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Diffusion probability models have shown significant promise in offline reinforcement learning by directly modeling trajectory sequences. However, existing approaches primarily focus on time-domain features while overlooking frequency-domain…
This paper is devoted to the derivation of a pleasingly parallel Galerkin method for the time-independent $N$-body Schr\"odinger equation, and its time-dependent version modeling molecules subject to an external electric field. In this…
The study of high-dimensional differential equations is challenging and difficult due to the analytical and computational intractability. Here, we improve the speed of waveform relaxation (WR), a method to simulate high-dimensional…
We establish the uniqueness of semi-wavefront solution for a non-local delayed reaction-diffusion equation. This result is obtained by using a generalization of the Diekman-Kaper theory for a nonlinear convolution equation. Several…
We design and analyze a Schwarz waveform relaxation algorithm for domain decomposition of advection-diffusion-reaction problems with strong heterogeneities. The interfaces are curved, and we use optimized Robin or Ventcell transmission…
The reaction-diffusion equation is one of the cornerstones equations in applied science and engineering. In the present study, a deep neural network has been trained in order to predict the solution of the equation with different…
Problems with localized nonhomogeneous material properties present well-known challenges for numerical simulations. In particular, such problems may feature large differences in length scales, causing difficulties with meshing and…
The paper is concerned with overlapping domain decomposition and exponential time differencing for the diffusion equation discretized in space by cell-centered finite differences. Two localized exponential time differencing methods are…
A modified method of functional constraints is used to construct the exact solutions of nonlinear equations of reaction-diffusion type with delay and which are associated with variable coefficients. This study considers a most generalized…
Discrete Wavelet Transform (DWT) has been widely explored to enhance the performance of image superresolution (SR). Despite some DWT-based methods improving SR by capturing fine-grained frequency signals, most existing approaches neglect…
We present a parareal in time algorithm for the simulation of neutron diffusion transient model. The method is made efficient by means of a coarse solver defined with large time steps and steady control rods model. Using finite element for…
Reaction-Diffusion systems arise in diverse areas of science and engineering. Due to the peculiar characteristics of such equations, analytic solutions are usually not available and numerical methods are the main tools for approximating the…
In this paper, we formulate and study substructuring type algorithm for the Cahn-Hilliard (CH) equation, which was originally proposed to describe the phase separation phenomenon for binary melted alloy below the critical temperature and…
In this paper, we propose a parallel-in-time algorithm for approximately solving parabolic equations. In particular, we apply the $k$-step backward differentiation formula, and then develop an iterative solver by using the waveform…
In this paper, we apply the Schwarz Waveform Relaxation (SWR) method to the one dimensional Schr{\"o}dinger equation with a general linear or a nonlinear potential. We propose a new algorithm for the Schr{\"o}dinger equation with time…
This paper employs the general time-space fractional diffusion equation to derive correlation time function for analyzing nuclear magnetic resonance (NMR) relaxation. Both the anomalous rotational and translational diffusion are treated.…
This article studies a dirichlet boundary value problem for singularly perturbed time delay convection diffusion equation with degenerate coefficient. A priori explicit bounds are established on the solution and its derivatives. For…
We propose an analytical method for finding the time domain solution for the problem of electronic relaxation of a molecule in solution. The relaxation process is modeled by the decay of a diffusing probability distribution through an…
Reaction-Diffusion equations can present solutions in the form of traveling waves. Such solutions evolve in different spatial and temporal scales and it is desired to construct numerical methods that can adopt a spatial refinement at…
This paper studies the shallow Ritz method for solving one-dimensional diffusion-reaction problems. The method is capable of improving the order of approximation for non-smooth problems. By following a similar approach to the one presented…