Backward Diffusion Research Articles

Dynamics of the modified Rotation-Camassa-Holm Equation with backward diffusion term

Dynamics of the modified Rotation-Camassa-Holm Equation with backward diffusion term

Sobolev‐type regularization method for the backward diffusion equation with fractional Laplacian and time‐dependent coefficient

This work is concerned with an ill‐posed problem of reconstructing the historical distribution of a backward diffusion equation with fractional Laplacian and time‐dependent coefficient in multidimensional space. The investigated problem is regularized by a Sobolev‐type equation method. Unlike previous works, to prove the convergence of the regularized solution to the exact one, we only require a very weak and natural a priori condition that the solution belongs to the standard Lebesgue space . This is done by suitably employing the Lebesgue‐dominated convergence theorem. If we go further to impose a stronger a priori condition, one may know how fast the convergence is. Finally, some MATLAB‐based numerical examples are provided to confirm the efficiency of the proposed method.

Deep learning approach for diffusion correction in Fricke hydrogel dosimeters

In recent years studies on Fricke gel dosimeters have demonstrated their potential in performing fast measurements of 3D spatial dose distributions. These dosimeters work by oxidising ferrous (Fe2+) ions to ferric (Fe3+) ions, which can be read by magnetic resonance or optical techniques. However, the accuracy of these measurements can be affected by the diffusion of Fe3+ ions within the gel matrix, causing image blurring. To overcome this issue, the study proposes a computational method using artificial intelligence (AI) and specifically deep learning (DL). The method is based on Physics Informed Neural Networks (PINNs), which are effective in solving partial differential equations, especially in inverse problems. The PINNs are optimised using physical equations as loss functions, allowing them to predict real-life phenomena like ion diffusion in Fricke gel. The idea is to solve the backward diffusion equation using a PINN model and predict the initial condition, providing as input the spatial distribution of Fe3+ ions at measurement time T, the boundary conditions and the diffusion coefficient D of Fe3+ ions inside the medium. The method was first tested using 1D simulated data in diffusion processes, achieving a mean squared error of 1−4×10−4a.u. in step-wise and rectangular distributions. It was then tested using experimental data, achieving a mean squared error of 2×10−4OD2. The technique here proposed is very promising for overcoming the limits due to the diffusion of Fe3+ ions in Fricke gel dosimeters.

Deep Neural Networks with Spacetime RBF for Solving Forward and Inverse Problems in the Diffusion Process

Deep Neural Networks with Spacetime RBF for Solving Forward and Inverse Problems in the Diffusion Process

Quantum Image Cryptography of Continuous Chaotic Maps by Using Pixel Shuffeling

Images are important data carriers because, compared to text data, they are harder to transfer or store securely and include higher volumes and redundancies of digital data. Images can be shielded against a variety of risks with security, including eavesdropping and illegal copying and alteration. Because of the potential quantum risk to the existing cryptographic encryption methods and the quick advancement towards the development of quantum computers, quantum image encryption algorithms have recently drawn increasing amounts of attention. The majority of quantum image encryption techniques such as diffusion and scrambling, involve two separate rounds. In this model, the three different chaotic maps are used separately for scrambling the images to determine the performance of the quantum image cryptography with different combination of the model. At first, the hash256 algorithm is used for generating the quantum key and the forward diffusion takes place for diffusing the first pixel to final pixel of the input image information. Then, the three different chaotic maps such as pixel permutation, Chen attractor and Lorenz attractor are used for scrambling the input image. Finally, the bit-level permutation and backward diffusion process are considered for the scrambled image. For evaluating the performance of the quantum image cryptography based on the three different chaotic maps, the NPCR, UACI, Entropy, SSIM, correlation characteristics and histogram analysis are determined. The attained NPCR, UACI, Entropy and SSIM of the CASIA2 dataset for Lorenz attractor are improved than the pixel permutation and Chen attractor. Thus, from the attained values, the Quantum Image Cryptography Based on Continuous Chaotic Map such as Lorenz attractor performs better for the statistical and differential analysis than the other chaotic maps.

