Diffusion Partial Differential Equation Research Articles

Product of Gaussian Mixture Diffusion Models

AbstractIn this work, we tackle the problem of estimating the density $$ f_X $$ f X of a random variable $$ X $$ X by successive smoothing, such that the smoothed random variable $$ Y $$ Y fulfills the diffusion partial differential equation $$ (\partial _t - \Delta _1)f_Y(\,\cdot \,, t) = 0 $$ ( ∂ t - Δ 1 ) f Y ( · , t ) = 0 with initial condition $$ f_Y(\,\cdot \,, 0) = f_X $$ f Y ( · , 0 ) = f X . We propose a product-of-experts-type model utilizing Gaussian mixture experts and study configurations that admit an analytic expression for $$ f_Y (\,\cdot \,, t) $$ f Y ( · , t ) . In particular, with a focus on image processing, we derive conditions for models acting on filter, wavelet, and shearlet responses. Our construction naturally allows the model to be trained simultaneously over the entire diffusion horizon using empirical Bayes. We show numerical results for image denoising where our models are competitive while being tractable, interpretable, and having only a small number of learnable parameters. As a by-product, our models can be used for reliable noise level estimation, allowing blind denoising of images corrupted by heteroscedastic noise.

Modeling anomalous transport in fractal porous media: A study of fractional diffusion PDEs using numerical method

Abstract Fractional diffusion partial differential equation (PDE) models are used to describe anomalous transport phenomena in fractal porous media, where traditional diffusion models may not be applicable due to the presence of long-range dependencies and non-local behaviors. This study presents an efficient hybrid meshless method to the compute numerical solution of a two-dimensional multiterm time-fractional convection-diffusion equation. The proposed meshless method employs multiquadric-cubic radial basis functions for the spatial derivatives, and the Liouville-Caputo derivative technique is used for the time derivative portion of the model equation. The accuracy of the method is evaluated using error norms, and a comparison is made with the exact solution. The numerical results demonstrate that the suggested approach achieves better accuracy and computationally efficient performance.

Structural identifiability analysis of linear reaction–advection–diffusion processes in mathematical biology

Effective application of mathematical models to interpret biological data and make accurate predictions often requires that model parameters are identifiable. Approaches to assess the so-called structural identifiability of models are well established for ordinary differential equation models, yet there are no commonly adopted approaches that can be applied to assess the structural identifiability of the partial differential equation (PDE) models that are requisite to capture spatial features inherent to many phenomena. The differential algebra approach to structural identifiability has recently been demonstrated to be applicable to several specific PDE models. In this brief article, we present general methodology for performing structural identifiability analysis on partially observed reaction–advection–diffusion PDE models that are linear in the unobserved quantities. We show that the differential algebra approach can always, in theory, be applied to such models. Moreover, despite the perceived complexity introduced by the addition of advection and diffusion terms, consideration of spatial analogues of non-spatial models cannot exacerbate structural identifiability. We conclude by discussing future possibilities and the computational cost of performing structural identifiability analysis on more general PDE models.

Accuracy Measurement of Hyperspectral Image Classification in Remote Sensing with the Light Spectrum-based Affinity Propagation Clustering-based Segmentation

The area of remote sensing and computer vision includes the challenge of hyperspectral image classification. It entails grouping pixels in hyperspectral pictures into several classes according to their spectral signature. Hyperspectral photographs are helpful for a variety of applications, including vegetation study, mineral mapping, and mapping urban land use, since they include information on an object's reflectance in hundreds of small, contiguous wavelength bands. This task's objective is to correctly identify and categorize several item categories in the image. Many approaches have been stated by several researchers in this field to enhance the accuracy of the segmentation and accuracy. However, fails to attain the optimal accuracy due to the intricate nature of the images. To tackle these issues, we propose a novel Modified Extreme Learning machine (M-ELM) approach for the credible hyperspectral image classification outcomes with the publicly available Kaggle datasets. Before the classification, the input images are segmented using the Light Spectrum-based modified affinity propagation clustering technique (LSO-MAPC). In the beginning, the images are pre-processed using the non-linear diffusion partial differential equations technique which effectively pre-processed the image spatially. Experiments are effectuated to analyze the performance of the proposed method and compared it with state-of-art works in a quantitative way. The proposed approach ensures a classification accuracy of 96%.

