Ignacio Muga
YOU?
Author Swipe
View article: Deep Fourier Residual method for solving time-harmonic Maxwell's equations
Deep Fourier Residual method for solving time-harmonic Maxwell's equations Open
Solving PDEs with machine learning techniques has become a popular alternative to conventional methods. In this context, Neural networks (NNs) are among the most commonly used machine learning tools, and in those models, the choice of an a…
View article: Automatic stabilization of finite-element simulations using neural networks and hierarchical matrices
Automatic stabilization of finite-element simulations using neural networks and hierarchical matrices Open
Petrov–Galerkin formulations with optimal test functions allow for the stabilization of finite element simulations. In particular, given a discrete trial space, the optimal test space induces a numerical scheme delivering the best approxim…
View article: Learning quantities of interest from parametric PDEs: An efficient neural-weighted Minimal Residual approach
Learning quantities of interest from parametric PDEs: An efficient neural-weighted Minimal Residual approach Open
The efficient approximation of parametric PDEs is of tremendous importance in science and engineering. In this paper, we show how one can train Galerkin discretizations to efficiently learn quantities of interest of solutions to a parametr…
View article: Adaptive stabilized finite elements via residual minimization onto bubble enrichments
Adaptive stabilized finite elements via residual minimization onto bubble enrichments Open
The Adaptive Stabilized Finite Element method (AS-FEM) developed in Calo et. al. combines the idea of the residual minimization method with the inf-sup stability offered by the discontinuous Galerkin (dG) frameworks. As a result, the discr…
View article: A Deep Double Ritz Method (D<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e3266" altimg="si244.svg"><mml:msup><mml:mrow/><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>RM) for solving Partial Differential Equations using Neural Networks
A Deep Double Ritz Method (DRM) for solving Partial Differential Equations using Neural Networks Open
Residual minimization is a widely used technique for solving Partial Differential Equations in variational form. It minimizes the dual norm of the residual, which naturally yields a saddle-point (min–max) problem over the so-called trial a…
View article: Automatic stabilization of finite-element simulations using neural networks and hierarchical matrices
Automatic stabilization of finite-element simulations using neural networks and hierarchical matrices Open
Petrov-Galerkin formulations with optimal test functions allow for the stabilization of finite element simulations. In particular, given a discrete trial space, the optimal test space induces a numerical scheme delivering the best approxim…
View article: Neural control of discrete weak formulations: Galerkin, least squares & minimal-residual methods with quasi-optimal weights
Neural control of discrete weak formulations: Galerkin, least squares & minimal-residual methods with quasi-optimal weights Open
There is tremendous potential in using neural networks to optimize numerical methods. In this paper, we introduce and analyse a framework for the neural optimization of discrete weak formulations, suitable for finite element methods. The m…
View article: A Deep Double Ritz Method (D$^2$RM) for solving Partial Differential Equations using Neural Networks
A Deep Double Ritz Method (D$^2$RM) for solving Partial Differential Equations using Neural Networks Open
Residual minimization is a widely used technique for solving Partial Differential Equations in variational form. It minimizes the dual norm of the residual, which naturally yields a saddle-point (min-max) problem over the so-called trial a…
View article: A Deep Fourier Residual Method for solving PDEs using Neural Networks
A Deep Fourier Residual Method for solving PDEs using Neural Networks Open
When using Neural Networks as trial functions to numerically solve PDEs, a key choice to be made is the loss function to be minimised, which should ideally correspond to a norm of the error. In multiple problems, this error norm coincides …
View article: An adaptive superconvergent finite element method based on local residual minimization
An adaptive superconvergent finite element method based on local residual minimization Open
We introduce an adaptive superconvergent finite element method for a class of mixed formulations to solve partial differential equations involving a diffusion term. It combines a superconvergent postprocessing technique for the primal vari…