Scalable Motion Style Transfer with Constrained Diffusion Generation

Current training of motion style transfer systems relies on consistency losses across style domains to preserve contents, hindering its scalable application to a large number of domains and private data. Recent image transfer works show the potential of independent training on each domain by leveraging implicit bridging between diffusion models, with the content preservation, however, limited to simple data patterns. We address this by imposing biased sampling in backward diffusion while maintaining the domain independence in the training stage. We construct the bias from the source domain keyframes and apply them as the gradient of content constraints, yielding a framework with keyframe manifold constraint gradients (KMCGs). Our validation demonstrates the success of training separate models to transfer between as many as ten dance motion styles. Comprehensive experiments find a significant improvement in preserving motion contents in comparison to baseline and ablative diffusion-based style transfer models. In addition, we perform a human study for a subjective assessment of the quality of generated dance motions. The results validate the competitiveness of KMCGs.

A Symmetry of Boundary Functions Method for Solving the Backward Time-Fractional Diffusion Problems

In the paper, we develop three new methods for estimating unknown initial temperature in a backward time-fractional diffusion problem, which is transformed to a space-dependent inverse source problem for a new variable in the first method. Then, the initial temperature can be recovered by solving a second-order boundary value problem. The boundary functions and a unique zero element constitute a group symmetry. We derive energetic boundary functions in the symmetry group as the bases to retrieve the source term as an unknown function of space and time. In the second method, the solution bases are energetic boundary functions, and then by collocating the governing equation we obtain the expansion coefficients for retrieving the entire solution and initial temperature. For the first two methods, boundary fluxes are over-specified to retrieve the initial condition. In the third method, we give two boundary conditions and a final time temperature to construct the bases in another symmetry group; the governing equation is collocated to a linear system to obtain the whole solution (initial temperature involved). These three methods are assessed and compared by numerical experiments.

Double-parameter regularization for solving the backward diffusion problem with parallel-in-time algorithm

We propose a double-parameter regularization scheme for dealing with the backward diffusion process. Considering the smoothing effect of Yosida approximation for PDE, we propose to regularize this ill-posed problem by modifying original governed system in terms of a pseudoparabolic equation together with a quasi-boundary condition simultaneously, which consequently contains two regularizing parameters. Theoretically, we establish the optimal error estimates between the regularizing solution and the exact one in terms of suitable choice strategy for the regularizing parameters, under a-priori regularity assumptions on the exact solution. The a-posteriori choice strategy for the regularizing parameters based on the discrepancy principle is also studied. To weaken the heavy computational cost for solving the discrete nonsymmetric linear regularizing system by finite difference scheme, especially in higher spatial dimensional cases, the block divide-and-conquer method together with the properties of the Schur complement is applied to decompose the linear system into two half-size linear systems, one of which can be solved by the diagonalization technique, and consequently an efficient parallel-in-time algorithm originally developed for direct problem is applicable. Our proposed method is of much lower complexity than the standard solver for the corresponding linear system. Finally, some numerical examples are presented to verify the efficiency of our proposed method.

Traveling waves in a quintic BBM equation under both distributed delay and weak backward diffusion

In this paper, we focus on the existence of periodic and solitary waves for a quintic Benjamin–Bona–Mahony (BBM) equation with distributed delay and diffused perturbation. The corresponding traveling wave equation is transformed into a three-dimensional dynamical system, which is regarded as a singularly perturbed system for small perturbation parameter. By geometric singular perturbation theory, the three-dimensional dynamical system can be reduced to a near-Hamiltonian planar system and a locally invariant manifold diffeomorphic to the critical manifold with normally hyperbolicity can be constructed. Further, the existence of periodic wave and solitary wave are established by proving the persistence of periodic and homoclinic orbits of the near-Hamiltonian planar system for sufficiently small perturbation, and their existence conditions for the delayed quintic BBM equation have also been obtained. The uniqueness of periodic wave is established by analyzing the monotonicity of ratios of Abelian integrals form a Chebyshev set. Moreover, the monotonicity of wave speed is proved, and the supremum and infimum of wave speed are gotten. Numerical simulations are in complete agreement with the theoretical predictions.

