon measures and continuous functions. Spaces of continuous functions, Riesz Theorem.

Pre-requisites:

Basic notions of functional analysis, functions of complex values, standard properties of classical solutions of semilinear first order equations, heat equation, wave equation, Laplace and Poisson's equations.

Reading list:

- L. Ambrosio, G. Da Prato, A. Mennucci. Introduction to measure theory and integration. Edizioni della Normale.

- L. Ambrosio, N. Fusco, D. Pallara. Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs.

- V.I. Bogachev. Measure theory, Volume I, Springer.

- H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext, Springer.

- P. Cannarsa, T. D’Aprile. Introduction to Measure Theory and Functional Analysis. Springer.

- C.M. Dafermos. Hyperbolic Conservation Laws in Continuum Physics, Springer.

- L.C. Evans. Partial Differential Equations. Graduate Studies in Mathematics, Vol. 19, AMS.

- L. Evans, R. Gariepy. Measure Theory and Fine Properties of Functions, CRC Press. Revised Edition.

- G.B. Folland. Real analysis: Modern techniques and their applications. New York Wiley

- G. Gilardi. Analisi 3. McGraw–Hill.

- L. Grafakos, Classical Fourier Analysis. Springer.

- V.S. Vladimirov. Equations of Mathematical Physics. Marcel Dekker, Inc

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of L'Aquila   |   Language: English   |   

Unit Coordinator: Donatella Donatelli

Aims:

Content:

- Derivation of the governing equations: Euler and Navier-Stokes
- Eulerian and Lagrangian description of fluid motion; examples of fluid flows
- Poiseulle and Couette fluid types
- Vorticity equation in 2D and 3D
- Dimensional analysis: Reynolds number, Mach Number, Frohde number.
- From compressible to incompressible models
- Existence of solutions for viscid and inviscid fluids
- Fluid dynamic modeling in various fields: mixture of fluids, combustion, astrophysics, geophysical fluids (atmosphere, ocean)
- Modeling for biofluids: hemodynamics, cerebrospinal fluids, cancer modelling, animal locomotion, bioconvection for swimming microorganisms.

Pre-requisites:

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of L'Aquila   |   Language: English   |   

Unit Coordinator: Donato Pera

Aims:

The aim of the course is the study of analytical, numerical and computational methods (on parallel computing structures), for the solution of partial differential equations considered as basic elements for the construction of mathematical models for natural disasters. During the course will be introduced basic concepts related to the analytical and numerical solution for wave equations, elastodynamics equations and advection-reaction-diffusion systems.
In addition to the classical analytical and numerical approach, some basic elements for the simulation of the studied problems on parallel computing structures will be introduced, with reference to the programming of Shared Memory, Distributed Memory and/or GPU computing architectures.
The course activities are consistent with the professional profiles proposed by the Mathematical Engineering master course in relation to the acquisition of programming skills for complex computing structures and the solution of theoretical models.

Learning outcomes

At the end of the course the student should be able to:

1) Know the basic aspects related to the analytical and numerical solutions of the proposed models.
2) Use parallel computing codes related to the models proposed during the course.
3) Propose solutions related to problems similar to the models proposed during the course.
4) Understand technical-scientific texts on related topics.

Content:

Wave equations analytical and numerical methods, elastodynamic equations analytical and numerical methods.
Diffusion equations analytical and numerical methods.
Advection-reaction-diffusion systems analytical and numerical methods.
Introduction to parallel computing architectures:
Shared memory systems, distributed memory systems and GPU computing.
Models for performance evaluation Speedup, Efficiency and Amdahl's law.
Introduction to Linux/Unix operating systems and scheduling for HPC applications.
Message passing interface (MPI) programming
MPI basic notions, point-to-point communications, collective communications

Pre-requisites:

Basic knowledge of mathematical analysis, numerical analysis and scientific programming

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of L'Aquila   |   Language: English   |   

Unit Coordinator: Federica Di Michele

Aims:

The course aims to introduce students to the study and modelling of natural disasters using artificial intelligence techniques. After an initial introduction to the programming language used (Python), the student learns how to prepare a dataset and how to build, validate and optimise machine learning based models. The concepts of supervised, semi-supervised and unsupervised machine learning are introduced and the basics of building neural network-based models for image recognition (convolution neural network (CNN)) and signal analysis (recurrent neural network (RNN) ) are provided.
All datasets used during the laboratory phase are related to seismic and environmental risk assessment and mitigation. 

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of L'Aquila   |   Language: English   |   

Unit Coordinator: Marcello Di Risio

Aims:

This course aims to give a general overview of numerical modeling approaches to reproducing water-related natural hazards. Analytical solutions are outlined for validation/verification purposes. Theoretical aspects will be then applied to a series of test cases, in particular related to (i) tsunami generation, propagation, and interaction with water body boundaries, (ii) wind driven waves propagation and interaction with coastal and harbour structures, (iii) river flow and (iii) detailed simulation (i.e. CFD) of wave-structure interaction. At the end of the course the students will be able to design and perform numerical investigations in typical real cases within the frame of water-related hazard assessment studies.

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: Hamburg University of Technology   |   Language: English   |   

Unit Coordinator: Dr. Jens-Peter M. Zemke

Content:

• Basics: analogy; layout of neural nets, universal approximation, NP-completeness
• Feedforward nets: backpropagation, variants of Stochastistic Gradients
• Deep Learning: problems and solution strategies
• Deep Belief Networks: energy based models, Contrastive Divergence
• CNN: idea, layout, FFT and Winograds algorithms, implementation details
• RNN: idea, dynamical systems, training, LSTM
• ResNN: idea, relation to neural ODEs
• Standard libraries: Tensorflow, Keras, PyTorch
• Recent trends

Reading list:

Skript
Online-Werke:

• http://neuralnetworksanddeeplearning.com/
• https://www.deeplearningbook.org/

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: Hamburg University of Technology   |   Language: English   |   

Unit Coordinator: Martin Burger

Aims:

This course gives an introduction to mathematical and numerical methods of medical imaging, where particular focus is placed on computer(ized) tomography (CT).

