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In this paper, we study deep neural networks (DNNs) for solving high-dimensional evolution equations with oscillatory solutions. Different from deep least-squares methods that deal with time and space variables simultaneously, we propose a…
This paper constitutes our initial effort in developing sparse grid discontinuous Galerkin (DG) methods for high-dimensional partial differential equations (PDEs). Over the past few decades, DG methods have gained popularity in many…
A large class of initial-boundary value problems of linear evolution partial differential equations formulated on the half-line is analyzed via the unified transform method. In particular, explicit formulae are presented for the generalized…
In this paper we present new versions of previously published numerical programs for solving the dipolar Gross-Pitaevskii (GP) equation including the contact interaction in two and three spatial dimensions in imaginary and in real time,…
We propose a general dual ascent framework for Lagrangean decomposition of combinatorial problems. Although methods of this type have shown their efficiency for a number of problems, so far there was no general algorithm applicable to…
We present a new method to interpret the $\gamma$-ray data of our inner Galaxy as measured by the Fermi Large Area Telescope (Fermi LAT). We train and test convolutional neural networks with simulated Fermi-LAT images based on models tuned…
The paper presents an algebraic technique for derivation of fast discrete cosine transform (DCT) algorithms. The technique is based on the algebraic signal processing theory (ASP). In ASP a DCT associates with a polynomial algebra…
We use the recent measurement of the isotropic $\gamma$-ray background (IGRB) by Fermi LAT and analysis of the contribution of unresolved point $\gamma$-ray sources to IGRB to build constraints on the models of ultra-high cosmic rays…
In this paper, the conformable Laguerre and associated Laguerre differential equations are solved using the Laplace transform. The solution is found to be in exact agreement with that obtained using the power series. In addition some of…
We establish generalized Gaussian bounds and local limit theorems with Gaussian-type error for the convolution powers of certain complex-valued functions on $\mathbb{Z}^d$. These global space-times estimates/error, which are sharp in…
We present a dissipative algorithm for solving nonlinear wave-like equations when the initial data is specified on characteristic surfaces. The dissipative properties built in this algorithm make it particularly useful when studying the…
We propose an algorithm to find a solution to an integro-differential equation of the DGLAP type for all the orders in the running coupling $\alpha$ with splitting functions given at a fixed order in $\alpha.$ Complex analysis is…
In this paper, we suggest a new way to identify single bright sources of Ultra High Energy Cosmic Rays (UHECR) on the sky, on top of background. We look for doublets of events at the highest energies, E > 6 x 10^19 eV, and identify low…
We present a calculation of nuclei propagation with energies above 1 EeV in the intergalactic photon field. The calculation is based on a Monte Carlo approach for the nucleus-photon interaction as well as the intergalactic magnetic field.…
We consider the possibility that ultra-high energy cosmic rays originate from the annihilation of relic superheavy dark-matter particles. We find that a cross section of <sigma_A v> ~ 10^{-26}cm^2 (M_X/10^{12}GeV)^{3/2} is required to…
In recent years, there has been an increasing interest in using deep learning and neural networks to tackle scientific problems, particularly in solving partial differential equations (PDEs). However, many neural network-based methods, such…
We study the deflection of ultra high energy cosmic ray protons in different models of the regular galactic magnetic field. Such particles have gyroradii well in excess of 1 kpc and their propagation in the galaxy reflects only the large…
The $hp$ local discontinuous Galerkin (LDG) method proposed by Castillo et al. [Math. Comp.,~71 (238): 455-478, 2002] has been shown to be an efficient approach for solving convection-diffusion equations. However, theoretical analysis…
In this paper, the discontinuous Galerkin based high-order gas-kinetic schemes (DG-HGKS) are developed for the three-dimensional Euler and Navier-Stokes equations. Different from the traditional discontinuous Galerkin (DG) methods with…
We consider spectral discretizations of hyperbolic problems on unbounded domains using Laguerre basis functions. Taking as model problem the scalar advection equation, we perform a comprehensive stability analysis that includes strong…