Related papers: A Natural Primal-Dual Hybrid Gradient Method for A…
Efficient numerical solvers for partial differential equations empower science and engineering. One of the commonly employed numerical solvers is the preconditioned conjugate gradient (PCG) algorithm which can solve large systems to a given…
Adversarial examples, crafted by adding imperceptible perturbations to natural inputs, can easily fool deep neural networks (DNNs). One of the most successful methods for training adversarially robust DNNs is solving a nonconvex-nonconcave…
Partial Differential Equations (PDEs) are central to modeling complex systems across physical, biological, and engineering domains, yet traditional numerical methods often struggle with high-dimensional or complex problems. Physics-Informed…
Primal-dual algorithms for the resolution of convex-concave saddle point problems usually come with one or several step size parameters. Within the range where convergence is guaranteed, choosing well the step size can make the difference…
In this paper, we propose a stochastic Primal-Dual Hybrid Gradient (PDHG) approach for solving a wide spectrum of regularized stochastic minimization problems, where the regularization term is composite with a linear function. It has been…
We present an online preconditioning technique for the primal-dual hybrid gradient (PDHG) algorithm for linear programming (LP). The method adaptively updates primal and dual preconditioners using an online optimization framework. To…
We study the convergence behaviors of primal-dual hybrid gradient (PDHG) for solving linear programming (LP). PDHG is the base algorithm of a new general-purpose first-order method LP solver, PDLP, which aims to scale up LP by taking…
We study a block-structured class of convex-concave saddle-point problems in which both the primal and dual variables admit natural separable decompositions. Motivated by large-scale applications where a full update on either side can be…
Image restoration remains a challenging task in image processing. Numerous methods tackle this problem, often solved by minimizing a non-smooth penalized co-log-likelihood function. Although the solution is easily interpretable with…
In this work, we study physics-informed neural networks (PINNs) constrained by partial differential equations (PDEs) and their application in approximating PDEs with two characteristic scales. From a continuous perspective, our formulation…
The least absolute shrinkage and selection operator (Lasso) is widely recognized across various fields of mathematics and engineering. Its variant, the generalized Lasso, finds extensive application in the fields of statistics, machine…
In this work we propose a new primal-dual algorithm with adaptive step-sizes. The stochastic primal-dual hybrid gradient (SPDHG) algorithm with constant step-sizes has become widely applied in large-scale convex optimization across many…
In the recent years, Physics Informed Neural Networks (PINNs) have received strong interest as a method to solve PDE driven systems, in particular for data assimilation purpose. This method is still in its infancy, with many shortcomings…
Natural Gradient Descent (NGD) has emerged as a promising optimization algorithm for training neural network-based solvers for partial differential equations (PDEs), such as Physics-Informed Neural Networks (PINNs). However, its practical…
In this work, we present a hybrid numerical method for solving evolution partial differential equations (PDEs) by merging the time finite element method with deep neural networks. In contrast to the conventional deep learning-based…
The primal-dual hybrid gradient (PDHG) algorithm for solving convex optimization problems that arise in tomographic imaging is revisited. In particular, simplification of the selection of step-size parameters is developed for optimization…
The use of neural networks to approximate partial differential equations (PDEs) has gained significant attention in recent years. However, the approximation of PDEs with localised phenomena, e.g., sharp gradients and singularities, remains…
The primal-dual hybrid gradient method (PDHG) is useful for optimization problems that commonly appear in image reconstruction. A downside of PDHG is that there are typically three user-set parameters and performance of the algorithm is…
Recently, neural networks have been widely applied for solving partial differential equations (PDEs). Although such methods have been proven remarkably successful on practical engineering problems, they have not been shown, theoretically or…
Line search procedures are often employed in primal-dual methods for bilinear saddle point problems, especially when the norm of the linear operator is large or difficult to compute. In this paper, we demonstrate that line search is…