Related papers: Design and convergence analysis of numerical metho…
Machine-learning (ML) based discretization has been developed to simulate complex partial differential equations (PDEs) with tremendous success across various fields. These learned PDE solvers can effectively resolve the underlying solution…
In this paper, we propose an adaptive approach, based on mesh refinement or parametric enrichment with polynomial degree adaption, for numerical solution of convection dominated equations with random input data. A parametric system emerged…
In this paper we make a study of a partial integral differential equation with $p$-Laplacian using a mixed finite element method. Two stable and convergent fixed point schemes are proposed to solve the nonlinear algebraic system. Using the…
We present a general abstract framework for the systematic numerical approximation of dissipative evolution problems. The approach is based on rewriting the evolution problem in a particular form that complies with an underlying energy or…
We study the problem of non-convex optimization using Stochastic Gradient Langevin Dynamics (SGLD). SGLD is a natural and popular variation of stochastic gradient descent where at each step, appropriately scaled Gaussian noise is added. To…
There has been significant interest in generalizations of the Nesterov accelerated gradient descent algorithm due to its improved performance guarantee compared to the standard gradient descent algorithm, and its applicability to large…
We consider a distributed learning problem in which the computation is carried out on a system consisting of a master node and multiple worker nodes. In such systems, the existence of slow-running machines called stragglers will cause a…
Stochastic gradient descent (SGD) method is popular for solving non-convex optimization problems in machine learning. This work investigates SGD from a viewpoint of graduated optimization, which is a widely applied approach for non-convex…
The gradient descent (GD) method -- is a fundamental and likely the most popular optimization algorithm in machine learning (ML), with a history traced back to a paper in 1847 (Cauchy, 1847). It was studied under various assumptions,…
Nonsmooth nonconvex optimization problems broadly emerge in machine learning and business decision making, whereas two core challenges impede the development of efficient solution methods with finite-time convergence guarantee: the lack of…
Nonlinear multi-scale problems are ubiquitous in materials science and biology. Complicated interactions between nonlinearities and (nonseparable) multiple scales pose a major challenge for analysis and simulation. In this paper, we study…
Stochastic gradient descent (SGD) provides a simple and efficient way to solve a broad range of machine learning problems. Here, we focus on distribution regression (DR), involving two stages of sampling: Firstly, we regress from…
Classical machine learning models such as deep neural networks are usually trained by using Stochastic Gradient Descent-based (SGD) algorithms. The classical SGD can be interpreted as a discretization of the stochastic gradient flow. In…
In this paper we analyze the behaviour of the stochastic gradient descent (SGD), a widely used method in supervised learning for optimizing neural network weights via a minimization of non-convex loss functions. Since the pioneering work of…
A local discontinuous Galerkin (LDG) method for approximating large deformations of prestrained plates is introduced and tested on several insightful numerical examples in our previous computational work. This paper presents a numerical…
We investigate numerical behaviour of a convection diffusion equation with random coefficients by approximating statistical moments of the solution. Stochastic Galerkin approach, turning the original stochastic problem to a system of…
The subject of this work is an adaptive stochastic Galerkin finite element method for parametric or random elliptic partial differential equations, which generates sparse product polynomial expansions with respect to the parametric…
As a counterpoint to recent numerical methods for crystal surface evolution, which agree well with microscopic dynamics but suffer from significant stiffness that prevents simulation on fine spatial grids, we develop a new numerical method…
We propose a multiscale spectral generalized finite element method (MS-GFEM) for discontinuous Galerkin (DG) discretizations. The method builds local approximations on overlapping subdomains as the sum of a local source solution and a…
A numerical dynamical low-rank approximation (DLRA) scheme for the solution of the Vlasov-Poisson equation is presented. Based on the formulation of the DLRA equations as Friedrichs' systems in a continuous setting, it combines recently…