Related papers: Dirichlet Boundary Value Correction using Lagrange…
We develop a linearized boundary control method for the inverse boundary value problem of determining the damping coefficient in the damped wave equation. The objective is to reconstruct an unknown perturbation in a known background damping…
We propose a new method to deal with the essential boundary conditions encountered in the deep learning-based numerical solvers for partial differential equations. The trial functions representing by deep neural networks are…
In this paper, we study the deep Ritz method for solving the linear elasticity equation from a numerical analysis perspective. A modified Ritz formulation using the $H^{1/2}(\Gamma_D)$ norm is introduced and analyzed for linear elasticity…
We present an accelerated, or 'look-ahead' version of the Newton-Dinkelbach method, a well-known technique for solving fractional and parametric optimization problems. This acceleration halves the Bregman divergence between the current…
We prove a number of \textit{a priori} estimates for weak solutions of elliptic equations or systems with vertically independent coefficients in the upper-half space. These estimates are designed towards applications to boundary value…
We show that the full symbol of the Dirichlet to Neumann map of the k-form Laplace's equation on a Riemannian manifold (of dimension greater than 2) with boundary determines the full Taylor series, at the boundary, of the metric. This…
We study the Dirichlet problem for a class of curvature equations arising from conformal geometry on Riemannian manifolds $(M^n, g)$ with boundary where $n \geq 3$. We prove there exists a unique solution using the continuity method which…
We propose a monotone, and consistent numerical scheme for the approximation of the Dirichlet problem for the normalized Infinity Laplacian, which could be related to the family of so--called two--scale methods. We show that this method is…
In this paper we give lower bounds for the representation of real univariate polynomials as sums of powers of degree 1 polynomials. We present two families of polynomials of degree d such that the number of powers that are required in such…
We investigate the convergence of a nonlinear approximation method introduced by Ammar et al. (J. Non-Newtonian Fluid Mech. 139:153-176, 2006) for the numerical solution of high-dimensional Fokker-Planck equations featuring in…
The augmented Lagrange method is employed to address the optimal control problem involving pointwise state constraints in parabolic equations. The strong convergence of the primal variables and the weak convergence of the dual variables are…
This paper is concerned with approximations and related discretization error estimates for the normal derivatives of solutions of linear elliptic partial differential equations. In order to illustrate the ideas, we consider the Poisson…
We propose a simple boundary condition regularization strategy to reduce error propagation in pressure field reconstruction from corrupted image velocimetry data. The core idea is to replace the canonical Neumann boundary conditions with…
We study expansions near the boundary of solutions to the Dirichlet problem for the constant mean curvature equation in the hyperbolic space. With a characterization of remainders of the expansion by multiple integrals, we establish optimal…
The Durand-Kerner algorithm is a widely used iterative technique for simultaneously finding all the roots of a polynomial. However, its convergence heavily depends on the choice of initial approximations. This paper introduces two novel…
We consider time discretization methods for abstract parabolic problems with inhomogeneous linear constraints. Prototype examples that fit into the general framework are the heat equation with inhomogeneous (time dependent) Dirichlet…
Using the argument of Geiss, Montgomery-Smith and Saksman \cite{GMSS}, and a new martingale inequality, the $L^p$--norms of certain Fourier multipliers in $\R^d$, $d\geq 2$, are identified. These include, among others, the second order…
We discuss several optimization procedures to solve finite element approximations of linear-quadratic Dirichlet optimal control problems governed by an elliptic partial differential equation posed on a 2D or 3D Lipschitz domain. The control…
Modeling the chemical, electric, and thermal transport as well as phase transitions and the accompanying mesoscale microstructure evolution within a material in an electronic device setting involves the solution of partial differential…
A Neural Network (NN) based numerical method is formulated and implemented for solving Boundary Value Problems (BVPs) and numerical results are presented to validate this method by solving Laplace equation with Dirichlet boundary condition…