Related papers: On improving the efficiency of ADER methods
We present a novel approach that redefines the traditional interpretation of explicit and implicit discretization methods for solving a general class of advection-diffusion equations (ADEs) featuring nonlinear advection, diffusion…
In the fields of statistics, machine learning, image science, and related areas, there is an increasing demand for decentralized collection or storage of large-scale datasets, as well as distributed solution methods. To tackle this…
An efficient approximate version of implicit Taylor methods for initial-value problems of systems of ordinary differential equations (ODEs) is introduced. The approach, based on an approximate formulation of Taylor methods, produces a…
This chapter provides an overview of state-of-the-art adaptive finite element methods (AFEMs) for the numerical solution of second-order elliptic partial differential equations (PDEs), where the primary focus is on the optimal interplay of…
In this work we explore the fidelity of numerical approximations to the analytic spectra of hyperbolic partial differential equation systems with variable coefficients. We are particularly interested in the ability of discrete methods to…
The main aim of this study is to introduce a 2-layered Artificial Neural Network (ANN) for solving the Black-Scholes partial differential equation (PDE) of either fractional or ordinary orders. Firstly, a discretization method is employed…
In this work, we provide a deep investigation of a family of arbitrary high order numerical methods for hyperbolic partial differential equations (PDEs), with particular emphasis on very high order versions, i.e., with order higher than 5.…
The multiscale complexity of modern problems in computational science and engineering can prohibit the use of traditional numerical methods in multi-dimensional simulations. Therefore, novel algorithms are required in these situations to…
We present an adaptive algorithm for effectively solving rough differential equations (RDEs) using the log-ODE method. The algorithm is based on an error representation formula that accurately describes the contribution of local errors to…
This paper presents a sequence of deferred correction (DC) schemes built recursively from the implicit midpoint scheme for the numerical solution of general first order ordinary differential equations (ODEs). It is proven that each scheme…
We propose and analyze reliable and efficient a posteriori error estimators for an optimal control problem that involves a nondifferentiable cost functional, the Poisson problem as state equation and control constraints. To approximate the…
In standard adversarial training, models are optimized to fit one-hot labels within allowable adversarial perturbation budgets. However, the ignorance of underlying distribution shifts brought by perturbations causes the problem of robust…
In this paper, we consider the problem of accelerating the numerical simulation of time dependent problems by time domain decomposition. The available algorithms enabling such decompositions present severe efficiency limitations and are an…
Advection-Diffusion-Reaction (ADR) Partial Differential Equations (PDEs) appear in a wide spectrum of applications such as chemical reactors, concentration flows, and biological systems. A large number of these applications require the…
Within recent years, considerable progress has been made regarding high-performance solvers for Partial Differential Equations (PDEs), yielding potential gains in efficiency compared to industry standard tools. However, the latter largely…
We introduce the technique of adaptive discretization to design an efficient model-based episodic reinforcement learning algorithm in large (potentially continuous) state-action spaces. Our algorithm is based on optimistic one-step value…
Online optimization has emerged as powerful tool in large scale optimization. In this paper, we introduce efficient online algorithms based on the alternating directions method (ADM). We introduce a new proof technique for ADM in the batch…
This article aims to demonstrate and discuss the applications of automatic differentiation (AD) for finding derivatives in PDE-constrained optimization problems and Jacobians in non-linear finite element analysis. The main idea is to…
Iterative steady-state solvers are widely used in computational fluid dynamics. Unfortunately, it is difficult to obtain steady-state solution for unstable problem caused by physical instability and numerical instability. Optimization is a…
The concept of effective order is a popular methodology in the deterministic literature for the construction of efficient and accurate integrators for differential equations over long times. The idea is to enhance the accuracy of a…