Modelling count data with partial differential equation models in biology

Partial differential equation (PDE) models are often used to study biological phenomena involving movement-birth–death processes, including ecological population dynamics and the invasion of populations of biological cells. Count data, by definition, is non-negative, and count data relating to biological populations is often bounded above by some carrying capacity that arises through biological competition for space or nutrients. Parameter estimation, parameter identifiability, and making model predictions usually involves working with a measurement error model that explicitly relating experimental measurements with the solution of a mathematical model. In many biological applications, a typical approach is to assume the data are normally distributed about the solution of the mathematical model. Despite the widespread use of the standard additive Gaussian measurement error model, the assumptions inherent in this approach are rarely explicitly considered or compared with other options. Here, we interpret scratch assay data, involving migration, proliferation and delays in a population of cancer cells using a reaction–diffusion PDE model. We consider relating experimental measurements to the PDE solution using a standard additive Gaussian measurement error model alongside a comparison to a more biologically realistic binomial measurement error model. While estimates of model parameters are relatively insensitive to the choice of measurement error model, model predictions for data realisations are very sensitive. The standard additive Gaussian measurement error model leads to biologically inconsistent predictions, such as negative counts and counts that exceed the carrying capacity across a relatively large spatial region within the experiment. Furthermore, the standard additive Gaussian measurement error model requires estimating an additional parameter compared to the binomial measurement error model. In contrast, the binomial measurement error model leads to biologically plausible predictions and is simpler to implement. We provide open source Julia software on GitHub to replicate all calculations in this work, and we explain how to generalise our approach to deal with coupled PDE models with several dependent variables through a multinomial measurement error model, as well as pointing out other potential generalisations by linking our work with established practices in the field of generalised linear models.

Pulse-adding of temporal dissipative solitons: resonant homoclinic points and the orbit flip of case B with delay

We numerically investigate the branching of temporally localised, two-pulse solutions from one-pulse periodic solutions with non-oscillating tails in delay differential equations (DDEs) with large delay. Solutions of this type are commonly referred to as temporal dissipative solitons (TDSs) (Yanchuk et al 2019 Phys. Rev. Lett. 123 53901) in applications, and we adopt this term here. We show by means of a prototypical example that—analogous to travelling pulses in reaction–diffusion partial differential equations (Yanagida 1987 J. Differ. Equ. 66 243–62)—the branching of two-pulse TDSs from one-pulse TDSs with non-oscillating tails is organised by codimension-two homoclinic bifurcation points of a real saddle equilibrium (Homburg and Sandstede 2010 Handbook of Dynamical Systems Elsevier) in a corresponding profile equation. We consider a generalisation of Sandstede’s model (Sandstede 1997 J. Dyn. Differ. Equ. 9 269–88) (a prototypical model for studying codimension-two homoclinic bifurcation points in ordinary differential equations) with an additional time-shift parameter, and use Auto07p (Doedel 1981 Congr. Numer. 30 265–84; Doedel and Oldeman 2010 AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equations Concordia University) and DDE-BIFTOOL (Sieber et al 2014 arXiv:1406.7144) to compute numerically the unfolding of these bifurcation points in the resulting DDE. We then interpret this model as the profile equation for TDSs in a DDE with large delay by exploiting the reappearance of periodic solutions in DDEs (Yanchuk and Perlikowski 2009 Phys. Rev. E 79 046221). In doing so, we identify both the non-orientable resonant homoclinic bifurcation and the orbit flip bifurcation of case B as organising centres for the existence of two-pulse TDSs in the DDE with large delay. We study the bifurcation curves emanating from these codimension-two points beyond a local neighbourhood in parameter space. In this way, we are able to discuss how folds of homoclinic bifurcations in an extended system bound the existence region of TDSs in the DDE with large delay. We also discuss the relation between a reduced multivalued-map (in the limit of infinite delay) and the existence of TDSs.

