Mathematics - Numerical Analysis and Optimization

ECTS

SEMESTER

lectures

classes / seminars

practical work

integrative teaching

independent work

10h

20h

0h

0h

20 h

Language used

French

Course supervisor(s)

Key words

Numerical analysis, optimization, simulation, linear algebra, approximation, MATLAB

Prerequisites

Basic Mathematics (calculus, linear algebra)

This course is designed to provide a basic understanding of the study and analysis of numerical methods useful for engineering sciences. Various topics will be discussed in lectures and put into practice during tutorials in the Matlab environment. Applications on concrete problems will be developed in order to highlight the importance of mastering numerical simulation, from theory to practice, for the engineer.

Session 1: Solving nonlinear equations

Classical algorithms (dichotomy, secant, Newton, fixed point) in dimension 1 and their adaptation in higher dimension. Study of convergence and comparison of algorithms. Methods for accelerating convergence (Aitken, Steffensen).

Session 2: Interpolation and approximation of functions

Lagrange interpolation, Newton formulas (divided differences). Error formulas. Uniform and least squares approximation, cubic spline functions.

Session 3: Integration and numerical derivation

Composite methods of the Newton-Cotes type and Gauss methods. Error analysis. Numerical derivation and error formula. Application to the solution of partial differential equations by finite difference methods.

Sessions 4 and 5: Differential equations

Some theoretical reminders. Numerical solution of differential equations: one-step, explicit and implicit methods (Euler, Crank-Nicholson, Runge-Kutta). Multi-step methods (Adams and its variants). Step adjustment. Notion of stability and convergence. Shooting method for boundary problems.

Sessions 6 and 7: Solving linear systems

Direct methods: Gauss pivot, LU and Choleski factorization. QR factorization. Conditioning of a matrix and effect on the error. Iterative methods (Jacobi, Gauss-Seidel, relaxation, conjugate gradient with or without preconditioning). Convergence and comparison of the different methods.

Session 8 and 9: Optimization

Theoretical basis of optimization: existence of solutions, convexity, first and second order optimality conditions, Lagrange and Karush-Kuhn-Tucker multipliers. Unconstrained optimization, gradient and conjugate gradient methods. Constrained optimization: projection and penalization methods.

Levels

Description and operational verbs

Know

The fundamental aspects of numerical analysis and optimization for engineering and scientific calculus

Understand

Numerical simulations as a prediction tool, with its qualities as well as its limits

Apply 

Resolving the practical issues related to engineering development with algorithmic methods

Analyse 

Detect and deduce the properties of certain numerical phenomena using mathematical reasoning.

Summarise

Formulate and develop an answer to the problems posed, organize the results into a coherent, rigorous and clear whole.

Assess

Judge the relevance of a result and its veracity. Validate the correctness of a method and a reasoning.

Continuous assessment

Written test

Oral presentation / viva

Written report / project