Diffusion Operator Research Articles

Computationally efficient coarse-graining XDEM/CFD modeling of fixed-bed combustion of biomass

In the multi-scale modeling of a dense particle system, the particle phase and the gas phase can be modeled on vastly different scales. The coupling between the two models has a critical influence on the predictions obtained from the combined framework but can be accomplished in a variety of ways under different assumptions. In this work, a transient 3D model using a new coupling approach for fixed-bed combustion of biomass is presented. The developed model is formulated as an Eulerian-Lagrangian framework. A particle grid, generated based on the fluid grid, is applied as a transfer grid, and a diffusion operation is implemented to smooth the interactions between the gas phase and the particles. The interactions between gas and solid phases as well as the radiative heat transfer between particles are considered. The particle motion is resolved by the soft-sphere model, whereas the conversion is calculated based on a thermally thick particle model. All sub-models are optimized to enhance computational efficiency. The 3D model is validated by comparing the simulations with laboratory-scale experiments for a fixed-bed operated in counter-current combustion mode. The key simulation parameters are configured by sensitivity analysis. The simulation results are in good agreement with the experimental measurements, and the combustion regimes with different air inlet conditions are well captured. The coupling effects are discussed in detail. The particle grid size influences the prediction of the transient results, and the interplay between the heat transfer mechanisms inside the fixed-bed and the coupling scheme is thoroughly analyzed. Both inter-particle radiation and gas-to-particle convection play essential roles in the heat transfer inside the fuel bed, while the inter-particle heat conduction can be neglected.

The evolution dynamics of an impulsive hybrid population model with spatial heterogeneity

This paper is devoted to the study of evolution dynamics for an impulsive hybrid population model with nonlocal dispersal/random diffusion operators in a heterogeneous landscape. Since this hybrid model generates a time-periodic semiflow on the space of continuous functions, the problem reduces to the investigation of a discrete-time recursion system defined by the Poincaré period map. We obtain a threshold-type result on the global dynamics of the population in a single-patch landscape. Moreover, the recursion admits a unique positive steady state which is globally asymptotically stable. For the population spread in a periodic-patchy landscape, we establish the existence of spreading speed and its coincidence with the minimal wave speed for spatially periodic traveling waves. Simulations are presented to illustrate our analytic results.

Applicable Image Security Based on New Hyperchaotic System

Hyperchaotic systems are widely applied in the cryptography domain on account of their more complex dynamical behavior. In view of this, the greatest contribution of this paper is that a two-dimensional Sine coupling Logistic modulated Sine (2D-SCLMS) system is proposed based on Logistic map and Sine map. By a series of analyses, including Lyapunov index (LE), 0–1 test, two complexity analysis methods, and two entropy analysis methods, it is concluded that the new 2D-SCLMS map is hyperchaotic with a wider range of chaos and more complex randomness. The new system combined with two-dimensional Logistic-Sine Coupling Mapping (2D-LSCM) is further applied to an image encryption application. SHA-384 is used to generate the initial values and parameters of the two chaotic systems. Symmetric keys are generated during this operation, which can be applied to the proposed image encryption and decryption algorithms. The encryption process and the decryption process of the new image encryption approaches mainly include pixel scrambling, exclusive NOR (Xnor), and diffusion operations. Multiple experiments illustrate that this scheme has higher security and lower time complexity.

A stochastic controlled Schroedinger equation: convergence and robust stability for numerical solutions

In this paper we study the stochastic stability of numerical solutions of a stochastic controlled Schr¨odinger equation. We investigate the boundedness in second moment, the convergence and the stability of the zero solution for this equation, using two new definitions of almost sure exponential robust stability and asymptotic stability, for the Euler-Maruyama numerical scheme. Considering that the diffusion term is controlled, by using the method of Lyapunov functions and the corresponding diffusion operator associated, we apply techniques of X. Mao and A. Tsoi for achieve our task. Finally, we illustrate this method with a problem in Nuclear Magnetic Resonance (NMR).

Practical computation of the diffusion MRI signal based on Laplace eigenfunctions: permeable interfaces.