Three-Dimensional Analytical Mathematical Diffusion Model of Air Pollutant in a Mixing Layer with Chemical Reaction and Wet Deposition of Larger Particles of Pollutants Due to a Point Source

A three dimensional analytical mathematical diffusion model of air pollutant in a mixing layer with chemical reaction and wet deposition of larger particles of pollutants due to a point source at is furnished to investigate the primary pollutant concentration in a protected sector and it is above the earth surface layer of the earth’s atmosphere. The pollutants are assumed to be emitted only from a ceaseless source in surface layer. The physical phenomenon of the above problem is reduced to advection diffusion partial differential equation and is solved by using the method of Fourier’s technique. The effect of rate of chemical reaction of pollutants and rate of wet deposition of pollutants on the concentration of dispersed pollutants is investigatedfor both stable and neutral atmospheric situations. Concentration contours are plotted and the results are analysed for air pollutants in stable and neutral atmospheric situations. In the upper part of the mixing layer pollutants concentration is less in stable atmospheric condition for smaller roughness length and moderate geostrophic wind speed.

A hybrid technique based on Lucas polynomials for solving fractional diffusion partial differential equation

This paper presents a new numerical technique to approximate solutions of diffusion partial differential equations with Caputo fractional derivatives. We use a spectral collocation method based on Lucas polynomials for time fractional derivatives and a finite difference scheme in space. Stability and error analyses of the proposed technique are established. To demonstrate the reliability and efficiency of our new technique, we applied the method to a number of examples. The new technique is simply applicable, and the results show high efficiency in calculation and approximation precision.

Event-triggered boundary consensus control for multi-agent systems of fractional reaction–diffusion PDEs

The distributed consensus is considered for multi-agent systems (MASs), which characterized by fractional reaction–diffusion partial differential equations (RDPDEs) in this paper. Based on Lyapunov technique and linear matrix inequalities (LMIs) theory, the consensus can be realized via two novel event-triggered boundary control schemes. Firstly, a novel convergence principle subject to finite time is presented for the continuously differentiable function. Secondly, the cooling fin on surface of high-speed aerospace vehicle is remodeled by fractional RDPDEs system, and the well-posedness of presented system is discussed applying the monotone iterative approach. Thirdly, according to the presented static event-triggered boundary control strategy, the consensus criterion in finite time is addressed in the form of LMIs, in addition, the settling time is calculated accurately. Applying the dynamic event-triggered control protocol, the Mittag-Leffler (M-L) consensus condition is achieved. Moreover, the Zeno behaviors are ruled out for proposed event-triggered mechanisms. Finally, the high-speed aerospace vehicle model is presented to verify the effectiveness of the control performance.

Quantum algorithm for smoothed particle hydrodynamics

We present a quantum computing algorithm for the smoothed particle hydrodynamics (SPH) method. We use a normalization procedure to encode the SPH operators and domain discretization in a quantum register. We then perform the SPH summation via an inner product of quantum registers. Using a one-dimensional function, we test the approach in a classical sense for the kernel sum and first and second derivatives of a one-dimensional function, using both the Gaussian and Wendland kernel functions, and compare various register sizes against analytical results. Error convergence is exponentially fast in the number of qubits. We extend the method to solve the one-dimensional advection and diffusion partial differential equations, which are commonly encountered in fluids simulations. This work provides a foundation for a more general SPH algorithm, eventually leading to highly efficient simulations of complex engineering problems on gate-based quantum computers.

Learning-based local weighted least squares for algebraic multigrid method

Algebraic multigrid (AMG) is an effective iterative algorithm for solving large-scale linear systems. One challenge of constructing the AMG algorithm is the determination of the prolongation operator, which affects the convergence rate of AMG and is problem-dependent. In this paper, we propose a new Learning-based Local Weighted Least Squares (L-LWLS) method to construct the prolongation operator of AMG. Specifically, we construct the prolongation operator by solving the LWLS model with learned spatially-varying weights. We use the gradient descent algorithm to optimize the model with a learned initialization of the solution. Then the constructed prolongation operator is further corrected by a learned correction function to improve the convergence rate of AMG. We conduct experiments on solving graph Laplacian linear systems, diffusion partial differential equations, and Helmholtz equations. Experiments show that the proposed method can construct a better prolongation operator leading to a faster convergence rate than the compared methods, including the classical AMG, the smoothed aggregation AMG, the bootstrap AMG, and the learning-based AMG method. The results show that the proposed method can generalize well to different parameter distributions and problem sizes, i.e., the number of variables in the linear system.