Modeling Long-Distance Forward and Backward Diffusion Processes in Tracer Transport Using the Fractional Laplacian on Bounded Domains

Recent studies have emphasized the importance of the long-distance diffusion model in characterizing tracer transport occurring within both subsurface and surface environments, particularly in heterogeneous systems. Long-distance diffusion, often referred to as nonlocal diffusion, signifies that tracer particles may experience a considerably long distance in either the forward or backward direction along preferential channels during the transport. The classical advection–diffusion (ADE) model has been widely used to describe tracer transport; however, they often fall short in capturing the intricacies of nonlocal diffusion processes. The fractional operator has gained recognition among hydrologists due to its potential to capture distinct mechanisms of transport and storage for tracer particles exhibiting nonlocal dynamics. However, the hypersingularity of the fractional Laplacian operator presents considerable difficulties in its numerical approximation in bounded domains. This study focuses on the development and application of the fractional Laplacian-based model to characterize nonlocal tracer transport behavior, specifically considering both forward and backward diffusion processes on bounded domains. The Riesz fractional Laplacian provides a mathematical framework for describing tracer diffusion processes on a bounded domain, and a novel finite difference method (FDM) is introduced as an effective numerical solver for addressing the fractional Laplacian-based model. Applications reveal that the fractional Laplacian-based model can effectively capture the observed nonlocal tracer transport behavior in a heterogeneous system, and nonlocal tracer transport exhibits dynamic characteristics, influenced by the evolving heterogeneity of the media at various temporal scales.

Seawater pervaporation through carbon honeycomb membrane: A molecular dynamics study

In this work, the seawater desalination performance of carbon honeycomb (CHC) membrane via the pervaporation (PV) process is investigated by molecular dynamics simulations. The simulation results show that the CHC membrane can lead to superior evaporation water flux with complete salt rejection. Unintuitively, the evaporation water flux first increases and then decreases with the increase of membrane thickness, showing a nonmonotonic dependence. Detailed analysis of the density distribution and diffusion dynamics of the water molecules indicates that this nonmonotonic dependence is closely related to the density gradient in the vapor phase, backward diffusion of evaporated water molecules, and flow resistance of the liquid phase. Increasing the pore size of the CHC membrane can lead to a significant increase in the evaporation water flow. However, an excessively large pore size permits the permeation of Na+ and Cl− ions into the CHC membrane, which impedes the diffusion of water molecules and slows down the increasing trend. With the increase in temperature, the evaporation water flux increases as expected because the change in Gibbs free energy becomes increasingly negative. Overall, this work provides insights into the PV performance of CHC membranes and promotes their application in next-generation PV membranes.

Adjoint models with non-Fickian reactive transport to identify pollutant sources in water

Pollutant source identification (PSI) has remained a research topic in environmental sciences for decades for hazardous chemicals, with the assumption that the transport of pollutants is Fickian. Pollutant transport in natural media, however, is usually non-Fickian, characterized by slow retention or preferential movement, and PSI models remained unknown for non-Fickian, reactive transport. To fill this knowledge gap, an adjoint fractional advection-dispersion-reaction equation (fADRE) was derived in this study to identify the source location of reactive pollutants undergoing non-Fickian diffusion with multiple phases in subsurface water with any dimension and boundary conditions. Six fundamental chemical reactions varying from radioactive decay to bimolecular reactions were considered, whose backward-in-time rate equations were simulated by particle tracking combined with backward non-Fickian diffusion. Applications showed that the adjoint fADRE identified the source location of reactive tracer transport in “homogeneous” sand columns repacked in our laboratory and a heterogeneous alluvial aquifer in an actual field site, while the Fickian-based, classical PSI model overestimated by four times the source location of reactive pollutants undergoing strong non-Fickian transport. To expand the applicability of the adjoint fADRE for PSI, we also discussed the impacts of the immobile pollutant phase and reactive kinetics on PSI, pollutant release history recovery, and usage of PSI for complex chemical reactions. The results showed that the fundamental chemical reactions considered in this study can affect PSI by adjusting the location and value of the peak of the probability density functions under conditions such as a relatively large decay constant or a space-dependent reaction rate, and the immobile pollutants can also affect PSI by shortening the source location and elongating the release time for pollutants usually detected only in the mobile phase. Therefore, non-Fickian dynamics and pollutant phases (which were missed by the adjoint ADRE), as well as reaction kinetics, must be captured by the adjoint model, such as the adjoint fADRE derived in this study, for PSI in real-world geological media.