Content:

• X-rays
• Radon transform
• Back projection
• Central slice theorem
• Filtered back projection formula
• Discrete image reconstruction
• Algebraic reconstruction techniques
• Kernel-based image reconstructions

Pre-requisites:

Analysis, Linear Algebra, Numerical Analysis

Reading list:

• C.L. Epstein: Introduction to the Mathematics of Medical Imaging. Second Edition. SIAM, Philadelphia, 2008. 
• T.G. Feeman: The Mathematics of Medical Imaging. A Beginner's Guide. 2nd edition, Springer, New York, 2015.
• S. Helgason: The Radon Transform. Second Edition. Birkhäuser, Boston, 1999.
• A. Iske: Approximation Theory and Algorithms for Data Analysis. Texts in Applied Mathematics, volume 68, Springer, Cham, 2018.
• F. Natterer: The Mathematics of Computerized Tomography. SIAM, Philadelphia, 2001.
• F. Natterer and F. Wübbeling: Mathematical Methods in Image Reconstruction. SIAM, Philadelphia, 2001.

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: Hamburg University of Technology   |   Language: English   |   

Unit Coordinator: Floran Grüner, Erika Garutti

Aims:

At the end of the course, students will be familiar with modern methods in medical imaging and with the fundamental techniques for radiation-therapy. They will have learned the physics limits and how physics can bring an added value by pushing the limits further. During the journal club, these topics will be analyzed in view of the most modern developments in the fields. Students will also learn how to structure and how to discuss a scientific publication. 

Content:

• PET
• SPECT
• CT 
• Multimodal imaging
• Spatial resolution and sensitivity in imaging of tumorous tissue and/or medical diagnostic agents

Reading list:

• J.L. Prince and J.M. Links: Medical imaging: signals and systems, Prentice Hall, 2006.
• C.Grupen and I.Buvat: Handbook of Particle Detection and Imaging;
• W.R.Leo: Techniques for Nuclear and Particle Physics Experiments, Springer;

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: Hamburg University of Technology   |   Language: English   |   

Unit Coordinator: Marko Lindner

Aims:

Students are able to characterize and compare diffusion equations, explain elementary methods of image processing, explain methods of image segmentation and registration, sketch and interrelate basic concepts of functional analysis.

Content:

• Basic methods of image processing
• Smoothing filters
• The diffusion / heat equation
• Variational formulations in image processing
• Edge detection
• De-convolution
• Inpainting
• Image segmentation
• Image registration

Pre-requisites:

• Analysis: partial derivatives, gradient, directional derivative
• Linear Algebra: eigenvalues, least squares solution of a linear system

Reading list:

Will be announced in the lecture

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: Hamburg University of Technology   |   Language: English   |   

Unit Coordinator: Tobias Knopp

Aims:

After successful completion of the module, students are able to describe reconstruction methods for different tomographic imaging modalities such as computed tomography and magnetic resonance imaging.

They know the necessary basics from the fields of signal processing and inverse problems and are familiar with both analytical and iterative image reconstruction methods.

The students have a deepened knowledge of the imaging operators of computed tomography and magnetic resonance imaging.

Content:

Overview about different imaging methods, signal processing, inverse problems, computed tomography, magnetic resonance imaging, compressed sensing, magnetic particle imaging.

Pre-requisites:

Basic knowledge in linear algebra, numerics, and signal processing

Reading list:

• Introduction to the Mathematics of Medical Imaging; C. L.Epstein; Siam, Philadelphia, 2008
• Medical Image Processing, Reconstruction and Restoration; J. Jan; Taylor and Francis, Boca Raton, 2006
• Principles of Magnetic Resonance Imaging; Z.-P. Liang and P. C. Lauterbur; IEEE Press, New York, 1999

ECTS Credits: 6   |   Semester: 1   |   Year: 1   |   Campus: University of Côte d'Azur   |   Language: English

Unit Coordinator: Rémi Catellier

Aims:

The purpose of the course is to teach the basics of the theory of stochastic processes, which has become a standard tool in the modelling of biological neural networks.

The course will focus in particular on the Brownian motion, on stochastic calculus and on diffusion processes, with the integrate and fire model as a benchmark example.

Markov property and martingale theory will be also addressed in this framework, with possible extensions to some jump processes like those used in ion channel models.

Content:

•    Brownian motion
•    Stochastic Calculus
•    Diffusion Processes
•    Markov Property
•    Martingales
•    Integrate and fire model
•    Ion channel models

Pre-requisites:

Probability with measure theory

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of Côte d'Azur   |   Language: English

Unit Coordinator: Alice Guyon, Ingrid Bethus

Aims:

Neuronal and cognitive systems cannot be modeled without knowledge of the basics of Neurosciences, from the molecular to the integrated level, involved in cognition and behaviors.

The first part of the program focuses on elementary neurophysiology and neuroanatomy. What are the different subparts constituting the nervous system and what are their main roles? How are neurons constituted? How do they generate activity and communicate with other neurons?

The second part of the program explores, with an integrative perspective, the neurobiological basis for higher mental functions through several examples. Sensorimotor functions are at the root of all the other processes. So the study of feeding behaviors is a good way to learn about the bio-logic of elementary behaviors, starting from the physiology of the autonomic nervous system and ending with neuroethological issues. Learning and memory are the basic processes of higher mental functions and also hot topics with applications in many domains.

In addition to all these fundamentals, the course also explains the materials and methods used in cognitive neurosciences to obtain data at the different levels of organization of nervous, cognitive and behavioral systems. This course is taught by a teaching staff member of the Master Programme Mod4NeuCog at UCA.

