Related papers: Adaptive Learning on the Grids for Elliptic Hemiva…
We introduce an $r-$adaptive algorithm to solve Partial Differential Equations using a Deep Neural Network. The proposed method restricts to tensor product meshes and optimizes the boundary node locations in one dimension, from which we…
Bilevel learning has gained prominence in machine learning, inverse problems, and imaging applications, including hyperparameter optimization, learning data-adaptive regularizers, and optimizing forward operators. The large-scale nature of…
The recent development of novel aerial vehicles capable of physically interacting with the environment leads to new applications such as contact-based inspection. These tasks require the robotic system to exchange forces with…
We consider classification tasks in the regime of scarce labeled training data in high dimensional feature space, where specific expert knowledge is also available. We propose a new hybrid optimization algorithm that solves the elastic-net…
In this paper, we propose CI-VI an efficient and scalable solver for semi-implicit variational inference (SIVI). Our method, first, maps SIVI's evidence lower bound (ELBO) to a form involving a nonlinear functional nesting of expected…
Monotone finite difference methods provide stable convergent discretizations of a class of degenerate elliptic and parabolic Partial Differential Equations (PDEs). These methods are best suited to regular rectangular grids, which leads to…
This article proposes a hybrid adaptive numerical method based on the Dual Reciprocity Method (DRM) to solve problems with non-linear boundary conditions and large-scale problems, named Hybrid Adaptive Dual Reciprocity Method (H-DRM). The…
The multigrid algorithm is a multilevel approach to accelerate the numerical solution of discretized differential equations in physical problems involving long-range interactions. Multiresolution analysis of wavelet theory provides an…
Wavelet-based grid adaptation methods use multiresolution analysis for error estimation, offering a mathematically rigorous approach to adaptive grid refinement when solving Partial Differential Equations (PDEs). However, applying these…
We propose a hybrid inertial self-adaptive algorithm for solving the split feasibility problem and fixed point problem in the class of demicontractive mappings. Our results are very general and extend several related results existing in…
This paper is concerned with the approximation of the solution of partial differential equations by means of artificial neural networks. Here a feedforward neural network is used to approximate the solution of the partial differential…
We propose a deep learning based method, the Deep Ritz Method, for numerically solving variational problems, particularly the ones that arise from partial differential equations. The Deep Ritz method is naturally nonlinear, naturally…
Hierarchical optimization refers to problems with interdependent decision variables and objectives, such as minimax and bilevel formulations. While various algorithms have been proposed, existing methods and analyses lack adaptivity in…
In this paper, we propose a novel gradient recovery method for elliptic interface problem using body-fitted mesh in two dimension. Due to the lack of regularity of solution at interface, standard gradient recovery methods fail to give…
Full-waveform inversion (FWI) is a powerful geophysical imaging technique that infers high-resolution subsurface physical parameters by solving a non-convex optimization problem. However, due to limitations in observation, e.g., limited…
This article is concerned with the numerical solution of convex variational problems. More precisely, we develop an iterative minimisation technique which allows for the successive enrichment of an underlying discrete approximation space in…
We give a new deterministic algorithm that non-adaptively learns a hidden hypergraph from edge-detecting queries. All previous non-adaptive algorithms either run in exponential time or have non-optimal query complexity. We give the first…
In this paper, we design and analyze a new family of adaptive subgradient methods for solving an important class of weakly convex (possibly nonsmooth) stochastic optimization problems. Adaptive methods that use exponential moving averages…
In this paper we describe in detail the computational algorithm used by our parallel multigrid elliptic equation solver with adaptive mesh refinement. Our code uses truncation error estimates to adaptively refine the grid as part of the…
The present paper represents a continuation of our previous one. There, a continuous dependence result for the solution of an elliptic variational-hemivariational inequality was obtained and then used to prove the existence of optimal pairs…