These lectures will be devoted to the construction of numerical methods for partial differential equations (PDEs) based on splines. Splines are a ubiquitous tool in numerical analysis, and during this course, after an introduction to their main properties, we will define spline-based approximation techniques for differential problems. Starting from the simple  Laplace problems, we will also investigate the approximation of more complex problems, such as the inverse harmonic problem and the Stokes and Maxwell  equations. Moreover, if time permits, we will introduce the concept of adaptive approximation and discuss how to design adaptive spline approaches.

This course is devoted to inverse problems where the goal is to compute a fast reconstruction of the state of a physical system from available measurement observations and the knowledge of a physical PDE model. Due to their ill-posedness, these problems are often addressed with Bayesian approaches that consist in searching for the most plausible solution using sampling strategies of the posterior density. In view of their high numerical cost, especially in a high dimensional framework, reduced modeling of parametric PDEs has been proposed as a vehicle to reduce complexity and achieve near real time in the reconstructions. The course will give an overview of linear and nonlinear strategies for model reduction, and inverse problems. The concepts may be viewed as exploring alternatives to Bayesian inversion in favor of more deterministic notions of accuracy quantification.

This course will introduce students to solving  optimization problems. Because of the wide (and growing) use of optimization in science, engineering, economics, and industry, it is essential for students and  practitioners alike to develop an understanding of optimization algorithms. Knowledge of the capabilities and limitations of these algorithms leads to a  better understanding of their impact on various applications, and points the way to future research on improving and extending optimization algorithms and software.

We will study numerical methods for elliptic PDEs. Here, we will study finite element methods for Poisson and biharmonic problems. We will consider the analysis of such problems including error estimates. 

This course deals with the numerical treatment and analysis of Partial differential equations. We will consider advance methods for  optimal control problems. We will also consider optimization methods with example of monotonic algorithms. The rest of the course will cover numerical treatment of variational inequalities, finite element method and reduced basis method method. 

We introduce the concepts underlying ill-posedness and consider real-life problems leading to ill-posedness and the numerical techniques required to solve such problems. Particularly, we will consider as example an inverse problem that arises in the management of water resources and pertains to the analysis of surface water pollution by organic matter.

Parallel-in-time integration has been the focus of intensive research efforts over the past two decades due to the advent of massively parallel computer architectures and the scaling limits of purely spatial parallelization. The course is an introduction to the emerging field of time parallel time integration.

Numerical methods have been used to solve the one-dimensional advection-diffusion equation with constant coefficients. This partial differential equation is dissipative but not dispersive. We will study numerical methods applied to both scalar and system advection-diffusion problems.