Bilateral boundary control of an input delayed 2-D reaction–diffusion equation

In this paper, a delay compensation design method based on PDE backstepping is developed for a two-dimensional reaction–diffusion partial differential equation (PDE) with bilateral input delays. The PDE is defined in a rectangular domain, and the bilateral control is imposed on a pair of opposite sides of the rectangle. To represent the delayed bilateral inputs, we introduce two 2-D transport PDEs that form a cascade system with the original PDE. A novel set of backstepping transformations is proposed for delay compensator design, including one Volterra integral transformation and two affine Volterra integral transformations. Unlike the kernel equation for 1-D PDE systems with delayed boundary input, the resulting kernel equations for the 2-D system have singular initial conditions governed by the Dirac Delta function. Consequently, the kernel solutions are written as a double trigonometric series with singularities. To address the challenge of stability analysis posed by the singularities, we prove a set of inequalities by using the Cauchy–Schwarz inequality, the 2-D Fourier series, and the Parseval’s theorem. A numerical simulation illustrates the effectiveness of the proposed delay-compensation method.

An open-source pipeline for solving continuous reaction–diffusion models in image-based geometries of porous media

We present a versatile open-source pipeline for simulating inhomogeneous reaction–diffusion processes in highly resolved, image-based geometries of porous media with reactive boundaries. Resolving realistic pore-scale geometries in numerical models is challenging and computationally demanding, as the scale differences between the sizes of the interstitia and the whole system can lead to prohibitive memory requirements. The present pipeline combines a level-set method with geometry-adapted sparse block grids on GPUs to efficiently simulate reaction–diffusion processes in image-based geometries. We showcase the method by applying it to fertilizer diffusion in soil, heat transfer in porous ceramics, and determining effective diffusion coefficients and tortuosity. The present approach enables solving reaction–diffusion partial differential equations in real-world geometries applicable to porous media across fields such as engineering, environmental science, and biology.

Numeric Fem’s Solution for Space-Time Diffusion Partial Differential Equations with Caputo–Fabrizion and Riemann–Liouville Fractional Order’s Derivatives

Abstract In this paper, we use the finite element method to solve the fractional space-time diffusion equation over finite fields. This equation is obtained from the standard diffusion equation by replacing the first temporal derivative with the new fractional derivative recently introduced by Caputo and Fabrizion and the second spatial derivative with the Riemann–Liouville fractional derivative. The existence and uniqueness of the numerical solution and the result of error estimation are given. Numerical examples are used to support the theoretical results.

Spectral Stability of Shock-fronted Travelling Waves Under Viscous Relaxation

Reaction-nonlinear diffusion partial differential equations can exhibit shock-fronted travelling wave solutions. Prior work by Li et al. (Physica D 423:132916, 2021) has demonstrated the existence of such waves for two classes of regularizations, including viscous relaxation (see Li et al. in Physica D 423:132916, 2021). Their analysis uses geometric singular perturbation theory: for sufficiently small values of a parameter ε>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon > 0$$\\end{document} characterizing the ‘strength’ of the regularization, the waves are constructed as perturbations of a singular heteroclinic orbit. Here we show rigorously that these waves are spectrally stable for the case of viscous relaxation. Our approach is to show that for sufficiently small ε>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon >0$$\\end{document}, the ‘full’ eigenvalue problem of the regularized system is controlled by a reduced slow eigenvalue problem defined for ε=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon = 0$$\\end{document}. In the course of our proof, we examine the ways in which this geometric construction complements and differs from constructions of other reduced eigenvalue problems that are known in the wave stability literature.

Learning subgrid-scale models with neural ordinary differential equations

We propose a new approach to learning the subgrid-scale model when simulating partial differential equations (PDEs) solved by the method of lines and their representation in chaotic ordinary differential equations, based on neural ordinary differential equations (NODEs). Solving systems with fine temporal and spatial grid scales is an ongoing computational challenge, and closure models are generally difficult to tune. Machine learning approaches have increased the accuracy and efficiency of computational fluid dynamics solvers. In this approach neural networks are used to learn the coarse- to fine-grid map, which can be viewed as subgrid-scale parameterization. We propose a strategy that uses the NODE and partial knowledge to learn the source dynamics at a continuous level. Our method inherits the advantages of NODEs and can be used to parameterize subgrid scales, approximate coupling operators, and improve the efficiency of low-order solvers. Numerical results with the two-scale Lorenz 96 ODE, the convection–diffusion PDE, and the viscous Burgers’ PDE are used to illustrate this approach.

