Backward Diffusion Equation Research Articles

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.

The Backward Problem for Nonlinear Fractional Diffusion Equation with Time-Dependent Order

This paper investigates the nonlinear backward fractional diffusion equations (BFDEs) having a variable order and variable diffusion coefficient. It is well-known that the BFDEs are ill-posed in the sense of Hadamard. Moreover, we investigate the problem taking into account the disturbance both variable diffusion coefficient and variable order. This may lead to some common regularization strategy to failure. Therefore, we propose a regularization method for this kind of problem. Under some appropriate regularity assumptions of the exact solution, a convergence estimate of Holder-type is proved.

Iterated fractional Tikhonov regularization method for solving the spherically symmetric backward time-fractional diffusion equation

In this paper, we focus on a backward problem for an inhomogeneous time-fractional diffusion equation on a spherical symmetric domain. This problem is ill-posed in the sense of Hadamard. We propose an iterated fractional Tikhonov regularization method in both cases: the deterministic case and random noise case. The a-priori and the a-posteriori choice rules for regularization parameters are discussed and both rules yield the corresponding convergence rates. The iterated fractional Tikhonov regularization method goes beyond the saturation results of classical Tikhonov method, and iterative fractional Tikhonov regularization method is superior to classical iterative Tikhonov regularization method under the a-priori parameter choice rule. A numerical example is conducted for showing the validity of the proposed method.

Locating small inclusions in diffuse optical tomography by a direct imaging method

Abstract Optical tomography is a typical non-invasive medical imaging technique, which aims to reconstruct geometric and physical properties of tissues by passing near infrared light through tissues for obtaining the intensity measurements. Other than optical properties of tissues, we are interested in finding locations of small inclusions inside the object from boundary measurements, based on the time-dependent diffusion model. First, we analyze the asymptotic behavior of the boundary measurements weighted by the fundamental solution of a backward diffusion equation as the diameters of inclusions go to zero. Then, we derive an efficient algorithm for locating small inclusions by finite boundary measurements. This algorithm is direct, simple and easy to be implemented numerically, since it only involves matrix operations and has no iteration process. Finally, some numerical results are presented to illustrate the feasibility and robustness of the algorithm. A new observation of the algorithm is that we can take the source points and test points independently and increase the resolution of numerical results by taking more test points.

Special Solutions to a Nonlinear Coarsening Model with Local Interactions

We consider a class of mass transfer models on a one-dimensional lattice with nearest-neighbour interactions. The evolution is given by the discrete backward fast diffusion equation, with exponent $\beta$ in the regime $(-\infty,0) \cup (0,1]$. Sites with mass zero are deleted from the system, which leads to a coarsening of the mass distribution. The rate of coarsening suggested by scaling is $t^\frac{1}{1-\beta}$ if $\beta \neq 1$ and exponential if $\beta = 1$. We prove that such solutions actually exist by an analysis of the time-reversed evolution. In particular we establish positivity estimates and long-time equililibrium properties for discrete parabolic equations with bounded initial data.

On one-dimensional forward–backward diffusion equations with linear convection and reaction

We study the initial-Neumann boundary value problem for a class of one-dimensional forward–backward diffusion equations with linear convection and reaction. The diffusion flux function is assumed to contain two forward-diffusion phases. We prove that for all smooth initial data with derivative value lying in certain phase transition regions one can construct infinitely many Lipschitz solutions that exhibit instantaneous phase transitions between the two forward phases. Furthermore, we introduce a notion of transition gauge for such solutions and prove that the transition gauge of all such constructed solutions can be arbitrarily close to a certain fixed constant. The results are new even for the pure forward–backward diffusion problem without convection and reaction. Our primary approach relies on a combination of the convex integration and Baire's category methods for a related nonlocal differential inclusion.

On a space fractional backward diffusion problem and its approximation of local solution

This article deals with a backward diffusion problem for an inhomogeneous backward diffusion equation with fractional Laplacian in R : u t ( x , t ) + − Δ α u ( x , t ) = f ( x , t ) , ( x , t ) ∈ R × [ 0 , T ] , u ( x , T ) = g ( x ) , x ∈ R , lim x → ± ∞ u ( x , t ) = 0 . This problem is an ill-posed problem due to the instability in solution. The goal of this paper is not only to provide a simple but effective regularization scheme to obtain the Hölder convergence rate, but also to give an approximation of solution of the equation with fractional diffusion to the one of the equation with Laplacian in both L 2 ( R ) and L p ( R ) setting. This result holds, in particular, when f ( x , t ) is spatially compactly supported, in which the difficulties due to the fractional Laplacian have been successfully overcome thanks to an additional condition on Fourier transform of f . We further study the convergence of solution of inhomogeneous problem to that of the homogeneous problem. Finally, numerical simulations, with finite difference schemes, based on Discrete Fourier Transform (DFT) algorithm are also presented to illustrate the theoretical results.

Hysteresis and Phase Transitions in a Lattice Regularization of an Ill-Posed Forward–Backward Diffusion Equation

We consider a lattice regularization for an ill-posed diffusion equation with trilinear constitutive law and study the dynamics of phase interfaces in the parabolic scaling limit. Our main result guarantees for a certain class of single-interface initial data that the lattice solutions satisfy asymptotically a free boundary problem with hysteretic Stefan condition. The key challenge in the proof is to control the microscopic fluctuations that are inevitably produced by the backward diffusion when a particle passes the spinodal region.

