Related papers: A Shallow Ritz Method for Elliptic Problems with S…

We introduce Neural Radiosity, an algorithm to solve the rendering equation by minimizing the norm of its residual similar as in traditional radiosity techniques. Traditional basis functions used in radiosity techniques, such as piecewise…

The main focus of this paper is the characterization and exploitation of the asymptotic spectrum of the saddle--point matrix sequences arising from the discretization of optimization problems constrained by elliptic partial differential…

This paper focuses on proposing a deep learning initialized iterative method (Int-Deep) for low-dimensional nonlinear partial differential equations (PDEs). The corresponding framework consists of two phases. In the first phase, an…

We present a novel energy-based numerical analysis of semilinear diffusion-reaction boundary value problems. Based on a suitable variational setting, the proposed computational scheme can be seen as an energy minimisation approach. More…

We explore a new way to handle flux boundary conditions imposed on level sets. The proposed approach is a diffuse interface version of the shifted boundary method (SBM) for continuous Galerkin discretizations of conservation laws in…

The solution to partial differential equations using deep learning approaches has shown promising results for several classes of initial and boundary-value problems. However, their ability to surpass, particularly in terms of accuracy,…

Overdetermined systems of first kind integral equations appear in many applications. When the right-hand side is discretized, the resulting finite-data problem is ill-posed and admits infinitely many solutions. We propose a numerical method…

We propose in this paper a multilevel correction method to solve optimal control problems constrained by elliptic equations with the finite element method. In this scheme, solving optimization problem on the finest finite element space is…

We investigate additive Schwarz methods for semilinear elliptic problems with convex energy functionals, which have wide scientific applications. A key observation is that the convergence rates of both one- and two-level additive Schwarz…

We propose an extension of the discretization approaches for multilayer shallow water models, aimed at making them more flexible and efficient for realistic applications to coastal flows. A novel discretization approach is proposed, in…

This paper proposes to solve the Total Variation regularized models by finding the residual between the input and the unknown optimal solution. After analyzing a previous method, we developed a new iterative algorithm, named as Residual…

We develop an adjoint approach for recovering the topographical function included in the source term of one-dimensional hyperbolic balance laws. We focus on a specific system, namely the shallow water equations, in an effort to recover the…

We propose a conservative energy method based on neural networks with subdomains for solving variational problems (CENN), where the admissible function satisfying the essential boundary condition without boundary penalty is constructed by…

The phenomenon of implicit regularization has attracted interest in recent years as a fundamental aspect of the remarkable generalizing ability of neural networks. In a nutshell, it entails that gradient descent dynamics in many neural…

We prove that the global minimum of the backpropagation (BP) training problem of neural networks with an arbitrary nonlinear activation is given by the ridgelet transform. A series of computational experiments show that there exists an…

In the context of unfitted finite element discretizations the realization of high order methods is challenging due to the fact that the geometry approximation has to be sufficiently accurate. Recently a new unfitted finite element method…

In this paper we investigate an adaptive discretization strategy for ill-posed linear prob- lems combined with a regularization from a class of semiiterative methods. We show that such a discretization approach in combination with a…

In this work, we propose an innovative iterative direct sampling method to solve nonlinear elliptic inverse problems from a limited number of pairs of Cauchy data. It extends the original direct sampling method (DSM) by incorporating an…

We present and analyze a novel regularized form of the gradient clipping algorithm, proving that it converges to global minima of the loss surface of deep neural networks under the squared loss, provided that the layers are of sufficient…

We estimate the error of the Deep Ritz Method for linear elliptic equations. For Dirichlet boundary conditions, we estimate the error when the boundary values are imposed through the boundary penalty method. Our results apply to arbitrary…