Abstract: In these five lectures, we present the main numerical methods used to solve optimal transport problems.

Lectures plan: 

Numbering below roughly corresponds to a little less than one lecture.

1. Introduction to optimal transport, OT. Monge’s problem and Kantorovich relaxation. Costs and different types of transport problems.

2. Discrete OT, linear programming, assignment, auction, and sinkhorn method. Example: Obstacle problem.

3. Continuous OT. Monge-Ampere PDE and discretization. Fluid flow formulation and discretization. Separable problems. Part I.

4. Continuous OT. Comparison. Part II.

5. Semi-discrete OT.

6. More semi-discrete OT and applications.

Abstract: Effective problem solving begins with precise problem formulation, highlighting the importance of the initial problem-setting phase. Without a clearly defined problem, identifying suitable tools and techniques for resolution becomes arduous and often futile. This transition from problem setting to problem solving is pivotal within the broader framework of knowledge advancement. Despite the remarkable progress of AI tools, they remain reliant on the groundwork laid by human intelligence. Mathematicians, leveraging their adeptness in discerning patterns and relationships within data and variables, play a crucial role during this phase.

This course will introduce fundamental mathematical concepts encompassing both traditional machine learning and scientific machine learning. The latter offers an optimal platform for the harmonious fusion of problem setting and problem solving, bolstered by profound domain expertise. Throughout the course, emphasis will be placed on a reference application centered around the development of a mathematical simulator for cardiac function.

Lectures Plan:

1. From Digital Models to Digital Twins (AQ): Basic concepts behind digital models (physics informed), AI and ML (data driven), scientific machine learning, and digital twins.

2. Machine Learning (AQ): Definitions, tasks, algorithms, strengths and weaknesses, basic elements of mathematical theory.

3. Scientific Machine Learning (AQ): Definition, goal, physics-informed machine learning for PDEs, examples.

4. Scientific Machine learning for the iHEART simulator - part I (FR): Overview of cardiac modelling, need for surrogate models, surrogate models in variable shape domains.

5. Scientific Machine learning for the iHEART simulator - part II (FR): Learning time-dependent models, applications to multiscale and multiphysics problems, learning space-time processes (Latent Dynamics Neural Networks).

Abstract: We present and discuss gradient-based optimization methods for solving minimization problems arising in machine learning applications. The methods analysed employ stochastic estimators of the  objective function's gradient. We will focus on the stochastic gradient method  and some of its widely used adaptive variants. We also give the main ingredients of machine learning models, including Neural Networks, and show the role played by automatic differentiation to compute the stochastic gradient. We finally introduce modern adaptive stochastic optimization methods, which have the potential to offer significant computational savings when training large-scale systems.

Lectures Plan:

1. Deterministic gradient method and its complexity.

2. ML models and the stochastic gradient.

3. Reducing the noise: stochastic gradients variants.

4. Adaptive stochastic optimization methods.

Abstract: The Virtual Element Method (VEM) is a technology introduced in 2013 for the discretization of partial differential equations that follows a similar paradigm to classical Finite Elements, but with important differences. By avoiding the explicit integration of the discrete shape functions and introducing an innovative construction of the stiffness matrixes, the VEM acquires interesting properties and advantages with respect to more standard methods. For instance, the VEM easily allows for general polygonal/polyhedral meshes, even non-conforming and with non-convex elements.

The present short course will be an introduction to VEM. In the first part we will introduce Virtual Elements taking the step from classical Finite Elements, focusing on a simple model problem. We will show the main features of the method such as explaining why the discrete space is "virtual", what are suitable degrees of freedom, and the trademark discretization approach of VEM based on projection plus stabilization. In the second part we will tackle more complex equations and show how more involved VEM spaces can be constructed and used to handle problems, for instance, in electromagnetism or fluid dynamics.