Dynamical analysis of a novel chaotic system and its application to image encryption

This study presents a novel four-dimensional chaotic system derived from the Lorenz-Haken equation. The paper investigates the system’s dissipative properties, equilibrium point, Poincaré map, 0–1 test, Lyapunov exponent, complexity, and bifurcation diagram to introduce the system’s dynamics. The research also examines the chaotic behavior of the system under various parameters, demonstrating the existence of symmetric attractors and offset boosting. To demonstrate the system’s feasibility, circuit simulation is performed using Multisim software, and the hardware implementation is carried out using a Field Programmable Gate Array (FPGA). An image encryption algorithm is designed by integrating matrix dislocation, backward diffusion, and deoxyribonucleic acid (DNA) encryption with the discretized chaotic system. The simulation results show that the algorithm has a good encryption effect. NIST tests show that the sequence has good randomness, correlation coefficient, and information entropy tests show that the encrypted image has strong randomness, and key space and key sensitivity prove that the system has a good ability to resist exhaustive attacks. The use of simulated differential attacks, cropping attacks, and noise attacks has demonstrated the high robustness of the system. The use of peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) index proves that the encryption algorithm does not affect image quality. Further proving the excellent performance of the proposed encryption algorithm.

The game model with multi-task for image denoising and edge extraction

Abstract Image denoising and edge extraction are two main tasks in image processing. In this paper, a game model is proposed to solve the image denoising and edge extraction, which combines an adaptive improved total variation (AdITV) model for image denoising and a global sparse gradient (GSG) model for edge extraction. The AdITV model is a forward-and-backward diffusion model. In fact, forward diffusion is applied to the homogeneous region to denoise, and backward diffusion is applied to the edge region to enhance the edge. A unified explicit discrete scheme is established in this paper to solve the AdITV model, which is compatible to forward diffusion and backward diffusion. The stability of the scheme is proved. On the other hand, GSG is a functional model based on sparse representation, which is robust to extract edges under the influence of noise. AdITV and GSG are chosen as two components of the game model. The alternate iteration method is used to solve the game problem. The convergence of the algorithm is proved and numerical experiments show the effectiveness of the model.

Enhanced total variation minimization for stable image reconstruction

The total variation (TV) regularization has phenomenally boosted various variational models for image processing tasks. We propose to combine the backward diffusion process in the earlier literature on image enhancement with the TV regularization, and show that the resulting enhanced TV minimization model is particularly effective for reducing the loss of contrast. The main purpose of this paper is to establish stable reconstruction guarantees for the enhanced TV model from noisy subsampled measurements with two sampling strategies, non-adaptive sampling for general linear measurements and variable-density sampling for Fourier measurements. In particular, under some weaker restricted isometry property conditions, the enhanced TV minimization model is shown to have tighter reconstruction error bounds than various TV-based models for the scenario where the level of noise is significant and the amount of measurements is limited. The advantages of the enhanced TV model are also numerically validated by preliminary experiments on the reconstruction of some synthetic, natural, and medical images.

Existence of Periodic Waves in a Perturbed Generalized BBM Equation

In this paper, a perturbed quintic BBM equation with weak backward diffusion and dissipation effects is investigated. By applying geometric singular perturbation theory and analyzing the perturbations of a Hamiltonian system with a hyper-elliptic Hamiltonian of degree six, we prove the existence of isolated periodic wave solutions with certain wave speed in an open interval. It is also shown that isolated periodic wave solutions persist for any energy parameter [Formula: see text] in an open interval under small perturbation. Furthermore, we prove that the wave speed [Formula: see text] of periodic wave is strictly monotonically increasing with respect to [Formula: see text] by analyzing Abelian integral having three generating elements. Moreover, the upper and lower bounds of the limiting wave speed are obtained. Our analysis is mainly based on Melnikov theory, Chebyshev criteria, and symbolic computation, which may be useful for other problems.

