Dipartimento di Matematica
Guido Castelnuovo

Lunedì 29 aprile alle ore 12:00, in Sala di Consiglio, si terrà il Seminario di Dipartimento del dott. Alessandro Alla, risultato vincitore di una procedura selettiva per una posizione da PA nel SC 01/A5, SSD MAT08.

Speaker: Alessandro Alla

Titolo: A computational framework for large-scale dynamics: model reduction, data-driven modeling and optimal control

Abstract: 
This talk provides an overview of my research interests which rely on the theoretical and computational aspects of optimal control problems, with a particular emphasis on the Hamilton-Jacobi-Bellman(HJB) equations, model order reduction, and data-driven modeling. Many complex mathematical models encountered in real-world scenarios pose challenges in numerical simulations due to their complexity and large scale. To tackle this, model order reduction replaces the original problem with a surrogate model, identifying a subspace that captures the essential dynamics of the underlying nonlinear Partial Differential Equations (PDEs) and projecting these PDEs onto that subspace. By reducing the problem dimension, the original nonlinear PDEs can be replaced by smaller systems of ordinary differential equations, enabling efficient and accurate solution of the approximate problem. Applications to Turing patterns and fluid dynamics will be shown. 

I have also engaged in data-driven modeling using classical data science techniques such as data mining, machine learning and bigdata. My focus is on discovering rigorous mathematical models behind experimental data such that we can use it to make predictions or reconstruct solutions for missing data within a required time frame. Nowadays, we can deal with a huge amount of data that describe unknown dynamical systems. Dynamic Mode Decomposition (DMD) is an example of data-driven modeling. Sparse optimization methods can also be employed to reconstruct nonlinear differential equations. Applications to TIM data will be provided.

Furthermore, I have delved into both infinite and finite horizon optimal control problems for nonlinear high-dimensional dynamical systems. Nonlinear feedback laws can be computed via the value function characterized as the unique viscosity solution to the corresponding Hamilton-Jacobi-Bellman (HJB) equation, stemming from the dynamic programming (DP) approach. However, the primary challenge lies in the curse of dimensionality due to its exponential growth in computational cost, making HJB equations solvable only in a relatively small dimension. To address this challenge, my contributions focus on the stationary HJB approach for high dimensional problems. I developed an accelerated method, coupling value iteration and policy iteration to enhance the computational efficiency of the numerical scheme. Subsequently, I shifted my focus to finite horizon optimal control problems and computed the value function using a DP algorithm on a tree structure algorithm (TSA) constructed by the time-discrete dynamics. In this way, there was no need to build a fixed space triangulation and to project on it. Applications to the control of PDEs will be presented.

To conclude, I will provide insights into my current works and outline future research directions.