Content:

• Neuronal and cognitive systems
• Neurophysiology and neuroanatomy
• Neurobiological basis for higher mental functions

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of Côte d'Azur   |   Language: English

Unit Coordinator: Sylvain Rubenthaler

Aims:

•    This course addresses the basic methods used for simulating random variables and implementing Monte-Carlo and Quasi Monte-Carlo methods.
•    Simulation of stochastic processes used in neuroscience, such as Brownian motion and solutions to stochastic differential equations, will be addressed.
•    The course will introduce sampling methods in finite dimension, discretization of diffusion processes, strong and weak errors.

Content:

• Simulation
• Monte-Carlo methodes
• Discretization schemes
• Error analysis

Pre-requisites:

Probability with measure theory, stochastic calculus, programming 

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of Côte d'Azur   |   Language: English

Unit Coordinator: Patricia Reynaud and Etienne Tanré

Aims:

This course aims at providing a deep understanding of the main stochastic models used in neurocognition (Markov Chains, Integrate and Fire, point processes) and addressed (at least for some of them) in the Stochastic Calculus course and to study their mathematical properties.

The pros and cons of each of them is discussed, especially in terms of the modeling and the statistical inference of real data.

This course is taught by a teaching staff member of the Master Programme Mod4NeuCog at UCA

Content:

•    Stochastic models in neurocognition, statistical inference
•    Statistical inference

Pre-requisites:

Statistics 

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of Côte d'Azur   |   Language: English

Unit Coordinator: François Delarue

Aims:

The purpose of this course is to address some mathematical and computational aspects of machine learning and of related large networks of artificial and biological neurons. 

The first part of the course will be an introduction to stochastic optimization, control and game theories. This includes stochastic gradient descents, stochastic gradient Langevin dynamics, dynamic programming principle, first order conditions, Hamilton-Jacobi equations and the notion of Nash equilibria. 

The second part will be dedicated to mathematical tools for large networks, including the concept of mean-field models from statistical physics. Illustration will be given in the framework of over-parametrized artificial networks and biological spiking neural networks. 

The third part will address computational aspects of some artificial networks, like generative adversarial networks and deep networks. This will include programming sessions.  

Content:

•           Stochastic optimisation and control 

•           Stochastic gradient descent and Langevin dynamics

•           Dynamic programming principle

•           Mean field models

•           Generative adversarial networks

•           Deep learning

Pre-requisites:

Probability with measure theory, optimization, stochastic calculus

ECTS Credits: 8   |   Semester: 2   |   Year: 1   |   Campus: Vienna University of Technology   |   Language: English   |   Code: DT0641

Unit Coordinator: Markus Faustmann, Claudia Blaas-Schenner

Aims:

Scientific Programming
  - formulate (certain) mathematical problems in algorithmic form,
  - explain the difference between imperative and object-oriented programming,
  - implement mathematical algorithms in Matlab, C, and C++,
  - present and explain own solutions, and
  - constructively discuss and analyze own solutions as well as those of other students.

Parallel Programming
  - understand and apply the main concepts of parallel programming
  - master the basic skills to write parallel programs using MPI and OpenMP
  - parallelize serial programs using basic features of MPI and OpenMP
  - be familiar with the components of an high-performance computing cluster
  - know the principles to take advantage of shared and distributed memory systems as well as accelerators and how to exploit the capabilities of modern high-performance computing systems

Content:

Scientific Programming:
  - Introduction to Matlab, C, and C++.
  - Representation of integer and floating point numbers.
  - Conditioning of given problems.
  - Computational cost of algorithms.
  - Variables and standard data types.
  - Pointers.
  - Loops and if-else.
  - Functions and recursion.
  - Call by value vs. call by reference.
  - Objects and classes (resp. structures),
  - Operator overloading, Inheritance.
  - Templates.
  - Visualization in MATLAB.
  - Programming exercises. 

Parallel Programming:
  - Basic features of parallel programming with MPI (Message Passing Interface) and OpenMP (Open Multi-Processing) using C
  - A look at CUDA to offload parts of the computation to GPUs
  - Students will do the hands-on labs directly on the Vienna Scientific Cluster, the high-performance computing facility of Austrian universities, and hence will learn about and get some experience in high-performance computing.  

Pre-requisites:

Basic skills in programming in C (e.g., as learnt during the lecture "Scientific Programming for Interdisciplinary Mathematics") as well as Linux command line and usage of an editor (vi or nano).

Reading list:

• Scientific programming in mathematics:

lecture notes

• Programming with MATLAB:

Otto and Denier, An Introduction to Programming and Numerical Methods in MATLAB

Brian Hahn, Essential MATLAB for Engineers and Scientists

Stormy Attaway, Matlab: A Practical Introduction to Programming and Problem Solving

•  Basics of Parallel Computing:

    Rauber, Rünger: Parallel programming. Second Edition, Springer 2013.

    Schmidt, Gonzalez-Dominguez, Hundt, Schlarb: Parallel Programming. Concepts and Practice. Morgan Kaufmann 2018.

ECTS Credits: 6   |   Semester: 2   |   Year: 1   |   Campus: Vienna University of Technology   |   Language: English   |   Code: DT0371

Unit Coordinator: Kevin Sturm, Phillip Baumann

Aims:

To understand, analyze, formulate and graphically or mathematically solve basic static and dynamic optimization problems. Students will know about the theory, the mathematically principles and various methods for an exact or iterative solution of optimization problems.

They can moreover differentiate between unconstrained and constrained optimization problems and they can select and apply the specifically appropriate solution methods. 

Content:

• Fundamentals of optimization: existence of minima and maxima, gradient, Hessian, convexity, convergence
• Unconstrained static optimization: optimality conditions, computer-aided optimization, line search methods, choice of the step length, principle of nested intervals, Armijo condition, Wolfe condition, gradient method, Newton method, conjugate gradient method, Quasi-Newton method, Gauss-Newton-method, trust region method, Nelder-Mead method
• Static optimization with constraints: equality and inequality constraints, sensitivity considerations, active set method, gradient projection method, reduced gradient method, penalty and barrier functions, sequential quadratic programming (SQP), local SQP, globalization of SQP

Pre-requisites:

• Analysis,
• Linear algebra,
• Numerical mathematics,
• Differential equations.

