# Applied Mathematics

Curriculum: HYRIS + ROSE

Term: 1st year, 1st Semester

CFU: 6

SSD: MATH-05/A

Duration and Schedule: available here

OBJECTIVES: To provide advanced mathematical tools that will be used throughout the rest of the program.

DESCRIPTION:
The course is divided into 1+3 chapters as follows.
1. Prerequisites. Complex numbers: cartesian, polar and exponential representations; properties and operations. Linear algebra: matrix operations (sum, multiplication, determinant); eigenvalues and eigenvectors. Calculus: differentiation and integration of N-variate real functions.
2. Optimization of N-variate functions. Free and constrained optimization of N-variate functions. Lagrange multipliers and KKT conditions. Optimization algorithms (Gradient, Newton, finite differences)
3. Ordinary Differential Equations (ODE) Scalar ODEs and system of ODEs. Analytic solutions of linear systems of ODEs (exponential matrix). Study of the harmonic oscillator (damped and with external force). Equilibria of linear and non-linear systems (linearization, Lyapunov’s function).
4. Function approximation and Fourier. Space of square-summable functions, orthonormal bases and Parseval’s identity, Fourier and Legendre expansions, interpolation and least squares approximation. Fourier transform, Dirac’s delta.
MATLAB will be used during the classes to provide examples of the discussed topics.

REQUIREMENTS:
• Basics of linear algebra
• Basics of multivariate calculus and ODE solution
• Elementary programming skills not necessary but highly welcome

REFERENCES:
Class notes made available during the course. For backup and further readings:
Optimization of N-variate functions (Ch. 1): J. Nocedal, S.Wright. Numerical Optimization. Springer;
Ordinary Differential Equations (Ch. 2): G. Teschl, Ordinary Differential Equations and Dynamical Systems, American Mathematical Society; Blanchard, Devaney, Hall. Differential Equations, Cengage Learning.
Function approximation, transforms (Ch. 3): A. Quarteroni, R. Sacco, F. Saleri. Numerical Mathematics. Springer; D. Kammler, A First Course in Fourier Analysis, Cambridge University Press.
MATLAB: MATLAB Primer (https://it.mathworks.com/help/pdf_doc/matlab/learn_matlab.pdf), MATLAB Programming Fundamentals (https://www.mathworks.com/help/pdf_doc/matlab/matlab_prog.pdf).
Italian-speaking students can also use this book:
Chapters 1, 2, 3: Analisi Matematica 2, M. Bramanti, C. Pagani, S. Salsa, Zanichelli ed.

ASSESSMENT:
The final grade will be given after a written exam over the content of the class (theory, exercises).
INSTRUCTOR: Massimiliano Martinelli: official webpage and CV and personal webpage

INSTITUTION: Consiglio Nazionale delle Ricerche – Istituto di Matematica Applicata e Tecnologie Informatiche “E. Magenes”  (CNR-IMATI)

E-mail:

Voice:  +39 0382 548234

BioI am a researcher at CNR – IMATI “E. Magenes” (Pavia). My research interests are: Tensor Decompositions: algorithms for Tensor-Train and Functional-Tensor-Train decompositions; applications of Tensor Decompositions to the multi-dimensional numerical integration and to the numerical solution of ODEs and PDEs. Isogeometric Analysis: algortihms for efficient quadrature of hierarchical isogeometric spaces; data structures and efficient quadrature algorithms for trimmed volumes in IgA. Development and optimization of scientific and high performance codes.