Related papers: Dual Variational Neural Network for the $p$-Laplac…
In this paper, we aim to solve high dimensional convex quadratic programming (QP) problems with a large number of quadratic terms, linear equality and inequality constraints. In order to solve the targeted {\bf QP} problems to a desired…
A fruitful approach for solving signal deconvolution problems consists of resorting to a frame-based convex variational formulation. In this context, parallel proximal algorithms and related alternating direction methods of multipliers have…
While variational methods have been among the most powerful tools for solving linear inverse problems in imaging, deep (convolutional) neural networks have recently taken the lead in many challenging benchmarks. A remaining drawback of deep…
We propose a two-scale neural network method for optimal control problems governed by convection-dominated convection-diffusion-reaction equations. Building on two-scale architectures developed for singularly perturbed forward problems, we…
We design inexact proximal augmented Lagrangian based decomposition methods for convex composite programming problems with dual block-angular structures. Our methods are particularly well suited for convex quadratic programming problems…
The 'exact subgraph' approach was recently introduced as a hierarchical scheme to get increasingly tight semidefinite programming relaxations of several NP-hard graph optimization problems. Solving these relaxations is a computational…
The task of projecting onto $\ell_p$ norm balls is ubiquitous in statistics and machine learning, yet the availability of actionable algorithms for doing so is largely limited to the special cases of $p = \left\{ 0, 1,2, \infty \right\}$.…
In this paper we study nonlinear second-order differential inclusions involving the ordinary vector $p$-Laplacian, a multivalued maximal monotone operator and nonlinear multivalued boundary conditions. Our framework is general and unifying…
In this paper we study probabilistic and neural network approximations for solutions to Poisson equation subject to Holder data in general bounded domains of $\mathbb{R}^d$. We aim at two fundamental goals. The first, and the most…
Super-Resolution convolutional neural networks have recently demonstrated high-quality restoration for single images. However, existing algorithms often require very deep architectures and long training times. Furthermore, current…
Recently, progress has been made in the application of neural networks to the numerical analysis of partial differential equations (PDEs). In the latter the variational formulation of the Poisson problem is used in order to obtain an…
This paper is concerned with a novel deep learning method for variational problems with essential boundary conditions. To this end, we first reformulate the original problem into a minimax problem corresponding to a feasible augmented…
In this paper, we denoise a given noisy image by minimizing a smoothness promoting function over a set of local similarity measures which compare the mean of the given image and some candidate image on a large collection of subboxes. The…
Although the Laplace approximation offers a simple route to uncertainty quantification in deep neural networks, its reliance on inverting large Hessian matrices has motivated a range of computationally feasible low-dimensional or sparse…
We develop a weak adversarial approach to solving obstacle problems using neural networks. By employing (generalised) regularised gap functions and their properties we rewrite the obstacle problem (which is an elliptic variational…
For the Poisson problem in two dimensions, posed on a domain partitioned into axis-aligned rectangles with up to one hanging node per edge, we envision an efficient error reduction step in an instance-optimal $hp$-adaptive finite element…
A new algorithm for solving large-scale convex optimization problems with a separable objective function is proposed. The basic idea is to combine three techniques: Lagrangian dual decomposition, excessive gap and smoothing. The main…
Optimization problems are ubiquitous in our societies and are present in almost every segment of the economy. Most of these optimization problems are NP-hard and computationally demanding, often requiring approximate solutions for…
Training of neural networks amounts to nonconvex optimization problems that are typically solved by using backpropagation and (variants of) stochastic gradient descent. In this work we propose an alternative approach by viewing the training…
We propose a general dual ascent framework for Lagrangean decomposition of combinatorial problems. Although methods of this type have shown their efficiency for a number of problems, so far there was no general algorithm applicable to…