arXiv subjects

Ferenc Izsák

Publications and source records attributed to Ferenc Izsák.

5 recordsLinked to original sources

arXiv2026

Neural Network-Assisted Refinement of Traditional Schemes for One-Dimensional Scalar Conservation Laws

A clear link is established between conventional numerical methods and neural network approximations for solving one-dimensional scalar conservation laws. The focus is on the construction of an appropriate flux term in the case of convex flux functions for improving the classical schemes. The first neural network developed here is able to rediscover Godunov's method, while the second one emulates the behavior of a second-order slope-limiter function. In this way, by merging them, second-order reconstruction-based schemes can be developed. The networks presented here employ a minimal number of parameters, significantly reducing the complexity compared to previous approaches. These networks can also be linked consecutively to get a deep one corresponding to multiple time steps. Training them with an appropriate loss leads to stable schemes, improving even the classical methods without increasing their complexity.

math.NA↗

arXiv2026

Tolerance Proportionality and Computational Stability in Adaptive Parallel-in-Time Runge-Kutta Methods

In this paper, we investigate how adaptive time-integration strategies can be effectively combined with parallel-in-time numerical methods for solving systems of ordinary differential equations. Our focus is particularly on their influence on tolerance proportionality. We examine various grid-refinement strategies within the multigrid reduction-in-time (MGRIT) framework. Our results show that a simple adjustment to the original refinement factor can substantially improve computational stability and reliability. Through numerical experiments on standard test problems using the XBraid library, we demonstrate that parallel-in-time solutions closely match their sequential counterparts. Moreover, with the use of multiple processors, computing time can be significantly reduced.

math.NA↗

arXiv2020

Fractional order elliptic problems with inhomogeneous Dirichlet boundary conditions

Fractional-order elliptic problems are investigated in case of inhomogeneous Dirichlet boundary data. The boundary integral form is proposed as a suitable mathematical model. The corresponding theory is completed by sharpening the mapping properties of the corresponding potential operators. Also a mild condition is provided to ensure the existence of the classical solution of the boundary integral equation.

math.AP↗

arXiv2014

Numerical solution of fractional order diffusion problems with Neumann boundary conditions

A finite difference numerical method is investigated for fractional order diffusion problems in one space dimension. For this, a mathematical model is developed to incorporate homogeneous Dirichlet and Neumann type boundary conditions. The models are based on an appropriate extension of the initial values. The well-posedness of the obtained initial value problems is proved and it is pointed out that the extensions are compatible with the above boundary conditions. Accordingly, a finite difference scheme is constructed for the Neumann problem using the shifted Grünwald--Letnikov approximation of the fractional order derivatives, which is based on infinite many basis points. The corresponding matrix is expressed in a closed form and the convergence of an appropriate implicit Euler scheme is proved.

math.NA↗

arXiv2014

Energy norm error estimates for averaged discontinuous Galerkin methods: multidimensional case

A mathematical analysis is presented for a class of interior penalty (IP) discontinuous Galerkin approximations of elliptic boundary value problems. In the framework of the present theory one can derive some overpenalized IP bilinear forms in a natural way avoiding any heuristic choice of fluxes and penalty terms. The main idea is to start from bilinear forms for the local average of discontinuous approximations which are rewritten using the theory of distributions. It is pointed out that a class of overpenalized IP bilinear forms can be obtained using a lower order perturbation of these. Also, error estimations can be derived between the local averages of the discontinuous approximations and the analytic solution in the $H^1$-seminorm. Using the local averages, the analysis is performed in a conforming framework without any assumption on extra smoothness for the solution of the original boundary value problem.

math.NA↗