Reading list:

• lecture notes,
• J. Macki und A. Strauss: Introduction to Optimal Control Theory. New York, Springer, 1982.
• Jorge Nocedal, Stephen J. Wright: Numerical Optimization, Springer 2006

ECTS Credits: 6   |   Semester: 2   |   Year: 1   |   Campus: Vienna University of Technology   |   Language: English   |   Code: DT0639

Unit Coordinator: Lothar Nannen

Aims:

Being able to check if the numerical solution of an ordinary differential equation makes sense. Furthermore, depending on the inherent structure of the differential equation, selecting the appropriate numerical integrator.

Content:

Initial and boundary value problems for ordinary and partial differential equations, one-step and multi-step methods, adaptivity, structure perserving integrators, introduction to numerical methods for partial differential equations.

Pre-requisites:

Basic lectures of analysis, linear algebra and numerical analysis.

Reading list:

Lecture notes will be provided

ECTS Credits: 7   |   Semester: 2   |   Year: 1   |   Campus: Vienna University of Technology   |   Language: English   |   Code: DT0640

Unit Coordinator: Markus Faustmann

Aims:

Being able to solve stationary partial differential equations numerically, analyse the quality of numerical solutions, select proper methods and implement them in a computer program.

Content:

- Variational formulations
- Sobolev spaces, H(div), H(curl)
- Finite element spaces (h, p, hp)
- Mixed formulations,
- Discontinuous Galerkin Methods,
- Time-dependent problems

Pre-requisites:

Applied Mathematics Foundations

Reading list:

• Lecture notes,
• Dietrich Braess: Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, Cambridge University Press, 2007
• Cleas Johnson: Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge Univ. Press, 1987, Dover 2009
• Susanne Brenner & Ridgway Scott: The Mathematical Theory of Finite Elements, Springer 2008
• Alexandre Ern & Jean-Luc Guermond: Theory and Practice of Finite Elements, Springer, 2010 

ECTS Credits: 3   |   Semester: 2   |   Year: 1   |   Campus: Vienna University of Technology   |      |   Code: DT0663

Unit Coordinator: Innes Institue

Aims:

Learn German at level A1 (speak, read, write)

Content:

• Learn German at level A1 (speak, read, write)
• Social & cultural program, waltz workshop

Reading list:

Lecture notes by Innes

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of L'Aquila   |   Language: English   |   Code: DT0113

Unit Coordinator: Corrado Lattanzio

Aims:

LEARNING OBJECTIVES.
The course aims at providing advanced mathematical notions used in the field of (applied) mathematical analysis and their applications to a variety of topics, including the basic equations of mathematical physics and some current research topics about linear and nonlinear partial differential equations.
Those objectives contribute to the learning goals of the entire course of studies, as the inner coherence of the master degree in Mathematics was verified at the time of the planning of the master program.
LEARNING OUTCOMES.
At the end of the course, the student should:
1. know the advanced mathematical notions used in the field of (applied) mathematical analysis, as measure theory, Sobolev Spaces, distributions, and their applications to the theory of linear and non-linear partial differential equations;
2. understand and be able to explain thesis and proofs in advanced mathematical analysis;
3. have strengthened the logic and computational skills;
4. be able to read and understand other mathematical texts on related topics.

Content:

Distributions. Locally integrable functions. The space of test function D(Ω). Distributions associated to locally integrable functions. Singular distributions. Examples. Operations on distributions: sum, products times functions, change of variables, restrictions, tensor product. Differentiation and his properties; comparison with classical derivatives. Differentiation of jump functions. Partition of unity. Support of a distribution; compactly supported distributions. Fourier transform and tempered distributions. Convolution between distributions and regularization of distributions. Denseness of D(Ω) in D′(Ω).

Sobolev spaces. Definition of weak derivatives and his motivation. Sobolev spaces Wk,p(Ω) and their properties. Interior and global approximation by smooth functions. Extensions. Traces. Embeddings theorems: Gagliardo–Nirenberg–Sobolev inequality and embedding theorem for p < n. Hölder spaces. Morrey inequality. Embedding theorem for p > n. Sobolev inequalities in the general case. Compact embeddings: Rellich–Kondrachov theorem, Poincaré inequalities. Embedding theorem for p = n. Characterization of the dual space H-1.

Second order parabolic equations. Definition of parabolic operator. Weak solutions for linear parabolic equations. Existence of weak solutions: Galerkin approximation, construction of approximating solutions, energy estimates, existence and uniqueness of solutions.

First order nonlinear hyperbolic equations. Scalar conservation laws: derivation, examples. Weak solutions, Rankine-Hugoniot conditions, entropy conditions. L1 stability, uniqueness and comparison for weak entropy solutions. Convergence of the vanishing viscosity and existence of the weak, entropy solution. Riemann problem.

Measures. System of sets, Positive Measures, Outer Measures, Construction of Measures, Signed Measures, Borel and Radon Measures
Integration. Measurable Functions, Simple Functions, Convergence Almost Everywhere, Integral of Measurable Functions, Convergences of Integrals, Fubini-Tonelli Theorems.
Differentiation. The Radon-Nikodym Theorem, Differentiation on Euclidean space, Differentiation of the real line.
Radon measures and continuous functions. Spaces of continuous functions, Riesz Theorem.

Pre-requisites:

Basic notions of functional analysis, functions of complex values, standard properties of classical solutions of semilinear first order equations, heat equation, wave equation, Laplace and Poisson's equations.

Reading list:

- L. Ambrosio, G. Da Prato, A. Mennucci. Introduction to measure theory and integration. Edizioni della Normale.

- L. Ambrosio, N. Fusco, D. Pallara. Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs.

- V.I. Bogachev. Measure theory, Volume I, Springer.

- H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext, Springer.

- P. Cannarsa, T. D’Aprile. Introduction to Measure Theory and Functional Analysis. Springer.

- C.M. Dafermos. Hyperbolic Conservation Laws in Continuum Physics, Springer.

