Matteo Calafà

This work aims at presenting a discontinuous Galerkin (DG) formulation employing a spectral basis for two important models employed in cardiac electrophysiology, namely the monodomain and bidomain models. The use ... of DG methods is motivated by the characteristic of the mathematical solution of such equations which often corresponds to a highly steep wavefront. Hence, the built-in flexibility of discontinuous methods in developing adaptive approaches, combined with the high-order accuracy, can well represent the underlying physics. The choice of a semi-implicit time integration allows for a fast solution at each time step. The article includes some numerical tests to verify the convergence properties and the physiological behaviour of the numerical solution. Also, a pseudo-realistic simulation turns out to fully reconstruct the propagation of the electric potential, comprising the phases of depolarization and repolarization, by overcoming the typical issues related to the steepness of the wave front.

The angular coefficient method represents a valid and efficient strategy to estimate the distribution of molecules in ultra-high vacuum systems. The problem is described through a Fredholm integral equation of ... the second kind that is usually solved with standard numerical methods, e.g., the finite element method or the Nyström quadrature method. In this work, we aim to rigorously study the underlying integral equation in order to verify some fundamental mathematical properties and justify the application and behaviour of such numerical methods. In particular, we address to the general scenario where domains are not globally smooth. In such context, boundary corners entail poorly regular integral kernels, which in turn lead to non-Lipschitz solutions and require the adoption of non-standard analysis techniques. By introducing the concept of vacuum-connection, we can establish a methodology to prove the well-posedness of the underlying problem. Furthermore, the undermined regularity of the analytical solution and the consequent lower numerical convergence rate are proved analytically and verified through simple numerical tests.

We propose physics-informed holomorphic neural networks (PIHNNs) as a method to solve boundary value problems where the solution can be represented via holomorphic functions. Specifically, we consider the case of ... plane linear elasticity and, by leveraging the Kolosov–Muskhelishvili representation of the solution in terms of holomorphic potentials, we train a complex-valued neural network to fulfill stress and displacement boundary conditions while automatically satisfying the governing equations. This is achieved by designing the network to return only approximations that inherently satisfy the Cauchy-Riemann conditions through specific choices of layers and activation functions. To ensure generality, we provide a universal approximation theorem guaranteeing that, under basic assumptions, the proposed holomorphic neural networks can approximate any holomorphic function. Furthermore, we suggest a new tailored weight initialization technique to mitigate the issue of vanishing/exploding gradients. Compared to the standard PINN approach, noteworthy benefits of the proposed method for the linear elasticity problem include a more efficient training, as evaluations are needed solely on the boundary of the domain, lower memory requirements, due to the reduced number of training points, and $C^\infty$ regularity of the learned solution. Several benchmark examples are used to verify the correctness of the obtained PIHNN approximations, the substantial benefits over traditional PINNs, and the possibility to deal with non-trivial, multiply-connected geometries via a domain-decomposition strategy.

We illustrate a time and memory efficient application of Runge-Kutta discontinuous Galerkin (RKDG) methods for the simulation of the ultrasounds advection in moving fluids. In particular, this study addresses to ... the analysis of transit-time ultrasonic meters which rely on the propagation of acoustic waves to measure fluids flow rate. Accurate and efficient simulations of the physics related to the transport of ultrasounds are therefore crucial for studying and enhancing these devices. Starting from the description of the linearized Euler equations (LEE) model and presenting the general theory of explicit-time DG methods for hyperbolic systems, we then motivate the use of a spectral basis and introduce a novel high-accuracy method for the imposition of absorbing and resistive walls which analyses the incident wave direction across the boundary surface. The proposed implementation is both accurate and efficient, making it suitable for industrial applications of acoustic wave propagation.