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Publications
Publications (42)
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 and test spaces. In the context of neural networks, we can address this min–max approach by employing one networ...
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 approximation in terms of a problem-dependent energy norm. This ideal approach has two shortcomings: first, we need to...
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 and test spaces. Such min-max problem is highly non-linear, and traditional methods often employ different mixed...
There is tremendous potential in using neural networks to optimize numerical methods. In this paper, we introduce and analyze a framework for the neural optimization of discrete weak formulations, suitable for finite element methods. The main idea of the framework is to include a neural-network function acting as a control variable in the weak form...
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 with--or is equivalent to--the $H^{-1}$-norm of the residual; however, it is often difficult to accurately comp...
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 variable with an adaptive finite element method via residual minimization. Such a residual minimization procedure i...
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 main idea of the framework is to include a neural-network function acting as a control variable in the weak form...
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In order to construct regularizations of continuous linear functionals acting on Sobolev spaces such as \(W_0^{1,q}(\varOmega )\), where \(1<q<\infty \) and \(\varOmega \) is a Lipschitz domain, we propose a projection method in negative Sobolev spaces \(W^{-1,p}(\varOmega )\), p being the conjugate exponent satisfying \(p^{-1} + q^{-1} = 1\). Our...
In this paper, we propose a novel computational implicit method, which we call Isogeometric Residual Minimization (iGRM) with direction splitting. The method mixes the benefits resulting from isogeometric analysis, implicit dynamics, residual minimization, and alternating direction solver. We utilize tensor product B-spline basis functions in space...
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 pathogens in the air. We can represent the concentration of pathogens in the air by "contamination" propagating fr...
In this paper, we introduce a stable isogeometric analysis discretization of the Stokes system of equations. We use this standard constrained problem to demonstrate the flexibility and robustness of the residual minimization method on dual stable norms [16], which unlocks the extraordinary approximation power of isogeometric analysis [44]. That is,...
In order to construct regularizations of continuous linear functionals acting on Sobolev spaces such as $W_0^{1,q}(\Omega)$, where $1<q<\infty$ and $\Omega$ is a Lipschitz domain, we propose a projection method in negative Sobolev spaces $W^{-1,p}(\Omega)$, $p$ being the conjugate exponent satisfying $p^{-1} + q^{-1} = 1$. Our method is particularl...
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-spline basis functions. We then seek to minimize the isogeometric residual over a spline space built on a tensor pr...
We show that it is possible to obtain a linear computational cost FEM-based solver for non-stationary Stokes and Navier-Stokes equations. Our method employs a technique developed by Guermond & Minev [20], which consists of singular perturbation plus a splitting scheme. While the time-integration schemes are implicit, we use finite elements to discr...
We introduce the concept of machine-learning minimal-residual (ML-MRes) finite element discretizations of partial differential equations (PDEs), which resolve quantities of interest with striking accuracy, regardless of the underlying mesh size. The methods are obtained within a machine-learning framework during which the parameters defining the me...
We investigate a residual minimization (RM) based stabilized isogeometric finite element method (IGA) for the Stokes problem. Starting from an inf-sup stable discontinuous Galerkin (DG) formulation, the method seeks for an approximation in a highly continuous trial space that minimizes the residual measured in a dual norm of the discontinuous test...
We investigate a residual minimization (RM) based stabilized isogeometric finite element method (IGA) for the Stokes problem. Starting from an inf-sup stable discontinuous Galerkin (DG) formulation, the method seeks for an approximation in a highly continuous trial space that minimizes the residual measured in a dual norm of the discontinuous test...
We design and analyze a new adaptive stabilized finite element method. We construct a discrete approximation of the solution in a continuous trial space by minimizing the residual measured in a dual norm of a discontinuous test space that has inf–sup stability. We formulate this residual minimization as a stable saddle-point problem, which delivers...
This preprint has been published as
A machine-learning minimal-residual (ML-MRes) framework for goal-oriented finite element discretizations
https://doi.org/10.1016/j.camwa.2020.08.012
https://www.sciencedirect.com/science/article/abs/pii/S0898122120303199
We investigate the application of the residual minimization method (RM) to stabilize the non-stationary Stokes problem. We discretize the trial and test spaces with higher continuity B-spline basis functions from isogeometric analysis (IGA) on a regular patch of elements. We first consider the RM with IGA to stabilize H^1_0, L^2_0 formulation of th...
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 the residual measured in a dual norm of a discontinuous test space that has inf-sup stability. We formulate this...
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 in the Lebesgue {L^{p}} -space, {1<p<\infty} . The greater generality of this weak setting is natural when dea...
While it is classical to consider the solution of the convection-diffusion-reaction equation in the Hilbert space {H_{0}^{1}(\Omega)} , the Banach Sobolev space {W^{1,q}_{0}(\Omega)} , {1<q<{\infty}} , is more general allowing more irregular solutions. In this paper we present a well-posedness theory for the convection-diffusion-reaction equation i...