- L.C. Evans. Partial Differential Equations. Graduate Studies in Mathematics, Vol. 19, AMS.

- L. Evans, R. Gariepy. Measure Theory and Fine Properties of Functions, CRC Press. Revised Edition.

- G.B. Folland. Real analysis: Modern techniques and their applications. New York Wiley

- G. Gilardi. Analisi 3. McGraw–Hill.

- L. Grafakos, Classical Fourier Analysis. Springer.

- V.S. Vladimirov. Equations of Mathematical Physics. Marcel Dekker, Inc

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of L'Aquila   |   Language: English   |   

Unit Coordinator: Simone Fagioli

Aims:

• To learn the basics in the mathematical modelling of population dynamics.
• To provide a mathematical description of ODE models in population dynamics and the interpretation of the qualitative behaviour of the solutions
• To get the basic notions in mathematical models in epidemiology and reaction kinetics.
• To learn the mathematical modelling of population models in heterogeneous environment, described by partial differential equations.
• To deal with advanced models in biology such as chemotaxis models and structured dynamics equations.
• To get a sound background in reaction diffusion phenomena, Turing instability, and pattern formation.

Content:

Continuous Population Models for Single Species. Continuous Growth Models. Delay models. Linear Analysis of Delay Population Models: Periodic Solutions.
Continuous models for Interacting Populations. Predator-Prey Models: Lotka-Volterra Systems. Realistic Predator–Prey Models. Competition Models: Principle of Competitive Exclusion. Mutualism or Symbiosis Time-space dependent models: PDEs in biology. Diffusion equations. Diffusion and Random walk. The gaussian distribution. Smoothing and decay properties of the diffusion operator. Nonlinear diffusion- Reaction–diffusion models for one single species. Diffusive Malthus equation and critical patch size. Travelling waves. Fisher–Kolmogoroff equation.
Reaction–diffusion systems. Multi species waves in Predator-Prey Systems. Turing instability and spatial patterns.
Chemotaxis modelling. Diffusion vs. Chemotaxis: stability vs. instability. Diffusion vs. Chemotaxis: stability and blow–up. Chemotaxis with nonlinear diffusion. Models with maximal density
Matematical models for tumor growth.
Structured population dynamics. An example in ecology: competition for resources. Continuous traits. Evolutionary stable strategy in a continuous model.

Pre-requisites:

Basic calculus and analysis (differential and integral calculus with functions of many variables).

Ordinary differential equations.

Basics in finite dimensional dynamical systems.

Elementary methods for the solution of linear partial differential equations (separation of the variables).

Reading list:

Murray - Mathematical Biology I&II

Lecturers lecture notes

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of L'Aquila   |   Language: English   |   

Unit Coordinator: Marco Di Francesco, Antonio Esposito

Aims:

At the end of the course, the student will be familiar with multi-agent systems models of discrete type (both deterministic and stochastic), with their meso-scopic formulation, with their continuum formulation, both of first and second order, and with their formulation on a graph. The students will acquire the mathematical techniques to solve those models suitably and will be able to use those models in various interdisciplinary applications and to adapt them to specific problems in contexts of interest in health-care systems such as diagnostics and imaging, neural networks, genetics, epidemiology, dynamic data management, and biological aggregation phenomena in physiology.

Content:

• Discrete particle systems for interacting agents. Models with external field. Models with nonlocal aggregation/repulsion forces. Models with alignment, self-propulsion and friction. Swarms models (Vicsek) . Opinion models (Sznajd, Krause). Examples of asymptotic behaviour. The stochastic case.
• Control for discrete models. Mean-field games. Application to optimisation problems. Many species models and models with species transitions. Applications in genetics, imaging, and data science.
• Complementary topics of abstract measure theory. Measure topologies. Transport of measures.
• Mesoscopic models of Vlasov type. Derivation as mean field limits from discrete particle models. Formal derivation of continuum second order models. Derivation of first order models in friction dominated regimes.
• Derivation of linear and nonlinear diffusion models from particle systems. Existence of solutions to the nonlinear diffusion equation. Asymptotic self-similar behavior. 
• Aggregation-diffusion equations. Existence of solutions with linear diffusion. Existence in the  diffusion-less case and formation of clusters in finite time. Stationary states with quadratic diffusion. The case of many species. Application to epidemiology.
• Introduction to graph modelling of nonlocal type. Applications to neural networks.
• Probabilistic label-switching models and applications.

Pre-requisites:

Ordinary differential equations, real and functional analysis

Reading list:

Lecture notes will be provided.

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of L'Aquila   |   Language: English   |   

Unit Coordinator: Raffaele D'Ambrosio

Aims:

Learning objectives.
The aim of the course is to provide knowledge and skills in the field of numerical modelling of Health Care systems, creating an effective bridge between the understanding of the continuous model, its qualitative properties and their counterparts in the discrete.
Learning outcomes.
At the end of the course, the student should
•⁠  ⁠have a deep knowledge and understanding of the most relevant computational techniques for the treatment of models in the Health Care  field, together with aspects related to their implementation in accurate and efficient mathematical software;
•⁠  ⁠demonstrate the ability to evaluate the most appropriate discretization in relation to the problem to be solved and ability in theoretical analysis and mathematical software design;
•⁠  ⁠demonstrate ability to read and understand other texts on related topics.

Content:

Introduction to mathematical models for Health Care based on ordinary, stochastic and partial differential equations;
For each theme, the presentation will be twofold: presenting the continuous model, its qualitative properties and, jointly, presenting the most appropriate numerical approach to the approximation of solutions and the discrete conservation of the properties of the model. 
Methods that preserve the positivity of solutions have particular relevance in this field.
In addition, advanced techniques of numerical linear algebra and optimization for large linear systems arising from semi-discretization of PDEs will be presented.

Pre-requisites:

Basic knowledge of Numerical Analysis, differential equations, linear algebra.

