Sergio Gómez

Sergio Gómez, Chiara Perinati, Paul Stocker et al.

We present and analyze an embedded Trefftz discontinuous Galerkin method for reaction-diffusion problems on anisotropic meshes. The method is constructed by imposing a relaxed local Trefftz condition via an embedding into a tensor-product DG space, yielding a reduced global system while preserving the approximation properties of the underlying high-order discretization. We prove stability and quasi-optimality on anisotropic, possibly curved, quadrilateral elements, and derive anisotropic a priori error estimates. Numerical experiments for $h$- and $hp$-refinement, including curved-domain examples, validate the theoretical results.

Paola F. Antonietti, Mattia Corti, Sergio Gómez et al.

We investigate a two-state conformational conversion system and introduce a novel structure-preserving numerical scheme that couples a local discontinuous Galerkin space discretization with the backward Euler time-integration method. The model is first reformulated in terms of auxiliary variables involving suitable nonlinear transformations, which allow us to enforce positivity and boundedness at the numerical level. Then, we prove a discrete entropy-stability inequality, which we use to show the existence of discrete solutions, as well as to establish the convergence of the scheme by means of some discrete compactness arguments. As a by-product of the theoretical analysis, we also prove the existence of global weak solutions satisfying the system's physical bounds. Numerical results validate the theoretical results and assess the capabilities of the proposed method in practice.

Sergio Gómez, David Hewett, Andrea Moiola

We formulate, analyse, and implement a discontinuous Galerkin finite element method (DG-FEM) for the approximation of the solution of an elliptic boundary value problem in a domain with fractal boundary. We consider the case of the Poisson equation in the Koch snowflake domain with zero Dirichlet boundary conditions, but our methodology can be generalised to other cases. Rather than first approximating the snowflake domain by a polygonal "prefractal'' and then applying a standard DG-FEM on the prefractal, we define a DG-FEM on the snowflake itself, using a geometry-conforming mesh (a fractal tiling) consisting of fractal elements, each similar to the original snowflake. Fluxes across inter-element boundaries, which are fractal curves, are represented in a weak way by integrals over element subdomains. We show how, for local polynomial basis functions, these integrals can be evaluated exactly using the similarity of the elements. We prove well-posedness and quasi-optimality of the method, and provide a partial convergence analysis. We present numerical results for piecewise linear and piecewise quadratic basis functions, which demonstrate the effectiveness of the method.

Matteo Ferrari, Sergio Gómez

We propose a space-time isogeometric finite element method for the linear Schrödinger equation, and establish its unconditional stability through a matrix-based analysis. Although maximal-regularity splines in time provide higher accuracy per degree of freedom compared to piecewise continuous polynomials, the nonlocal support of the spline bases precludes the use of standard variational arguments in the stability proofs. To overcome this, we show that the resulting scheme is governed by a family of nearly Toeplitz system matrices and, by studying the condition number of these matrices, we prove that the family is weakly well-conditioned, which guarantees the unconditional stability of the method. Furthermore, the discrete scheme preserves mass and energy at the final time. Numerical experiments confirm our theoretical findings and illustrate the optimal convergence behavior of the scheme. Finally, we exploit an algebraic connection between our formulation and a recent first-order-in-time space-time isogeometric method for the wave equation to derive a complete matrix-based stability analysis for the latter.

Alberto Fernández, Sergio Gómez

Agglomerative hierarchical clustering can be implemented with several strategies that differ in the way elements of a collection are grouped together to build a hierarchy of clusters. Here we introduce versatile linkage, a new infinite system of agglomerative hierarchical clustering strategies based on generalized means, which go from single linkage to complete linkage, passing through arithmetic average linkage and other clustering methods yet unexplored such as geometric linkage and harmonic linkage. We compare the different clustering strategies in terms of cophenetic correlation, mean absolute error, and also tree balance and space distortion, two new measures proposed to describe hierarchical trees. Unlike the $β$-flexible clustering system, we show that the versatile linkage family is space-conserving.

Joel Z. Leibo, Julien Cornebise, Sergio Gómez et al.

This paper describes a framework for modeling the interface between perception and memory on the algorithmic level of analysis. It is consistent with phenomena associated with many different brain regions. These include view-dependence (and invariance) effects in visual psychophysics and inferotemporal cortex physiology, as well as episodic memory recall interference effects associated with the medial temporal lobe. The perspective developed here relies on a novel interpretation of Hubel and Wiesel's conjecture for how receptive fields tuned to complex objects, and invariant to details, could be achieved. It complements existing accounts of two-speed learning systems in neocortex and hippocampus (e.g., McClelland et al. 1995) while significantly expanding their scope to encompass a unified view of the entire pathway from V1 to hippocampus.