The complex transverse water proton magnetization subject to diffusion-encoding magnetic field gradient pulses in a heterogeneous medium such as brain tissue can be modeled by the Bloch-Torrey partial differential equation. The spatial integral of the solution of this equation in realistic geometry provides a gold-standard reference model for the diffusion MRI signal arising from different tissue micro-structures of interest. A closed form representation of this reference diffusion MRI signal, called matrix formalism, which makes explicit the link between the Laplace eigenvalues and eigenfunctions of the tissue geometry and its diffusion MRI signal, was derived 20 years ago. In addition, once the Laplace eigendecomposition has been computed and saved, the diffusion MRI signal can be calculated for arbitrary diffusion-encoding sequences and b-values at negligible additional cost. In a previous publication, we presented a simulation framework that we implemented inside the MATLAB-based diffusion MRI simulator SpinDoctor that efficiently computes the matrix formalism representation for biological cells subject to impermeable membrane boundary conditions. In this work, we extend our simulation framework to include geometries that contain permeable cell membranes. We describe the new computational techniques that allowed this generalization and we analyze the effects of the magnitude of the permeability coefficient on the eigendecomposition of the diffusion and Bloch-Torrey operators. This work is another step in bringing advanced mathematical tools and numerical method development to the simulation and modeling of diffusion MRI.

On the lower bound of the principal eigenvalue of a nonlinear operator

We prove sharp lower bound estimates for the first nonzero eigenvalue of the non-linear elliptic diffusion operator L_p on a smooth metric measure space, without boundary or with a convex boundary and Neumann boundary condition, satisfying BE(kappa ,N) for kappa ne 0. Our results extends the work of Koerber Valtorta (Calc Vari Partial Differ Equ. 57(2), 49 2018) for case kappa =0 and Naber–Valtorta (Math Z 277(3–4):867–891, 2014) for the p-Laplacian.

Generation of analytic semigroups for some generalized diffusion operators in $$L^p$$-spaces

We consider five different abstract Cauchy problems where the operator is of fourth order. For each problem, using semigroups techniques and functional calculus, we study the invertibility and the spectral properties of the fourth order operator to show that it generates an analytic semigroup.

The novel image encryption scheme based on three- dimentional coupled chaotic system

In order to improve the complexity of chaotic mapping and the security of image encryption algorithms, this paper proposes a new coupled three-dimensional chaotic system based on sine mapping and lorenz mapping. Based on this chaotic system, a new color image encryption algorithm is proposed, which performs index position scrambling and Arnold scrambling on the bit plane of the plaintext image, and then performs an XOR diffusion operation on the scrambled ciphertext to obtain the final ciphertext image. Simulation experiments show that the algorithm has a large key space, a small image correlation coefficient, an information entropy value of 7.9921, and a good encryption effect.

Uniqueness of wave speeds in bistable reaction–diffusion equations

This article studies the uniqueness of wave speeds of traveling wave solutions in bistable reaction–diffusion equations, in which the degenerate condition at steady states is admissible and the diffusion operator may be the Laplacian, discrete lattice or integral form. By upper and lower solutions method, we prove the uniqueness of wave speeds and strict monotonicity of traveling wave solutions.

MULTIMODAL DATA VISUALIZATION AND DENOISING WITH INTEGRATED DIFFUSION.

We propose a method called integrated diffusion for combining multimodal data, gathered via different sensors on the same system, to create a integrated data diffusion operator. As real world data suffers from both local and global noise, we introduce mechanisms to optimally calculate a diffusion operator that reflects the combined information in data by maintaining low frequency eigenvectors of each modality both globally and locally. We show the utility of this integrated operator in denoising and visualizing multimodal toy data as well as multi-omic data generated from blood cells, measuring both gene expression and chromatin accessibility. Our approach better visualizes the geometry of the integrated data and captures known cross-modality associations. More generally, integrated diffusion is broadly applicable to multimodal datasets generated by noisy sensors collected in a variety of fields.