Reading list:

• F. Brauer et al., Mathematical Epidemiology, Springer-Verlag (2008).
• J. D. Lambert, Numerical methods for ordinary differential systems: the initial value problem, John Wiley (1991).
• ⁠A. Quarteroni, Numerical Models for Differential Problems, Springer (2017).
• ⁠E. Isaacson, H.Keller, Analysis of numerical methods, Dover Publications (1994).
• ⁠W. Hundsdorfer, J. G. Verwer - Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations
• ⁠D.J. Higham and P. E. Kloeden, An Introduction to the Numerical Simulation of Stochastic Differential Equations, SIAM, 2021.
• Walter Gander, Martin J. Gander, Felix Kwok. Scientific Computing An introduction using Maple and MATLAB. Springer.

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of L'Aquila   |   Language: English   |   

Unit Coordinator: Alessandro Borri

Aims:

Systems biology is an emerging research area, which aims at providing mathematical models helping to understand the dynamic interactions occurring within and among cells. This course provides the basic mathematical tools to model and analyze gene transcription as well as biochemical reaction networks: the most important network motifs are investigated exploiting both deterministic and stochastic approaches.

Content:

• Review of basic concepts of biology and probability; deterministic vs. stochastic approach.
• Transcription factors and transcription networks; gene expression: transcription and translation; separation of time scales and the Quasi-Steady-State Approximation (QSSA); activators and repressors; Hill and logic input functions; dynamics of simple regulation and switching production in ordinary differential equation (ODE) form; response time; mass action kinetics and conservation equation; the Michaelis-Menten kinetics; cooperativity; dependent and independent bindings; multi-dimensional input functions.
• Network Motifs: real and randomized networks; the Erdos-Renyi (ER) model; graph properties; negative autoregulation (NAR) and stability analysis; three-node network motifs: feedback loops (FBLs) and feedforward loops (FFLs); coherent and incoherent FFLs; sign-sensitive delay, output pulse generation, speed up of response time, biphasic response curves in I1-FFL; review of the other types of FFL; examples: Arabinose system, Galactose system, flagella motor genes in E. Coli.
• Introduction to Numerical Simulation in Systems Biology; MATLAB: introduction and basic notions; integration of non-stiff differential equations via ode45; randomized computation, Monte Carlo simulation, construction of randomized networks (E-R models) and digraphs.
• General Transcription and Interaction Networks: Single-Input Module (SIM), Multi-Output Feedforward Loop, LIFO and FIFO temporal programs, bi-fans and Densely Overlapping Regulons (DORs), interlocked FFLs; positive autoregulation (PAR) and response slow-down; two-node positive feedback: double-positive (lock-on) and double-negative (toggle-switch) circuits; three-node motives; protein-protein interaction (PPI) networks and hybrid motifs; network motifs in real-world networks: food webs, social networks, electrical circuits, neuronal networks. Examples: sporulation of bacterium Bacillus subtilis, phosphorylation, response to pain stimuli in humans.
• Biological Oscillations: the role of negative feedback and delay; time-separation, feedback strength and cooperativity; noise-induced and delay oscillations; bistability and hysteresis; robustness in biological circuits; models at a higher level of organization: tissues and organs; the glucose-insulin feedback loop.
• The stochastic approach in Systems Biology: motivation, Chemical Reaction Networks, Continuous-Time Markov Chains; the Chemical Master Equation (CME) and its properties, characterization of the CME stationary distribution, the macroscopic equation, one-step processes; stochastic simulation in Systems Biology, inverse sampling method, Gillespie Stochastic Simulation Algorithm (SSA); examples.
• Mesoscopic models and noise: extrinsic vs intrinsic noise, bursty reactions, the Chemical Langevin Equation, the Wiener Process, the Euler-Maruyama integration of Stochastic Differential Equations (SDEs).
• Biological and biomedical examples: metabolism, tumors, epidemics, pharmacokinetics and pharmacodynamics (PKPD), environmental pollution, ecology.

Pre-requisites:

Basic notions of mathematical analysis, dynamical systems and probability.

Reading list:

- Uri Alon, An introduction to systems biology: design principles of
biological circuits, CRC press, 2019.
- E. Klipp, W. Liebermeister, C. Wierling, and A. Kowald, Systems biology: a textbook, John Wiley & Sons, 2016.
- J. D. Murray, Mathematical biology, 3rd edition. New York: Springer, 2001.

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: Vienna University of Technology   |   Language: English

Unit Coordinator: Herbert Steinrück

Pre-requisites:

• Partial differential equations,
• Fluid mechanics,
• Numerical methods for fluid mechanics

Reading list:

• Lecture notes,   
• P. G. Drazin, Introduction to Hydrodynamic Stability, Cambridge University Press, Cambridge (2002)  
• S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Oxford University Press (1961) P. G. Drazin, Introduction to Hydrodynamic Stability, Cambridge University Press (2002) 

Unit Title: Computational fluid dynamics

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: Vienna University of Technology

Unit Coordinator: Manuel Garcia-Villalba

Aims:

To numerically simulate real fluid mechanical problems as well as to give an overview of the methods used in private and educational facilities.

Pre-requisites:

Partial differential equations, fluid mechanics

Reading list:

• Lecture notes,   
• C. A. J. Fletcher: Computational techniques for fluid dynamics, vol. 1, Springer (1991)
• J. J. Ferziger, M. Peric: Computational methods for fluid dynamics, Springer (2002)

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: Vienna University of Technology   |   Language: English

Unit Coordinator: Anton Arnold

Aims:

Students will be able to specify and apply a selection of PDE-applications in natural sciences and technology, and to discuss them from a mathematics perspective.

They are able to describe the modelling assumptions or restrictions, as well as their essential analytical and numerical properties.

