Related papers: Solving discrete constrained problems on de Rham c…
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…
This is not a new result. Purpose of this work is to describe a method to search the analytical expression of the general real solution of the two-dimensional Laplace differential equation. This thing is not easy to find in scientific…
We investigate finite-dimensional constrained structured optimization problems, featuring composite objective functions and set-membership constraints. Offering an expressive yet simple language, this problem class provides a modeling…
This paper studies the problem of reconstructing binary matrices that are only accessible through few evaluations of their discrete X-rays. Such question is prominently motivated by the demand in material science for developing a tool for…
This chapter does not deal with specific tools and techniques for managing complex systems, but proposes some basic concepts that help us to think and speak about complexity. We review classical thinking and its intrinsic drawbacks when…
In this work we address the problem of finding serendipity versions of approximate de Rham complexes with enhanced regularity. The starting point is a new abstract construction of general scope which, given three complexes linked by…
This note is about promoting singularity subtraction as a helpful tool in the discretization of singular integral operators on curved surfaces. Singular and nearly singular kernels are expanded in series whose terms are integrated on…
In this short article, we study different problems described as initial value problems of discrete differential equations and develop a a transform method called the sigma transform, a discrete version of the continuous Laplace transform to…
The Laplace transform is an algebraic method that is widely used for analyzing physical systems by either solving the differential equations modeling their dynamics or by evaluating their transfer function. The dynamics of the given system…
Given a domain $\Omega \subset \mathbb{R}^n$, the de Rham complex of differential forms arises naturally in the study of problems in electromagnetism and fluid mechanics defined on $\Omega$, and its discretization helps build stable…
We consider the problem of solving a large-scale Quadratically Constrained Quadratic Program. Such problems occur naturally in many scientific and web applications. Although there are efficient methods which tackle this problem, they are…
We build convergent discretizations and semi-implicit solvers for the Infinity Laplacian and the game theoretical $p$-Laplacian. The discretizations simplify and generalize earlier ones. We prove convergence of the solution of the Wide…
The biharmonic equation with Dirichlet and Neumann boundary conditions discretized using the mixed finite element method and piecewise linear (with the possible exception of boundary triangles) finite elements on triangular elements has…
Neural Ordinary Differential Equations model dynamical systems with ODEs learned by neural networks. However, ODEs are fundamentally inadequate to model systems with long-range dependencies or discontinuities, which are common in…
In mixed finite element approximations of Hodge Laplace problems associated with the de Rham complex, the exterior derivative operators are computed exactly, so the spatial locality is preserved. However, the numerical approximations of the…
In this paper, we study the Dirichlet problem for Laplace's equation in an open disk. The uniqueness of solutions is ensured by the well-known weak maximum principle. We introduce a novel approach to demonstrate the existence of a solution…
In this paper, we introduce a novel semi-analytical method for solving a broad class of initial value problems involving differential, integro-differential, and delay equations, including those with fractional and variable-order…
The probe and singular sources methods are two well-known classical direct reconstruction methods in inverse obstacle problems governed by partial differential equations. In this paper, by considering an inverse obstacle problem governed by…
The problem of discretization of Darboux integrable equations is considered. Given a Darboux integrable continuous equation, one can obtain a Darboux integrable differential-discrete equation, using the integrals of the continuous equation.…
We consider the Dirichlet problem for the Beltrami equation in some simply connected domain. We consider the class of all homeomorphic solutions of such a problem with a normalization condition and set-theoretic constraints on their complex…