Partial Differential Equations. Numerical Analysis.

COMPETENCES and SKILLS - Acquiring theoretical knowledge that allows the student to develop original ideas, in a research context, knowing how to translate industrial needs into I+D+i projects in the Industrial Mathematics field. - Being able to explain the results, along with the acquired knowledge, to experts and non-experts. - Being able to get deeper into a subject in an autonomous way, which will be very useful to obtain a Ph.D. - Being able to get quantitative and qualitative information from experimental data using numerical techniques. - Knowing how to select the appropriate techniques to solve a specific problem.

Introduction and Basic Notions - Direct and inverse problems - Well and ill-posed problems - Existence and uniqueness of the solution - Stability Least squares - Motivation and general idea - Applications Regularization - Motivation and general idea - Tikhonov, Lardy and Landweber algorithms - Morozov's discrepancy principle Singular Value Decomposition - Theoretical background, meaning and properties - Noise filtering and data reconstruction - Linear systems and regularization - Extensions Computed Axial Tomography - Radon transform and sinogram - Methods: back projection and algebraic reconstruction Topological Derivative - Theoretical background - Defects detection - Methods: multifrequency and iterative - Applications

Methodology: - In-person classes. - Homeworks and presentations. Tutorials: The students can ask questions via e-mail or during classes.