Related papers: Goal-Oriented Adaptivity and Multilevel Preconditi…
In this paper we have derived explicitly computable bounds on the error in energy norms for the fully nonlinear Poisson-Boltzmann equation. Together with the computable bounds, we have also obtained efficient error indicators which can…
A widely used electrostatics model in the biomolecular modeling community, the nonlinear Poisson-Boltzmann equation, along with its finite element approximation, are analyzed in this paper. A regularized Poisson-Boltzmann equation is…
Given a partial differential equation (PDE), goal-oriented error estimation allows us to understand how errors in a diagnostic quantity of interest (QoI), or goal, occur and accumulate in a numerical approximation, for example using the…
We propose and analyze optimal additive multilevel solvers for isogeometric discretizations of scalar elliptic problems for locally refined T-meshes. Applying the refinement strategy in Morgenstern and Peterseim (2015, Comput. Aided Geom.…
We describe a finite-volume method for solving the Poisson equation on oct-tree adaptive meshes using direct solvers for individual mesh blocks. The method is a modified version of the method presented by Huang and Greengard (2000), which…
We consider stopping criteria that balance algebraic and discretization errors for the conjugate gradient algorithm applied to high-order finite element discretizations of Poisson problems. Firstly, we introduce a new stopping criterion…
We show that the error achievable using physics-informed neural networks for solving systems of differential equations can be substantially reduced when these networks are trained using meta-learned optimization methods rather than to using…
We consider an automatic construction of locally optimal preconditioners for positive definite linear systems. To achieve this goal, we introduce a differentiable loss function that does not explicitly include the estimation of minimal…
Multiresolution provides a fundamental tool based on the wavelet theory to build adaptive numerical schemes for Partial Differential Equations and time-adaptive meshes, allowing for error control. We have introduced this strategy before to…
This paper proposes an efficient algorithm for solving the Hartree--Fock equation combining a multilevel correction scheme with an adaptive refinement technique to improve computational efficiency. The algorithm integrates a multilevel…
In this work, we consider multigoal-oriented error estimation for stationary fluid-structure interaction. The problem is formulated within a variational-monolithic setting using arbitrary Lagrangian-Eulerian coordinates. Employing the…
In this article we develop a convergence theory for goal-oriented adaptive finite element algorithms designed for a class of second-order semilinear elliptic equations. We briefly discuss the target problem class, and introduce several…
Creating impact in real-world settings requires artificial intelligence techniques to span the full pipeline from data, to predictive models, to decisions. These components are typically approached separately: a machine learning model is…
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…
In numerical simulations of many charged systems at the micro/nano scale, a common theme is the repeated solution of the Poisson-Boltzmann equation. This task proves challenging, if not entirely infeasible, largely due to the nonlinearity…
The Poisson-Boltzmann equation (PBE) is a relevant partial differential equation commonly used in biophysical applications to estimate the electrostatic energy of biomolecular systems immersed in electrolytic solutions. A conventional mean…
We consider the goal-oriented error estimates for a linearized iterative solver for nonlinear partial differential equations. For the adjoint problem and iterative solver we consider, instead of the differentiation of the primal problem, a…
In this work, we revisit the marking decisions made in the standard adaptive finite element method (AFEM). Experience shows that a na\"{i}ve marking policy leads to inefficient use of computational resources for adaptive mesh refinement…
Recent efforts to accelerate first-order methods have focused on convex optimization problems that satisfy a geometric property known as error-bound condition, which covers a broad class of problems, including piece-wise linear programs and…
This article presents a hardware architecture independent implementation of an adaptive mesh refinement Poisson solver that is integrated into the electrostatic Particle-In-Cell beam dynamics code OPAL. The Poisson solver is solely based on…