Hydrogen diffusion into caprock: A semi-analytical solution and a hydrogen loss criterion

Depleted gas reservoirs can provide a gigaton capacity and the necessary infrastructure for large scale hydrogen storage. Therefore, underground storage of hydrogen in these geological formations is considered a valuable option. However, the high molecular diffusion of hydrogen from these storage sites is a major concern. In this paper, the idea of hydrogen diffusion through the caprock during geological storage was investigated using a series of one- and three-dimensional numerical models. In these models, the caprock and overlying formations were considered as a semi-infinite system, while the interaction between the reservoir and the caprock was defined by the pressure and gas composition at the boundary. Hydrogen diffusion was based on chemical potential differences and the influence of pre-diffused hydrocarbon gas in the caprock was considered. Furthermore, based on the analogies between the related partial differential equations of diffusion and spontaneous imbibition processes, a simple yet accurate semi-analytical solution was proposed. The presented semi-analytical solution can be used to predict the cumulative hydrogen loss over time. This solution states that the hydrogen loss increases proportionally with the square root of time. Using the semi-analytical solution, a criterion for loss by molecular hydrogen diffusion was presented that indicates the fraction of hydrogen lost by diffusion at a given time. The results obtained based on the diffusion fluxes showed that the hydrogen loss can be underestimated if the existence of hydrocarbon gas in the caprock is ignored. The analysis also indicated that hydrogen loss is directly proportional to the interface between rock and reservoir exposed to hydrogen, the caprock porosity, the gas saturation of the caprock, the square root of the diffusion coefficient, and the square root of time. While the presence of hydrocarbon gas facilitates diffusion, the thermodynamic effects at high pressures lead to a comparatively lower molar density in the caprock than in the matrix. Thus, a lower final loss of the injected hydrogen can be expected at higher pressures.

A Model of Froth Flotation with Drainage: Simulations and Comparison with Experiments

The operation of a froth flotation column can be described by a nonlinear convection–diffusion partial differential equation that incorporates the solids–flux and drift–flux theories as well as a model of foam drainage. The resulting model predicts the bubble and (gangue) particle volume fractions as functions of height and time. The steady-state (time-independent) version of the model defines so-called operating charts that map conditions on the gas and pulp feed rates that allow for operation with a stationary froth layer. Operating charts for a suitably adapted version of the model are compared with experimental results obtained with a laboratory flotation column. Experiments were conducted with a two-phase liquid–bubble flow. The results indicate good agreement between the predicted and measured conditions for steady states. Numerical simulations for transient operation, in part for the addition of solid particles, are presented.

Solving a generalized fractional diffusion equation with variable fractional order and moving boundary by two numerical methods: FDM vs FVM

In this paper, both the finite difference method (FDM) and the finite volume method (FVM) are employed to solve the fractional partial differential diffusion equation with temporal dimension and one spatial dimension. In this case, the boundary on the right of the domain is moving with time, while the variable fractional order is depicted as a function of both time and space. Special technique has been proposed to deal with the moving boundary which not only involves the computational difficulty and also accumulates the error. The accuracy and computational resource consumption of the two methods are compared in four designed cases with different functions of moving boundaries and fractional orders. The results show that the computation cost of FDM and FVM is almost the same in problems with one-dimensional space, but the accuracy of the FDM is higher than that of the FVM. Besides, compared with linear cases, the computational accuracy of both methods decreases significantly with nonlinear functions of fractional derivative and moving boundary.

Stability and decoherence rates of a GKP qubit protected by dissipation⋆

We analyze an experimentally accessible Lindblad master equation for a quantum harmonic oscillator. It approximately stabilizes finite-energy periodic grid states called Gottesman-Kitaev-Preskill (GKP) states, that can be used to encode and protect a logical qubit. We give explicit upper bounds for the energy of the solutions of the Lindblad master equation. Using three periodic observables to define the Bloch sphere coordinates of a logical qubit, we show that their dynamics is governed by a diffusion partial differential equation on a 2D-torus with a Witten Laplacian. We show that the evolution of these logical coordinates is exponentially slow even in presence of small diffusive noise processes along the two quadratures of the phase space. Numerical simulations indicate similar results for other physically relevant noise processes.