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Connecting orbits are important invariant structures in the state space of nonlinear systems and various techniques are designed for their computation. However, a uniform analytic approximation of the whole orbit seems rare. Here, based on…
In the present work, we introduce a Self-Consistent Density-Functional Embedding technique, which leaves the realm of standard energy-functional approaches in Density Functional Theory and targets directly the density-to-potential mapping…
We focus on nonconvex and nonsmooth minimization problems with a composite objective, where the differentiable part of the objective is freed from the usual and restrictive global Lipschitz gradient continuity assumption. This longstanding…
Virtual element methods is a new promising finite element methods using general polygonal meshes. Its optimal a priori error estimates are well established in the literature. In this paper, we take a different viewpoint. We try to uncover…
We introduce a class of fully-connected neural networks whose activation functions, rather than being pointwise, rescale feature vectors by a function depending only on their norm. We call such networks radial neural networks, extending…
This paper proposes a novel method for learning highly nonlinear, multivariate functions from examples. Our method takes advantage of the property that continuous functions can be approximated by polynomials, which in turn are representable…
Optimization problems with composite functions consist of an objective function which is the sum of a smooth and a (convex) nonsmooth term. This particular structure is exploited by the class of proximal gradient methods and some of their…
In this paper, we consider an approximation method, and a novel general analysis, for second-order elliptic differential equations with heterogeneous multiscale coefficients. We obtain convergence of the Generalized Multi-scale Finite…
The suitable basis functions for approximating periodic function are periodic, trigonometric functions. When the function is not periodic, a viable alternative is to consider polynomials as basis functions. In this paper we will point out…
This paper proposes and develops a new Newton-type algorithm to solve subdifferential inclusions defined by subgradients of extended-real-valued prox-regular functions. The proposed algorithm is formulated in terms of the second-order…
This paper presents a nonconforming finite element approximation of the space of symmetric tensors with square integrable divergence, on tetrahedral meshes. Used for stress approximation together with the full space of piecewise linear…
The task of approximating a function of d variables from its evaluations at a given number of points is ubiquitous in numerical analysis and engineering applications. When d is large, this task is challenged by the so-called curse of…
This paper is devoted to the study of general (Laurent) polynomial modifications of moment functionals on the unit circle, i.e., associated with hermitian Toeplitz matrices. We present a new approach which allows us to study polynomial…
We study the finite element approximation of linear second-order elliptic partial differential equations in nondivergence form with highly heterogeneous diffusion and drift coefficients. A generalized Cordes condition is imposed to…
We present two new classes of orthogonal functions, log orthogonal functions (LOFs) and generalized log orthogonal functions (GLOFs), which are constructed by applying a $\log$ mapping to Laguerre polynomials. We develop basic approximation…
A posteriori estimates give bounds on the error between the unknown solution of a partial differential equation and its numerical approximation. We present here the methodology based on H1-conforming potential and H(div)-conforming…
The analysis of complex nonlinear systems is often carried out using simpler piecewise linear representations of them. A principled and practical technique is proposed to linearize and evaluate arbitrary continuous nonlinear functions using…
We discuss some aspects of approximating functions on high-dimensional data sets with additive functions or ANOVA decompositions, that is, sums of functions depending on fewer variables each. It is seen that under appropriate smoothness…
The relationship between mappings of sets and renormalization group transformations is established, and renormalization group invariants of such mappings are found. These results are valid both for continuous and discrete mappings and for…
Universal approximation theory offers a foundational framework to verify neural network expressiveness, enabling principled utilization in real-world applications. However, most existing theoretical constructions are established by…