View article: Neural Control of Discrete Weak Formulations: Galerkin, Least-Squares and Minimal-Residual Methods with Quasi-Optimal Weights
Neural Control of Discrete Weak Formulations: Galerkin, Least-Squares and Minimal-Residual Methods with Quasi-Optimal Weights Open
There is tremendous potential in using neural networks to optimize numerical methods. In this paper, we introduce and analyse a framework for the neural optimization of discrete weak formulations, suitable for finite element methods. The m…
View article: Three-dimensional simulations of the airborne COVID-19 pathogens using the advection-diffusion model and alternating-directions implicit solver
Three-dimensional simulations of the airborne COVID-19 pathogens using the advection-diffusion model and alternating-directions implicit solver Open
In times of the COVID-19, reliable tools to simulate the airborne pathogens causing the infection are extremely important to enable the testing of various preventive methods. Advection-diffusion simulations can model the propagation of pat…
View article: Projection in negative norms and the regularization of rough linear functionals
Projection in negative norms and the regularization of rough linear functionals Open
In order to construct regularizations of continuous linear functionals acting on Sobolev spaces such as $W_0^{1,q}(Ω)$, where $1
View article: Data-Driven Goal-Oriented Finite Element Methods: A~Machine-Learning Minimal-Residual (ML-MRes) Framework
Data-Driven Goal-Oriented Finite Element Methods: A~Machine-Learning Minimal-Residual (ML-MRes) Framework Open
We consider the data-driven acceleration of Galerkin-based finite element discretizations for the approximation of partial differential equations (PDEs). The aim is to obtain approximations on meshes that are very coarse, but nevertheless …
View article: Isogeometric Residual Minimization Method (iGRM) with direction splitting preconditioner for stationary advection-dominated diffusion problems
Isogeometric Residual Minimization Method (iGRM) with direction splitting preconditioner for stationary advection-dominated diffusion problems Open
In this paper, we introduce the isoGeometric Residual Minimization (iGRM) method. The method solves stationary advection-dominated diffusion problems. We stabilize the method via residual minimization. We discretize the problem using B-spl…
View article: Data-Driven Finite Elements Methods: Machine Learning Acceleration of Goal-Oriented Computations
Data-Driven Finite Elements Methods: Machine Learning Acceleration of Goal-Oriented Computations Open
We introduce the concept of data-driven finite element methods. These are finite-element discretizations of partial differential equations (PDEs) that resolve quantities of interest with striking accuracy, regardless of the underlying mesh…
View article: Discretization of Linear Problems in Banach Spaces: Residual Minimization, Nonlinear Petrov--Galerkin, and Monotone Mixed Methods
Discretization of Linear Problems in Banach Spaces: Residual Minimization, Nonlinear Petrov--Galerkin, and Monotone Mixed Methods Open
This work presents a comprehensive discretization theory for abstract linear operator equations in Banach spaces. The fundamental starting point of the theory is the idea of residual minimization in dual norms, and its inexact version usin…
View article: An adaptive stabilized finite element method based on residual minimization.
An adaptive stabilized finite element method based on residual minimization. Open
We devise and analyze a new adaptive stabilized finite element method. We illustrate its performance on the advection-reaction model problem. We construct a discrete approximation of the solution in a continuous trial space by minimizing t…
View article: The Discrete-Dual Minimal-Residual Method (DDMRes) for Weak Advection-Reaction Problems in Banach Spaces
The Discrete-Dual Minimal-Residual Method (DDMRes) for Weak Advection-Reaction Problems in Banach Spaces Open
We propose and analyze a minimal-residual method in discrete dual norms for approximating the solution of the advection-reaction equation in a weak Banach-space setting. The weak formulation allows for the direct approximation of solutions…
View article: The Convection-Diffusion-Reaction Equation in Non-Hilbert Sobolev Spaces: A Direct Proof of the Inf-Sup Condition and Stability of Galerkin’s Method
The Convection-Diffusion-Reaction Equation in Non-Hilbert Sobolev Spaces: A Direct Proof of the Inf-Sup Condition and Stability of Galerkin’s Method Open
While it is classical to consider the solution of the convection-diffusion-reaction equation in the Hilbert space H 0 1 ( Ω ) {H_{0}^{1}(\Omega)} , the Banach Sobolev space W 0 1 , q ( Ω ) {W^{1,q}_{0}(\Omega)} , 1 …