Content:

• Fluid dynamical models (Euler, Navier-Stokes, vortex models),
• Traffic flow models,
• Theory of elasticity,
• Hyperbolic conservation laws,
• Image processing models (nonlinear diffusion filter, shock filter),
• Models for pattern formation (reaktion-diffusion equations, Turing instability),
• Evolution of thin films,
• Collective behavior (kinetic equations)

Pre-requisites:

• Partial differential equations;
• Basic knowledge in physics / mechanics is helpful

Reading list:

• Lecture notes,
• R. J. LeVeque : Numerical Methods for Conservation Laws, Birkhäuser (1990).
• A.J. Chorin, J.E. Marsden: A mathematical introduction to fluid mechanics, Springer (1990).
• C. Marchioro, M. Pulvirenti: Mathematical theory of incompressible nonviscous fluids, Springer (1994).
• G. Aubert, P. Kornprobst: Mathematical Problems in Image Processing, Springer, New York (2006).

ECTS Credits: 7   |   Semester: 1   |   Year: 2   |   Campus: Vienna University of Technology   |   Language: English

Unit Coordinator: Ulrich Schmid

Aims:

•  Introduction to semiconductor physics and devices 360.241

Students will be able to have a basic understanding of the underlying physics involved in the operation of semiconductor devices and will have a consistent base level of comfort and familiarity with the topic before they move to more advanced courses. The primary focus of the course will be on the development of physics and concepts, however a basic introduction to the application of simulation and numerical calculation within the field of semiconductor devices will also be partially covered.

• Theory, modelling and simulation of MEMS & NEMS:

• Model the mechanical behavior of MEMS/NEMS by continuum mechanics.
• State and test the underlying assumptions of the theory of linear elasticity.
• Describe the interaction between MEMS/NEMS and fluids.
• Model the piezoelectric effect.
• Calculate the eigenmodes and eigenfrequencies of selected structures.
• Derive the differences between macroscopic and microscopic systems
• Predict the dynamics of resonators.
• Derive the basic theory of the method of finite elements.
• Explain the working principle of reference oscillators, mass und fluid sensors.
• Discuss novel concepts like phononic crystals and quantum MEMS/NEMS.
• Understand technical subject-specific technical terminology and critically evaluate relevant scientific publications.
• Use open source software for eigenmode analysis.

Content:

• Introduction to semiconductor physics and devices:

In this course we will learn how the quantum mechanics of solids and transport has been harnessed to build the Digital Age. We will explore the physics of semiconductors and semiconductor devices and the role they play in modern technology. We will also lay the ground-work for more advanced discussions of how computational techniques and simulation can be used to push knowledge in this field forward to new horizons.

* Basic Structure of Crystals and Solids, Quantum Mechanics of Solids, Physics of Semiconductors, Transport in Semiconductors, PN junction, PN diode, PN Junctions and Modern Technology (LEDs, Photovoltaics, etc.), MOS transistor, MOS capacitor, MOSFETs and Modern Technology (VLSI, IGFET Sensors, etc.), Metal/semiconductor interfaces, Heterostructures and Modern Technology (Schottky Diodes, 2D-FETs, etc.)

• Theory, modelling and simulation of MEMS & NEMS:

The design of micro- and nanoelectromechanical systems (MEMS/NEMS) is a highly interdisciplinary field which reflects in the variety of topics of this course. Starting from an introduction to continuum mechanics and piezoelectricity we investigate different aspects of the mechanics of basic MEMS/NEMS structures like membranes and beams. By understanding the interaction of MEMS/NEMS with their environment, we are able to understand the outstanding performance of MEMS/NEMS sensors for mass and fluid sensing. Another important aspect for the modelling of MEMS/NEMS is the representation of MEMS/NEMS with discrete lumped element models and we discuss the most important discrete models. For quantitative predictions often numerical methods need to be employed for which the FEM is the most known. We discuss the fundamental theory of the FEM and its limitations. Using the above theory, we study example applications like reference oscillators or fluid sensors. Additionally, we take a look at novel concepts like phononic crystals or quantum MEMS/NEMS. 

Pre-requisites:

• Some basic exposure to the concepts and equations in the physics of electromagnetism;
• (Vector) calculus,
• Differential equations especially. 

Reading list:

Lecture notes 

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: Vienna University of Technology   |   Language: English

Unit Coordinator: Josef Weinbub

Aims:

Students will be able to select and apply fundamental methods of scientific computing and to judge the challenges regarding computing time and implementation effort.

Furthermore, the students have solution skills for inter-disciplinary problems, are able to evaluate and analyze computational approaches, and are able to scientifically formulate and extensively analyze compute-intensive problems as well as develop appropriate approaches. 

Content:

• Computer Architectures
• Serial Optimization
• Numerical Derivatives and Integrals
• Finite Difference Discretization
• Numerical Linear Algebra
• Random Number Generation and Monte Carlo Methods
• Shared Memory Parallel Computing
• Algorithmic Complexity and Data Structures
• Mesh Generation and Visualization
• Software Engineering Principles for Scientific Computing

Pre-requisites:

Numerical mathematics, scientific programing.

Reading list:

Lecture notes

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: Autonomous University of Barcelona   |   Language: English

Unit Coordinator: Lluis Alseda Soler, Martin Hernan Campos Heredia, Judit Chamorro Servent, Susana Serna

Aims:

Analysis of case studies, and practice of team working dynamics and client-consultant relationship.

The cases will focus primarily on optimisation and logistics real problems, but may have a scope beyond the pure "decision making" setting.

The course will be organised and directed by a professor of the department of Mathematics and will include minicourses and presentations given by industrial collaborators and other departments' teaching staff.

Content:

Mathematical modelling, i.e. solving real-world problems by means of mathematics.

Pre-requisites:

Students must have mathematical and computational skills at the level of a science degree.

Reading list:

• Ch. Rousseau, Y. Saint-Aubin, 2008. Mathematics and Technology. Springer.
• P. Pevzner, R. Shamir, 2011. Bioinformatics for Biologists. Cambridge Univ. Press

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: Autonomous University of Barcelona   |   Language: English

Unit Coordinator: Albert Ruiz Cirera, Judit Chamorro Servent

Aims:

The objective of this course is to study and practise the paradigmatic problems  (Routing, Scheduling, Location, Packing), relevant in logistics, that lead to discrete and combinatorial optimisation models, intrinsically difficult in practice due to its huge input size. 

