Most stochastic gradient descent algorithms can optimize neural networks that are sub-differentiable in their parameters; however, this implies the network’s activation function must exhibit a degree of continuity which limits network model’s uniform approximation capacity to continuous functions. This paper focuses on case where discontinuities arise from distinct sub-patterns, each defined di...

We devise and analyze a simple numerical method for the approximation of large bending isometries. The discretization employs a discrete Kirchhoff triangle to deal with second order derivatives and convergence of discrete solutions to minimizers of the continuous formulation is proved. Unconditional stability and convergence of an iterative scheme for the computation of discrete minimizers that...

Given iid observations from an unknown absolute continuous distribution defined on some domain Ω, we propose a nonparametric method to learn a piecewise constant function to approximate the underlying probability density function. Our density estimate is a piecewise constant function defined on a binary partition of Ω. The key ingredient of the algorithm is to use discrepancy, a concept origina...

In this paper the piecewise level set method is combined with phase field method to solve the shape and topology optimization problem. First, the optimization problem is formed based on piecewise constant level set method then is updated using the energy term of phase field equations. The resulting diffusion equation which updates the level set function and optimization ...

We consider a piecewise continuous, piecewise monotone interval map and a weight of bounded variation, constant on homtervals and continuous at periodic points of the map. With these data we associate a sequence of weighted Milnor-Thurston kneading matrices, converging to a countable matrix with coeecients analytic functions. We show that the determinants of these matrices converge to the inver...

A second order elliptic problem with highly discontinuous coefficients has been considered. The problem is discretized by two methods: 1) continuous finite element method (FEM) and 2) composite discretization given by a continuous FEM inside the substructures and a discontinuous Galerkin method (DG) across the boundaries of these substructures. The main goal of this paper is to design and analy...

Reduced-order modeling represents an attractive framework for accelerating computationally expensive reservoir simulation applications. In this paper we introduce and apply a reduced-order modeling approach for history matching. The method considered, trajectory piecewise linearization (TPWL), has been used previously for production optimization problems, where it has provided large computation...