Parallel preconditioners and multigrid solvers for stochastic polynomial chaos discretizations of the diffusion equation at the large scale

2015 ◽  
Vol 23 (1) ◽  
pp. 5-36 ◽  
Author(s):  
Barry Lee
Keyword(s):  
Diffusion Equation ◽  
Large Scale ◽  
Polynomial Chaos ◽  
Multigrid Solvers ◽  
Parallel Preconditioners

The Local Singular Boundary Method for Solution of Two-Dimensional Advection–Diffusion Equation

2021 ◽  
pp. 2150041
Author(s):  
Karel Kovářík ◽  
Juraj Mužík
Keyword(s):  
Diffusion Equation ◽  
Large Scale ◽  
Sparse Matrix ◽  
Singular Boundary ◽  
Tracer Transport ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Boundary Method ◽  
Local Variant ◽  
Singular Boundary Method

This work focuses on the derivation of the local variant of the singular boundary method (SBM) for solving the advection-diffusion equation of tracer transport. Localization is based on the combination of SBM and finite collocation. Unlike the global variant, local SBM leads to a sparse matrix of the resulting system of equations, making it much more efficient to solve large-scale tasks. It also allows solving velocity vector variable tasks, which is a problem with global SBM. This paper compares the results on several examples for the steady and unsteady variant of the advection-diffusion equation and also examines the dependence of the accuracy of the solution on the density of the nodal grid and the size of the subdomain.

Hybrid Hermite polynomial chaos SBP-SAT technique for stochastic advection-diffusion equations

2020 ◽  
Vol 31 (09) ◽  
pp. 2050128
Author(s):  
Navjot Kaur ◽  
Kavita Goyal
Keyword(s):  
Diffusion Equation ◽  
Hermite Polynomial ◽  
Polynomial Chaos ◽  
Real Life ◽  
Computational Effort ◽  
Dirichlet Boundary Conditions ◽  
Diffusion Equations ◽  
Test Problems ◽  
Advection Diffusion Equation ◽  
Advection Diffusion

The study of advection–diffusion equation has relatively became an active research topic in the field of uncertainty quantification (UQ) due to its numerous real life applications. In this paper, Hermite polynomial chaos is united with summation-by-parts (SBP) – simultaneous approximation terms (SATs) technique to solve the advection–diffusion equations with random Dirichlet boundary conditions (BCs). Polynomial chaos expansion (PCE) with Hermite basis is employed to separate the randomness, then SBP operators are used to approximate the differential operators and SATs are used to enforce BCs by ensuring the stability. For each chaos coefficient, time integration is performed using Runge–Kutta method of fourth order (RK4). Statistical moments namely mean and variance are computed using polynomial chaos coefficients without any extra computational effort. The method is applied on three test problems for validation. The first two test problems are stochastic advection equations on [Formula: see text] without any boundary and third problem is stochastic advection–diffusion equation on [0,2] with Dirichlet BCs. In case of third problem, we have obtained a range of permissible parameters for a stable numerical solution.

scholarly journals Application of Image Denoising Method Based on Two-Way Coupling Diffusion Equation in Public Security Forensics

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Yiqun Wang ◽  
Changpeng He ◽  
Zhenjiang Li
Keyword(s):  
Diffusion Equation ◽  
Noise Level ◽  
Large Scale ◽  
Noise Removal ◽  
Video Image ◽  
Public Security ◽  
Order Partial Differential Equation ◽  
Clear Image ◽  
Coupled Diffusion ◽  
Live Broadcast