The modern metaheuristics for approximating the solutions of such problems, like Evolutionary Algorithms, Tabu Search, Particle Swarm, or Ant Colony will be introduced

Content:

• Combinatorial Algorithms for graphs and routing: Dijstra and A* algorithms.
• Optimisation on graphs.
• Deterministic optimization for nonlinear problems (constrained and non-constrained).
• Genetic Algorithms.
• Simulated Annealing.
• Ant colony optimisation algorithms.
• Particle swarm optimization.
• Neural Networks in optimization.
• Scheduling.
• Machine learning trough neural networks.

Pre-requisites:

Mathematical knowledge at the level of Science degree. Programming skills

Reading list:

• Judea Pearl, A* Algorithms and such: Heuristics: Intelligent Search Strategies for Computer Problem Solving, Addison-Wesley, 1984.
• Alfio Quarteroni, Riccardo Sacco, Fausto Saleri, Numerical Mathematics, , Texts in Applied Mathematicsm 37, Springer, 1991.
• Sean Luke, Essentials of Metaheuristics, 2009: http://cs.gmu.edu/∼sean/book/metaheuristics/
• David Beasley, David R. Bully and Ralph R. Martinz, An Overview of Genetic Algorithms (Part 1: Fundamentals and Part 2: Research Topics)
• S. Kirkpatrick, C. D. Gelatt Jr. and M. P. Vecchi, Optimization by Simulated Annealing, Science, May 1983, Vol. 220, no. 4598, pp. 671-680.
• William H. Press, Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery, Numerical Recipes in C.
• The Art of Scientific Computing (second edition)}, Cambridge University Press.
• Marco Dorigoa and Christian Blum, Ant colony optimizationtheory: A survey, Theoretical Computer Science 344 (2005) 243 - 278.
• Ronald L. Graham, Combinatorial Scheduling Theory
• R. Gary Parker, Deterministic Scheduling Theory, Chapman Hall.
• Peter Brucker, Scheduling Algorithms, Fourth Edition, Springer
• R.L. Graham, E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Khan, Optimization and approximation in deterministic sequencing and scheduling: a survey
• Peter Brucker, Scheduling Algorithms, Springer-Verlag, 2007, Berlin Heidelberg New York (ISBN 978-3-540-69515-8).
• Jean-Yves Potvin, Kate A. Smith, Artificial Neural Networks for Combinatorial Optimization
• Kate Smith, Neural Networks for Combinatorial Optimization: A Review of More Than a Decade of Research
• Kate Smith, Marimuthu Palaniswami and Mohan Krishnamoorthy. Neural Techniques for Combinatorial Optimization with Applications
• Lecture notes provided by the teacher.

ECTS Credits: 9   |   Semester: 1   |   Year: 2   |   Campus: Autonomous University of Barcelona   |   Language: English

Unit Coordinator: Juan-José Ramos Gonzalez

Aims:

To deepen in the modelling methods and quantitative techniques aimed to support the decision making process in logistics and supply chain management.

To practice the workflow from the recognition and analysis of a real case problem, to the identification of the feasible activities, and to the selection of good solutions.

Practical problems studied include demand forecasting, production planning, warehouse and inventory management, and transport networks.

Content:

• Introduction to decision making.
• Linear and integer programming, constraint programming, artificial intelligence methods in optimisation.
• Production planning.
• Scheduling problems (job sequencing, resource allocation).
• Heuristics in transport planning.

Reading list:

Main textbooks:

• Hartmurt Stadlert and Cristoph Kilger (Eds.) Supply Chain Management and Advanced Planning. Third Edition. Springer, 2005.
• Ioannis T. Christou. Quantitative Methods in Supply Chain Management. Models and Algorithms. Springer, 2012.
• H. Paul Williams. Model Building in Mathematical Programming. Wiley. 2013 (5th edition)

Further readings:

• Joseph Geunes, Panos M. Pardalos and H. Edwin Romeijn (Eds.) Supply Chain Management: Models, Applications, and Research Directions. Kluwer Academic Publishers, 2002. 
• F. Robert Jacobs, William L. Berry, D. Clay Waybark and Thomas E. Vollmann. Manufacturing Planning and Control for Supply Chain Management. McGraw-Hill, 2011 (6th edition).
• F. Robert Jacobs and Richard B. Chase. Operations and Supply Chain management. McGraw-Hill Irwing, 2011 (13 th edition).

ECTS Credits: 3   |   Semester: 1   |   Year: 2   |   Campus: Autonomous University of Barcelona   |   Language: English

Unit Coordinator: Ramon Anton Piera Eroles

Aims:

To introduce students to the knowledge, processes, skills, tools and techniques suitable for project management, such that the application of them satisfy the requirements specified for project development, and may have a significant impact on its success. Specifically: learning the terminology and basic concepts of project management area and understanding the relationship between logistics and supply chain management and project management.

Content:

• Introduction to Project Management
• System Development Cycle
• Feasibility Study
• Project Planning
• Graphs-based Programming Methods
• Cost Analysis
• Risk Management
• Project Control

Reading list:

• Heagney, Joseph. Fundamentals of Project Management, 5th edition. 2016.
• Martinelli, Russ, et al. Program Management for Improved Business Results, 2nd edition, 2014.
• Nicholas, John M. Project management for business and technology: principles and practice , 2nd edition. Prentice Hall, 2001.
• Nicholas, John M., Steyn, H. Project management for business and technology: principles and practice, 3rd edition. Elsevier, 2008.
• Nicholas, John M. and Steyn, H. Project management for engineering, business, and technology, 4th edition. Routledge, 2012.
• A Guide to the project management body of knowledge: (PMBOK® Guide), 6th edition. Project Management Institute, 2017.
• Lewis, James P. Fundamentals of project management: developing core competencies to help outperform the competition. Amacom, 2002.