In this paper, we propose a novel computational implicit method, which we call Isogeometric Residual Minimization (iGRM) with direction splitting. The method mixes the benefits resulting from isogeometric analysis, implicit dynamics, residual minimization, and alternating direction solver. We utilize tensor product B-spline basis functions in space...
In this paper, we propose the Isogeometric Residual Minimization (iGRM) with direction splitting. The method mixes the benefits resulting from isogeometric analysis, residual minimization, and alternating direction solver. Namely, we utilize tensor product B-spline basis functions, and an alternating direction methods. We apply a stabilized mixed m...
We show existence and uniqueness of the outgoing solution for the Maxwell problem with an impedance boundary condition of Leontovitch type in a half-space. Due to the presence of surface waves guided by an infinite surface, the established radiation condition differs from the classical one when approaching the boundary of the half-space. This speci...
In some geological formations, borehole resistivity measurements can be simulated using a sequence of 1D models. By considering a 1D layered media, we can reduce the dimensionality of the problem from 3D to 1.5D via a Hankel transform. The resulting formulation is often solved via a semi-analytic method, mainly due to its high performance. However,...
We describe a numerical study to quantify the influence of tool-eccentricity on wireline (WL) and logging-while-drilling (LWD) sonic logging measurements. Simulations are performed with a height-polynomial-adaptive (hp) Fourier finite-element method that delivers highly accurate solutions of linear visco-elasto-acoustic problems in the frequency do...
We consider the discontinuous Petrov-Galerkin (DPG) method, wher the test
space is normed by a modified graph norm. The modificatio scales one of the
terms in the graph norm by an arbitrary positive scaling parameter. Studying
the application of the method to the Helmholtz equation, we find that better
results are obtained, under some circumstances...
We study the properties of a novel discontinuous Petrov Galerkin (DPG) method for acoustic wave propagation. The method yields Hermitian positive definite matrices and has good pre-asymptotic stability properties. Numerically, we find that the method exhibits negligible phase errors (otherwise known as pollution errors) even in the lowest order cas...
The phase error, or the pollution effect in the finite element solution of wave propagation problems, is a well known phenomenon that must be confronted when solving problems in the high-frequency range. This paper presents a new method with no phase errors for one-dimensional (1D) time-harmonic wave propagation problems using new ideas that hold p...
Under a time-harmonic assumption, we prove existence and uniqueness results for the outgoing elastic wave in an isotropic half-plane, where the source is given by a local normal stress excitation of the free boundary. This is the starting point for problems of elastic wave scattering by locally perturbed flat surfaces. The main difficulty is that t...
In this Note we deduce an explicit Sommerfeld-type radiation condition which is convenient to prove the uniqueness for the time-harmonic outgoing wave problem in an isotropic elastic half-plane with free boundary condition. The expression is obtained from a rigorous asymptotic analysis of the associated Green's function. The main difficulty is that...
We obtain uniqueness and existence results of an outgoing solution for the Helmholtz equation in a half-space, or in a compact
local perturbation of it, with an impedance boundary condition. It is worth noting that these kinds of domains have unbounded
boundaries which lead to a non-classical exterior problem. The established radiation condition is...
In this article, we study the existence and uniqueness of outgoing solutions for the Helmholtz equation in locally perturbed
half-planes with passive boundary. We establish an explicit outgoing radiation condition which is somewhat different from
the usual Sommerfeld's one due to the appearance of surface waves. We work with the help of Fourier ana...
In this Note we obtain existence and uniqueness results for the Helmholtz equation in the half-space R+3 with an impedance or Robin boundary condition. Basically, we follow the procedure we have already used in the bi-dimensional case (the half-plane). Thus, we compute the associated Green's function with the help of a double Fourier transform and...
This Note gives answers to the uniqueness and existence questions for solutions of the Helmholtz equation in an half-plane with an impedance or mixed boundary condition. We deal with unbounded domains which boundaries are unbounded too. The radiation conditions are different from the ones that we found in an usual exterior problem due to the appear...
Projects
Projects (2)
The main goal of this Fondecyt postdoctoral project is the development of reliable numerical methods for solving differential equations (ODEs and PDEs) when we are interested in the accuracy of the approximate solution on a particular quantity of interest. To do this, we will consider two ideas: goal-oriented finite element spaces and deep learning.
The main objective of this Marie Curie RISE Action is to improve and exchange interdisciplinary knowledge on applied mathematics, high performance computing, and geophysics to be able to better simulate and understand the materials composing the Earth's subsurface. This is essential for a variety of applications such as CO2 storage, hydrocarbon extraction, mining, and geothermal energy production, among others. All these problems have in common the need to obtain an accurate characterization of the Earth's subsurface, and to achieve this goal, several complementary areas will be studied, including the mathematical foundations of various high-order Galerkin multiphysics simulation methods, the efficient computer implementation of these methods in large parallel machines and GPUs, and some crucial geophysical aspects such as the design of measurement acquisition systems in different scenarios. Results will be widely disseminated through publications, workshops, post-graduate courses to train new researchers, a dedicated webpage, and visits to companies working in the area. In that way, we will perform an important role in technology transfer between the most advanced numerical methods and mathematics of the moment and the area of applied geophysics.