A reduced basis method for fractional diffusion operators II

Abstract We present a novel numerical scheme to approximate the solution map s ↦ u(s) := 𝓛–s f to fractional PDEs involving elliptic operators. Reinterpreting 𝓛–s as an interpolation operator allows us to write u(s) as an integral including solutions to a parametrized family of local PDEs. We propose a reduced basis strategy on top of a finite element method to approximate its integrand. Unlike prior works, we deduce the choice of snapshots for the reduced basis procedure analytically. The integral is interpreted in a spectral setting to evaluate the surrogate directly. Its computation boils down to a matrix approximation L of the operator whose inverse is projected to the s-independent reduced space, where explicit diagonalization is feasible. Exponential convergence rates are proven rigorously. A second algorithm is presented to avoid inversion of L. Instead, we directly project the matrix to the subspace, where its negative fractional power is evaluated. A numerical comparison with the predecessor highlights its competitive performance.

Strong solutions of semilinear SPDEs with unbounded diffusion

We prove a modification to the classical maximal inequality for stochastic convolutions in 2-smooth Banach spaces using the factorization method. This permits to study semilinear stochastic partial differential equations with unbounded diffusion operators driven by cylindrical Brownian motion via the mild solution approach. In the case of finite dimensional driving noise, provided sufficient regularity on the coefficients, we establish existence and uniqueness of strong solutions. In the case of “critical unboundedness”, we show how to use the stochastic compactness method to obtain a martingale solution.

Gradient flows in metric random walk spaces

Recently, motivated by problems in image processing, by the analysis of the peridynamic formulation of the continuous mechanic and by the study of Markov jump processes, there has been an increasing interest in the research of nonlocal partial differential equations. In the last years and with these problems in mind, we have studied some gradient flows in the general framework of a metric random walk space, that is, a Polish metric space (X, d) together with a probability measure assigned to each xin X, which encode the jumps of a Markov process. In this way, we have unified into a broad framework the study of partial differential equations in weighted discrete graphs and in other nonlocal models of interest. Our aim here is to provide a summary of the results that we have obtained for the heat flow and the total variational flow in metric random walk spaces. Moreover, some of our results on other problems related to the diffusion operators involved in such processes are also included, like the ones for evolution problems of p-Laplacian type with nonhomogeneous Neumann boundary conditions.

Enhanced quantum signature scheme using quantum amplitude amplification operators.

Quantum signature is the use of the principles of quantum computing to establish a trusted communication between two parties. In this paper, a quantum signature scheme using amplitude amplification techniques will be proposed. To secure the signature, the proposed scheme uses a partial diffusion operator and a diffusion operator to hide/unhide certain quantum states during communication. The proposed scheme consists of three phases, preparation phase, signature phase and verification phase. To confuse the eavesdropper, the quantum states representing the signature might be hidden, not hidden or encoded in Bell states. It will be shown that the proposed scheme is more secure against eavesdropping when compared with relevant quantum signature schemes.

An explicit iterative scheme for 3D multicomponent heat conducting flow simulation

This paper presents the development and testing of time integration scheme for the system of hydrodynamic equations of gas mixture. These equations take into account the phenomena of multicomponent diffusion and heat transfer. The governing equation system is discretized by the finite volume approach on three-dimensional unstructured grids. According to the algorithm, computation of any single time step is split into the sequence of hyperbolic and parabolic stages. The hyperbolic subtask is solved using the Godunov-type scheme. The parabolic subtask is solved by the explicit iterative Chebyshev scheme, which is algorithmically simple and does not involve tuning parameters. This stage addresses dissipative fluxes (viscosity, multicomponent diffusion and thermal conductivity). The number of the explicit iterations is determined by the convective time step and by the upper bound for the discrete diffusion operator. The resulting scheme ensures the fulfillment of the conservation laws at the discrete level. The computer code can be used in highly parallel computing for large-scale simulation. The proposed approach is recommended for applied problems in which convective and dissipative processes are in close interaction, particularly for problems of plasma physics and astrophysics.

