Teacher

  • Christophe Vignat
  • Alexis Aravanis

Hours

24h CM

Control Signal Image Stat.

The objective of this course is to allow students to master the mathematical foundation necessary to tackle academic or industrial research subjects, in the fields of control, signal and image processing.

Optimization and differential equations

  • Optimization without and with constraints (gradient and Newton algorithms); Karush-Kuhn-Tucker conditions.
  • Non-linear ordinary differential equations (existence and uniqueness conditions).
  • Stability; Lyapunov functions.

Probability

  • Probabilized space, events, independence; random variables; sequences of events, extreme events, Borel-Cantelli.
  • Mean values, moments, correlation, function of a random variable, conditional expectation; distribution function, probability density, characteristic function, generating function of moments and their inversions; continuous and discrete classical distributions.
  • Markov, Chebyshev inequalities; convergence types (weak, in probability, almost certain), weak and strong laws of large numbers, dominated convergence theorem, central limit theorem, law of the iterated logarithm.

Algebra

  • Types of matrices: rectangular, square, symmetric, Hermitian, orthogonal, unitary, positive definite, Hankel, Toeplitz, circulant, VanderMonde, diagonal, triangular, block-diagonal.
  • Matrix algebras: product, vectorization, invariants (determinant, trace, rank), block multiplication; matrix factorizations: Jordan, diagonalizability, triangularizability, Schur, QR, Cholesky, LU, spectral decomposition, Singular Value Decomposition; least squares, pseudoinverse, normal equations.
  • Applications: matrix exponentials, systems of differential equations.