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CMUC Numerical Analysis and Optimization PhD Seminar
23rd March 2025, Coimbra, Portugal |
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Program
Poisson Hamiltonian Neural Networks: Structure-Preserving Learning of Dynamical Systems In this talk, we introduce Poisson Hamiltonian Neural Networks (PHNNs) as an extension of Hamiltonian Neural Networks to better capture the dynamics of Poisson-Hamiltonian systems. By incorporating structure-preserving numerical methods, PHNNs can learn a wider range of dynamical systems beyond traditional symplectic models. We explored different training strategies, comparing Explicit Euler (EE) and Poisson-Hamiltonian Integrators (PHI). Our results showed that, while Euler-trained models offer better short-term accuracy, PHI-trained models stand out for their long-term stability and preservation of geometric structures. A hybrid approach - training with EE and testing with PHI - proved to be the best balance between accuracy and stability. These results highlight the potential of combining machine learning with geometric numerical methods to model complex dynamic systems without the need for explicit governing equations. 12:00-12:30 Diogo Soares On the number of shellable arrangements of pseudolines An arrangement of pseudolines is a colection of unbounded x-monotone simple curves in the euclidean plane where each pair of curves intersects exactly once. Levi introduced them in the late 1920s as a natural generalization of line arrangements, relaxing the requirement of straightness while maintaining their topological properties. These structures are all over combinatorics, yet finding a closed formula to enumerate the set of non-isomorphic pseudoline arrangements remains an open problem. In this talk, we present a recursive formula to count a specific subset of pseudoline arrangements, known as shellable arrangements, and provide some bounds and asymptotics for their enumeration. 12:30-13:00 Carlos Correia Forward and inverse problems of elastography Aiming at producing a transient Optical Coherence Elastography (OCE) technique that potentially detects neurodegeneration during the presymptomatic stage through the assessment of retinal biomechanics, we are developing a numerical algorithm that will allow mapping the biomechanical properties of the tissue from the measured displacements. Our method includes a viscoelastic numerical model (forward problem) comprising a mixed Finite Element Method (FEM) rooted in the weak formulation of the time-dependent linear elasticity equation and is coupled with an iterative optimization algorithm (inverse problem) based on a trust region method. The robustness of the numerical model, as well as the ability of the inverse algorithm to reconstruct simulated elastic moduli within few iterations were attested, paving the way for the reconstruction of biomechanical properties with real OCE data. 13:00-14:30 Lunch break 14:30-15:00 Afonso Costa Discontinuous Galerkin finite element method for the Kelvin-Voigt Viscoelastic Mathematical Model This presentation introduces a space-time numerical scheme for simulating viscoelastic materials modelled by the Kelvin-Voigt equation. We begin by generalizing the well-known discontinuous Galerkin finite element method (DG-FEM) for elliptic problems to our viscoelastic model with non-homogeneous Dirichlet conditions, and establish its fundamental properties—coercivity and boundedness of the bilinear form. Building on this framework, we develop a coupled DG-FEM in space and finite difference method (FDM) in time for the viscoelastic model. Stability and theoretical error estimates are studied, demonstrating second-order convergence in space and first-order convergence in time. Numerical simulations have been conducted to validate the theoretical error estimates, confirming them and supporting the accuracy and effectiveness of the proposed scheme. 15:00-15:30 Milene Santos Discontinuous Galerkin method for curved boundary domains The ever-increasing complexity of real-world applications has raised important challenges in the search for accurate and efficient numerical methods for solving partial differential equations. In particular, we are interested in solving boundary value problems in curved boundary domains considering the Discontinuous Galerkin (DG) method. The question that arises concerns the reduction of the order of convergence of numerical methods when considering the approximation of the domain by a polygonal mesh. In [1] we developed a strategy called DG-ROD (Reconstruction for Off-site Data) method, which is based on a polynomial reconstruction of the boundary condition imposed on the computational domain. In this talk, we present a study on the existence and uniqueness of the solution and derive error estimates for a boundary-value problem with homogeneous Dirichlet boundary conditions. We prove that, under certain regularity conditions on the solution, the DG--ROD solution exhibits an optimal convergence rate in the L2-norm. [1] Santos, M., Araújo, A., Barbeiro, S., Clain, S., Costa, R., Machado, G.J. (2024). Very High-Order Accurate Discontinuous Galerkin Method for Curved Boundaries with Polygonal Meshes. Journal of Scientific Computing 100, 66. 15:30-16:00 Coffee break 16:00-16:30 Augusto Fernandes A Keller-Segel-Flow model for a viscoelastic tissue: Numerical analysis and simulation Cancerous tissues exhibit increased rigidity relative to normal or adjacent tissues, a phenomenon attributable to alterations in the extracellular matrix (ECM). This increase in stiffness is attributed to elevated collagen levels and enhanced cross-linking within the ECM. Additionally, the activity of stromal cells, such as fibroblasts, plays a crucial role in this process by facilitating the remodeling and stiffening of the ECM. This stiffening has been demonstrated to play a role in cancer progression by influencing various aspects of cell behavior, including proliferation, invasion, and metastasis. This talk focuses on the modeling and numerical analysis of cell migration by chemical signals and interstitial flow in viscoelastic tissues, two crucial factors in tumor metastasis. We consider a nonlinear Keller-Segel model that includes an elliptic equation based on Darcy's law for fluid flow and a hyperbolic equation for the tissue displacement. Following the MOL approach, we propose a space discretization that can be seen simultaneously as a FDM and a fully discrete FEM and we prove that the method has second-order accuracy with respect to a discrete H1-norm. Numerical simulations confirm the sharpness of the error analysis. 16:30-17:00 João Marcelino Discontinuous Galerkin method for curved boundary domains A Bayesian Network (BN) is a probabilistic graphical model that represents a set of variables and their conditional dependencies using a directed acyclic graph (DAG). There are several ways to represent a BN, such as parent sets and characteristic imsets. The score-based approach consists of finding the BN that maximizes a score-based quality criterion. When the quality score criterion is equivalent and additively decomposable, the BN learning problem can be formulated as an Integer Linear Programming (ILP) problem. Besides BNs, there are other interesting graphical models, such as decomposable models and homogeneous decomposable models, in this talk we will show how these graphical models can also be learned by solving an ILP, and give some preliminary findings in that direction. Venue Room 2.4 do Departament of Mathematics da University of Coimbra, Portugal. Contact Adérito Araújo
Address:
E-mail: alma@mat.uc.pt
Supporting institutions
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