Principles of modeling the fluorescence spectral dynamics of dye molecules in solutions

FSDS is Fluorescence Spectral Dynamics Simulator, designed to investigate photoinduced charge transfer and its manifestations in time-resolved fluorescence spectra of a dye in a solvent. The project is implemented in C, using the MPI library. The computer model simulates stationary absorption and fluorescence spectra, as well as the time evolution of population distributions along the solvent reaction coordinate, to calculate spectral dynamics. Excitation and relaxation of intramolecular high-frequency vibrations are described at the quantum level. FSDS can fit the physical parameters of the dyes to experimental data. The dye excitation is consistently described accounting for the finite duration of the pump pulse. The program code implements two evolutionary models: a spin-boson model within the harmonic potential and a stochastic model based on the Smoluchowski diffusion operator for an arbitrary free energy surface. Computer simulation shows the numerical coincidence of the calculations performed in the framework of these models for the harmonic free energy surface. The detail description of the code and numerical schemes are presented. Program summaryProgram Title: Fluorescence Spectral Dynamics Simulator (FSDS)CPC Library link to program files:https://doi.org/10.17632/d39rrrkvhr.1Code Ocean capsule:https://codeocean.com/capsule/6544976Licensing provisions: GPLv3Programming language: CSupplementary material: Sample configuration files, User ManualNature of problem: The time-resolved spectra contain reach information about photochemical processes. However, the chemically important information about separate elementary processes is entangled due to they proceed in parallel. Time-resolved spectra modeling allows extracting information about these processes. The presented program code makes it possible to determine the relaxation characteristics of the solvent and the times of intramolecular relaxation.Solution method: Finite-difference time-domain and Brownian dynamics simulation methods.

Fractional stochastic partial differential equation for random tangent fields on the sphere

This paper develops a fractional stochastic partial differential equation (SPDE) to model the evolution of a random tangent vector field on the unit sphere. The SPDE is governed by a fractional diffusion operator to model the Levy-type behaviour of the spatial solution, a fractional derivative in time to depict the intermittency of its temporal solution, and is driven by vector-valued fractional Brownian motion on the unit sphere to characterize its temporal long-range dependence. The solution to the SPDE is presented in the form of the Karhunen-Loeve expansion in terms of vector spherical harmonics. Its covariance matrix function is established as a tensor field on the unit sphere that is an expansion of Legendre tensor kernels. The variance of the increments and approximations to the solutions are studied and convergence rates of the approximation errors are given. It is demonstrated how these convergence rates depend on the decay of the power spectrum and variances of the fractional Brownian motion.

A 3D image encryption algorithm based on chaos and random cross diffusion

3D image is a kind of image closer to life, and 3D image has a stronger visual impact. In the past, researchers mostly protected flat images, but ignored the protection of 3D images. This paper combines the 2D chaotic system and uses random cross diffusion to propose a new encryption algorithm for 3D images which are STL format. Different from the traditional scrambling and diffusion operations, this paper uses only one iterative operation to achieve the effects of scrambling and diffusion, making the algorithm more secure. In the encryption algorithm, each value of the ciphertext is XORed with the value of the plaintext on the same coordinate axis and different from the coordinate where the ciphertext is located, the chaotic sequence, the ciphertext of different coordinate axes significantly reduces the correlation between the adjacent coordinate values. In addition, the proposed 3D image encryption algorithm is tested by statistical analysis. The experimental results show that this 3D image encryption algorithm has high security.

JDOI variance reduction method and the pricing of American-style options

This article revisits the Diffusion Operator Integral (DOI) variance reduction technique originally proposed in Heath and Platen (2002) and extends its theoretical concept to the pricing of American-style options under (time-homogeneous) Lévy stochastic differential equations. The resulting Jump Diffusion Operator Integral (JDOI) method can be combined with numerous Monte Carlo-based stopping-time algorithms, including the ubiquitous least-squares Monte Carlo (LSMC) algorithm of Longstaff and Schwartz (cf. Carriere (1996) and Longstaff and Schwartz (2001)). We exemplify the usefulness of our theoretical derivations under a concrete, though very general jump-diffusion stochastic volatility dynamics and test the resulting LSMC-based version of the JDOI method. The results provide evidence of a strong variance reduction when compared with a simple application of the LSMC algorithm and proves that applying our technique on top of Monte Carlo-based pricing schemes provides a powerful way to speed-up these methods.

On the Uniqueness of Solutions to Martingale Problems for Diffusion Operators with Progressively Measurable Random Coefficients

On the Uniqueness of Solutions to Martingale Problems for Diffusion Operators with Progressively Measurable Random Coefficients