Swarm formation as backward diffusion

Considerable progress has been made in understanding insect swarms—forms of collective animal behaviour that unlike bird flocks, fish schools and animal herds do not possess global order. Nonetheless, little is known about swarm formation. Here we posit a mechanism for the formation of insect swarms that is consistent with recent empirical observations reported by (Patel and Ouellette 2022). It correctly predicts new features of swarm formation that have not been reported on previously. Our simple analytically tractable model shows how harmonic potential wells, a characteristic feature of swarming, and so swarm cohesion, arise from diffusion and local fission-fusion dynamics and how, in accord with observations, these wells deepen over time. The overall form of these potential wells is predicted to depend on the number and spatial distribution of all individuals, making them manifestly a collective phenomenon. Finally, swarms are predicted to ‘cool’ (that is, condense) as they form.

Effect of distance of vertical barrier wall to the release nozzle on the hydrogen concentration profile induced by impingement of unignited release of cryogenic hydrogen

As a recognized clean energy source with advantages of non-pollution, high energy density and less energy loss during the cooling process, cryogenic hydrogen is widely used worldwide. The barrier wall is an effective means of preventing the backward diffusion of jets. However, the effect of distance of vertical barrier wall to the release nozzle on the hydrogen concentration profile is still unknown. In this work, cryogenic hydrogen is released with different distances of the barrier wall (0.20, 0.25, 0.30, 0.35, 0.40 m) and temperatures of cryogenic hydrogen (140, 180, 220, 260, 300 K) experimentally. 10 hydrogen concentration sensors are placed in the horizontal and vertical directions on the barrier wall, respectively, to record the hydrogen concentration. It is found that the maximum hydrogen concentration increases as the distance of the vertical barrier wall decreases. Models of the maximum hydrogen concentration in volume and mass fraction are proposed. The hydrogen concentration increases significantly in the horizontal and vertical directions as the vertical barrier wall is placed closer to the release nozzle. The hydrogen concentration in the horizontal and vertical directions of the vertical barrier wall follows the Gaussian and exponential distribution, respectively. The prediction models of hydrogen concentration in the horizontal and vertical directions are proposed as well. The research results can provide a theoretical basis and data support for the design of safety codes for cryogenic hydrogen storage and the safety distance of barrier walls in hydrogen refueling stations.

Existence of periodic wave for a perturbed MEW equation

<abstract><p>A perturbed MEW equation including small backward diffusion, dissipation and nonlinear term is considered by the geometric singular perturbation theory. Based on the monotonicity of the ratio of Abelian integrals, we prove the existence of periodic wave on a manifold for perturbed MEW equation. By Chebyshev system criterion, the uniqueness of the periodic wave is obtained. Furthermore, the monotonicity of the wave speed is proved and the range of the wave speed is obtained. Additionally, the monotonicity of period is given by Picard-Fuchs equation.</p></abstract>

EXISTENCE AND UNIQUENESS OF PERIODIC WAVES FOR A PERTURBED SEXTIC GENERALIZED BBM EQUATION

This paper is devoted to the existence and uniqueness of periodic waves for a perturbed sextic generalized BBM equation with weak backward diffusion and dissipation effects. By applying geometric singular perturbation theory and analyzing the perturbations of a Hamiltonian system with a hyper-elliptic Hamiltonian of degree seven, we prove the existence and uniqueness of periodic wave solutions with each wave speed in an open interval. It is also proved that the periodic wave solution persists for any energy parameter $h$ in an open interval and sufficiently small perturbation parameter. Furthermore, we prove that the wave speed $c_0(h)$ is strictly monotonically increasing with respect to $h$ by analyzing Abelian integral having three generating elements. Moreover, the upper and lower bounds of the limiting wave speed are obtained.