This paper uses the web live broadcast and on-demand platform based on the B/S architecture as the application side and designs a video image forensic system that can meet multiple police types and multiple application scenarios. The system uses mobile phones as the video image capture terminal to solve the problem of rapid response and concealment and uses 5G communication technology as the transmission medium to solve the problem of device mobility and link maintenance. The problem of diversification of the use and application modes of multiple police types is solved; the video image evidence is managed in a centralized storage, audit, and export method, and the security and authenticity of the evidence are solved. While the system realizes a series of functions such as the collection, transmission, storage, and application of video image evidence, it also realizes the application-side video image live broadcast function according to actual work needs and solves the large-scale case command and decision-making problem that has been plagued by public security organs. In order to remove the noise in the public security forensic images and to smooth the noise while retaining the details of the image, this paper proposes a denoising algorithm based on the two-way coupling diffusion equation. By improving the second-order partial differential equation, a new diffusion function with better diffusion effect than the original model is constructed. We combined the adaptive edge threshold and stop criterion to establish a new denoising algorithm model, which can get better denoising results. When the noise level is low, the PSNR value and SSIM value of several denoising methods are relatively ideal, and the result is at a higher level, the denoising picture effect is better, and there is no obvious incomplete noise removal or detail problems. As the noise level increases, the denoising results will gradually decrease, and the effects will also vary to different degrees. When the noise intensity increases, visually, it can be clearly seen that the two-way coupled diffusion equation and DnCNN have better denoising effects. When the noise level is high, the two-way coupled diffusion equation network is used to use the clear image and the denoised image for indistinguishable calculation. The method in this paper almost retains all the texture details in the clear image, and there are almost no artifacts and images. On the other hand, the color of the image after denoising by the method in this paper is more vivid, and it is closer to the target picture in terms of picture definition and tone, the denoising effect is ideal, and the generated image has a higher degree of restoration. Compared with the residual GAN, the two-way coupling diffusion equation network converges faster and the network performance is improved.

Long time, large scale properties of the noisy driven-diffusion equation

1994 ◽  
Vol 446 (1926) ◽  
pp. 67-85 ◽  
Keyword(s):  
Diffusion Equation ◽  
Large Scale ◽  
Noise Spectrum ◽  
Dynamical Equations ◽  
Scaling Exponents ◽  
Small Noise ◽  
Nonlinear Dynamical ◽  
Theoretical Approaches ◽  
Long Time ◽  
Scale Properties

We study the driven-diffusion equation, describing the dynamics of density fluc­tuations δρ(x → , t) in powders or traffic flows. We have performed quite detailed numerical simulations of this equation in one dimension, focusing in particular on the scaling behaviour of the correlation function <δρ(x → , t)δρ(0, 0)> . One of our motivations was to assess the validity of various theoretical approaches, such as Renormalization Group and different self consistent truncation schemes, to these nonlinear dynamical equations. Although all of them are seen to predict correctly the scaling exponents, only one of them (where the non-exponential nature of the relaxation is taken into account) is able to reproduce satisfactorily the value of the numerical prefactors. Several other interesting issues, such as the noise spectrum of the output current, or the statistics of distance between jams (showing a transition between a ‘laminar’ régime for small noise to a ‘jammed’ régime for higher noise) are also investigated.

TWO-SCALE DISTRIBUTION OF PRIME NUMBER AND ITS DIFFUSION EQUATION MODEL

Fractals ◽  
2017 ◽  
Vol 25 (02) ◽  
pp. 1750022 ◽  
Author(s):  
YINGJIE LIANG ◽  
WEN CHEN
Keyword(s):  
Brownian Motion ◽  
Diffusion Equation ◽  
Fractional Derivative ◽  
Natural Number ◽  
Prime Number ◽  
Power Law ◽  
Large Scale ◽  
Equation Model ◽  
Small Scale ◽  
Number Formula

This study reveals the two-scale characteristics in prime number distribution. It is observed a sub-diffusion process of power law decay at the small scale of the natural number [Formula: see text], but is found to obey the classical Brownian motion of an exponential decay at the large scale [Formula: see text]. Such two-scale mechanism gives rise to the multi-fractal scaling from the power law to the exponential law distributions in a transition region of the natural number [Formula: see text]. In the small range, the sub-diffusion of prime number distribution is well depicted by the fractional derivative equation model, and in the large scale, exponential decay distribution can accurately be described by a classical diffusion equation model. The Riemann diffusion equation proposed recently by the present authors can accurately model the prime distribution from small to moderate to large scales and is reduced to the fractional derivative sub-diffusion equation at small scale and the classical Brownian motion diffusion equation at large scale, respectively.

Sign in / Sign up

Export Citation Format

Share Document