Related papers: Diffusive Corrections to Pn Approximations

We address an original approach for the convergence analysis of a finite-volume scheme for the approximation of a stochastic diffusion-convection equation with multiplicative noise in a bounded domain of $\mathbb{R}^d$ (with $d=2$ or $3$)…

Approximation techniques have been historically important for solving differential equations, both as initial value problems and boundary value problems. The integration of numerical, analytic and perturbation methods and techniques can…

The impressive performance of deep neural networks (DNNs) has immensely strengthened the line of research that aims at theoretically analyzing their effectiveness. This has incited research on the reaction of DNNs to noisy input, namely…

The use of proximal point operators for optimization can be computationally expensive when the dimensionality of a function (i.e., the number of variables) is high. In this study, we sought to reduce the cost of calculating proximal point…

This work unifies pseudo-time and inexact regularization techniques for nonmonotone classes of partial differential equations, into a regularized pseudo-time framework. Convergence of the residual at the predicted rate is investigated…

Plug-and-Play (PnP) methods are a class of efficient iterative methods that aim to combine data fidelity terms and deep denoisers using classical optimization algorithms, such as ISTA or ADMM, with applications in inverse problems and…

Partial Differential Equations (PDEs) are central to science and engineering. Since solving them is computationally expensive, a lot of effort has been put into approximating their solution operator via both traditional and recently…

This paper proposes a data-driven framework to learn a finite-dimensional approximation of a Koopman operator for approximating the state evolution of a dynamical system under noisy observations. To this end, our proposed solution has two…

This paper presents a multiple length-scale asymptotic analysis for transport problems in 1-D diffusive random media. This analysis shows that the Levermore-Pomraning (LP) equations can be adjusted in order to achieve the correct asymptotic…

We investigate an optimization problem that arises when working within the paradigm of Data-Driven Computational Mechanics. In the context of the diffusion-reaction problem, such an optimization problem seeks for the continuous primal…

In this paper we present a new methodology for option pricing. The main idea consists to represent a generic probability distribution function (PDF) via a perturbative expansion around a given, simpler, PDF (typically a gaussian function)…

In this article, we introduce a finite element method designed for the robust computation of approximate signed distance functions to arbitrary boundaries in two and three dimensions. Our method employs a novel prediction-correction…

In recent work, Li et al.\ (Comm.\ Math.\ Sci., 7:81-107, 2009) developed a diffuse-domain method (DDM) for solving partial differential equations in complex, dynamic geometries with Dirichlet, Neumann, and Robin boundary conditions. The…

The attention mechanism is the computational core of modern Transformer architectures, but its quadratic complexity in the input sequence length is the bottleneck for large-scale inference. This has motivated a rapidly growing body of work…

Fractional diffusion equations have been an effective tool for modeling anomalous diffusion in complicated systems. However, traditional numerical methods require expensive computation cost and storage resources because of the memory effect…

The classical work of (Arora et al., 1999) provides a scheme that gives, for any $\epsilon>0$, a polynomial time $1-\epsilon$ approximation algorithm for dense instances of a family of $\mathcal{NP}$-hard problems, such as Max-CUT and…

Approximating the Koopman operator from data is numerically challenging when many lifting functions are considered. Even low-dimensional systems can yield unstable or ill-conditioned results in a high-dimensional lifted space. In this…

Deep Neural Networks (DNNs) approaches for the Optimal Power Flow (OPF) problem received considerable attention recently. A key challenge of these approaches lies in ensuring the feasibility of the predicted solutions to physical system…

In the present work, an approach to the moment closure problem on the basis of orthogonal polynomials derived from Gram matrices is proposed. Its properties are studied in the context of the moment closure problem arising in gas kinetic…

Rectifier (ReLU) deep neural networks (DNN) and their connection with piecewise affine (PWA) functions is analyzed. The paper is an effort to find and study the possibility of representing explicit state feedback policy of model predictive…