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In this paper, we study adaptive neuron enhancement (ANE) method for solving self-adjoint second-order elliptic partial differential equations (PDEs). The ANE method is a self-adaptive method generating a two-layer spline NN and a numerical…
In this paper we propose an accurate, and computationally efficient method for incorporating adaptive spatial resolution into weakly-compressible Smoothed Particle Hydrodynamics (SPH) schemes. Particles are adaptively split and merged in an…
Sparse parametric models are of great interest in statistical learning and are often analyzed by means of regularized estimators. Pathwise methods allow to efficiently compute the full solution path for penalized estimators, for any…
We propose an inertial variant of the strongly convergent inexact proximal-point (PP) method of Solodov and Svaiter (2000) for monotone inclusions. We prove strong convergence of our main algorithm under less restrictive assumptions on the…
This paper presents a methodology for using varying sample sizes in sequential quadratic programming (SQP) methods for solving equality constrained stochastic optimization problems. The first part of the paper deals with the delicate issue…
Reasoning about distance is indispensable for establishing or avoiding contact in manipulation tasks. To this end, we present an online approach for learning implicit representations of signed distance using piecewise polynomial basis…
This paper is devoted to studying the global and finite convergence of the semi-smooth Newton method for solving a piecewise linear system that arises in cone-constrained quadratic programming problems and absolute value equations. We first…
Semi-infinite programming can be used to model a large variety of complex optimization problems. The simple description of such problems comes at a price: semi-infinite problems are often harder to solve than finite nonlinear problems. In…
Many separable nonlinear optimization problems can be approximated by their nonlinear objective functions with piecewise linear functions. A natural question arising from applying this approach is how to break the interval of interest into…
In this paper, a piecewise quadratic nonconforming finite element method on rectangular grids for a fourth-order elliptic singular perturbation problem is presented. This proposed method is robustly convergent with respect to the…
The article is devoted to some adaptive methods for variational inequalities with relatively smooth and relatively strongly monotone operators. Starting from the recently proposed proximal variant of the extragradient method for this class…
Numerical relativity has traditionally been pursued via finite differencing. Here we explore pseudospectral collocation (PSC) as an alternative to finite differencing, focusing particularly on the solution of the Hamiltonian constraint (an…
Piecewise linearization is a key technique for solving nonlinear problems in transportation network design and other optimization fields, in which generating breakpoints is a fundamental task. This paper proposes an optimal breakpoint…
This paper is focused on the convergence analysis of an adaptive stochastic collocation algorithm for the stationary diffusion equation with parametric coefficient. The algorithm employs sparse grid collocation in the parameter domain…
In this paper, we propose an adaptive sieving (AS) strategy for solving general sparse machine learning models by effectively exploring the intrinsic sparsity of the solutions, wherein only a sequence of reduced problems with much smaller…
A simple greedy refinement procedure for the generation of data-adapted triangulations is proposed and studied. Given a function of two variables, the algorithm produces a hierarchy of triangulations and piecewise polynomial approximations…
An adaptive method for parabolic partial differential equations that combines sparse wavelet expansions in time with adaptive low-rank approximations in the spatial variables is constructed and analyzed. The method is shown to converge and…
In this paper, we introduce a technique to enhance the computational efficiency of solution algorithms for high-dimensional discrete simulation-based optimization problems. The technique is based on innovative adaptive partitioning…
This work explores a novel approach for adaptive, differentiable parametrization of large-scale non-stationary random fields. Coupled with any gradient-based algorithm, the method can be applied to variety of optimization problems,…
We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for…