A numerical method for determining a quasi solution of a backward time-fractional diffusion equation

In this study, we present existence and uniqueness theorems of a quasi solution to backward time-fractional diffusion equation. To do that, we consider a methodology, involving minimization of a least squares cost functional, to identify the unknown initial data. Firstly, we prove the continuous dependence on the initial data for the corresponding forward problem and then we obtain a stability estimate. Based on this, we give the existence theorem of a quasi solution in an appropriate class of admissible initial data. Secondly, it is shown that the cost functional is Fréchet-differentiable and its derivative can be formulated via the solution of an adjoint problem. These results help us to prove the convexity of cost functional and subsequently the uniqueness theorem of the quasi solution. In addition, in order to approximate the quasi solution, WEB-spline finite element method is used. Since the obtained system of linear equations is ill-posed, we apply the Levenberg-Marquardt regularization. Finally, a numerical example is given to show the validation of the introduced method.

An inverse problem for space‐fractional backward diffusion problem

In this paper, an inverse problem for space‐fractional backward diffusion equation, which is highly ill‐posed, is considered. This problem is obtained from the classical diffusion equation by replacing the second‐order space derivative with a Riesz–Feller derivative of order α ∈ (0,2]. We show that such a problem is severely ill‐posed, and further present a simplified Tikhonov regularization method to deal with this problem. Convergence estimate is presented under a priori choice of regularization parameter. Numerical experiments are given to illustrate the accuracy and efficiency of the proposed method. Copyright © 2013 John Wiley & Sons, Ltd.

Regularization by projection for a backward problem of the time-fractional diffusion equation

Abstract. Time-fractional diffusion equations, which frequently appear in the diffusion process, describe the continuous time random walk phenomena. In this paper we investigate a backward time-fractional diffusion equation where one wants to extract the initial temperature distribution from the observation data provided along the final time t = T ${t=T}$ . Based upon the eigenfunction expansion, an analytical solution is deduced. However, such an approach does not depend continuously on the observation data. Hence, we propose a regularization by projection method where the truncated level plays the role of the regularization parameter. Under appropriate regularity assumptions of the exact solution, a uniform error estimate with an optimal convergence rate between the reconstructed solution and the exact one is obtained both for a priori and a posteriori parameter choice strategies. Finally, numerical examples are presented to illustrate the validity and effectiveness of the proposed method.

NUMERICAL EXPLORATION OF A FORWARD–BACKWARD DIFFUSION EQUATION

We analyze numerically a forward–backward diffusion equation with a cubic-like diffusion function — emerging in the framework of phase transitions modeling — and its "entropy" formulation determined by considering it as the singular limit of a third-order pseudo-parabolic equation. Precisely, we propose schemes for both the second- and the third-order equations, we discuss the analytical properties of their semi-discrete counterparts and we compare the numerical results in the case of initial data of Riemann type, showing strengths and flaws of the two approaches, the main emphasis being given to the propagation of transition interfaces.

Young measure solutions of some nonlinear mixed-type equations

This contribution deals with measure-valued solutions to two types of nonlinear partial differential equations for which, in general, the results on the existence of classical or weak solutions fail. These are the potential equation for transonic flow and the associated unsteady problem (forward–backward diffusion equation). The solutions are constructed by an iteration scheme (Katchanov method) and additional time discretization (Rothe method) in the second case. The existence is proved in the sense of spatial gradient Young measures. Copyright © 2010 John Wiley & Sons, Ltd.

Singular solutions of the diffusion equation of population genetics

The forward diffusion equation for gene frequency dynamics is solved subject to the condition that the total probability is conserved at all times. This can lead to solutions developing singular spikes (Dirac delta functions) at the gene frequencies 0 and 1. When such spikes appear in solutions they signal gene loss or gene fixation, with the “weight” associated with the spikes corresponding to the probability of loss or fixation. The forward diffusion equation is thus solved for all gene frequencies, namely the absorbing frequencies of 0 and 1 along with the continuous range of gene frequencies on the interval ( 0 , 1 ) that excludes the frequencies of 0 and 1. Previously, the probabilities of the absorbing frequencies of 0 and 1 were found by appeal to the backward diffusion equation, while those in the continuous range ( 0 , 1 ) were found from the forward diffusion equation. Our unified approach does not require two separate equations for a complete dynamical treatment of all gene frequencies within a diffusion approximation framework. For cases involving mutation, migration and selection, it is shown that a property of the deterministic part of gene frequency dynamics determines when fixation and loss can occur. It is also shown how solution of the forward equation, at long times, leads to the standard result for the fixation probability.

Competitive Diffusion-Influenced Reaction of a Reactive Particle with Two Static Sinks

An investigation into the kinetics of reaction between a diffusing particle and a system of two static spherical sinks is presented. The backward diffusion equation is solved for the probability of reaction with each sink, using both absorbing and radiation boundary conditions. The rate constants for each reaction are also calculated. The reactivity of the sinks is shown to be subadditive, and if the sinks are asymmetric the less reactive sink is more strongly affected by the competition. Competitive effects are found to be modeled adequately by using effective reaction radii. The IRT method is shown to have serious defects for such a system because of the correlation of the two sinks. An application to the reaction of OH radicals with thymidine is presented.

On existence of weak solutions for one-dimensional forward–backward diffusion equations

We establish the existence of infinitely many weak solutions for the general one-dimensional forward–backward diffusion equation u t = σ ( u x ) x under the homogeneous Neumann boundary condition by rephrasing it as a first-order differential inclusion problem.

The Backward Stochastic Liouville Equation

The backward stochastic Liouville equation is formulated in a Dirac-type notation in order to emphasize the kinship with the backward diffusion equation and the Heisenberg picture of quantum mechanics. The backward equations are useful both for analytic treatments and for numerical methods since the solution contains the average value of the observable for all initial spin and spatial conditions. The crucial point in using the backward equations is the boundary and initial conditions, which are derived for an arbitrary observable A and corresponding rate of change Ȧ. The formalism is applied to derive explicit analytic expressions for the recombination yield of two radicals that performs a free diffusion in a liquid under high magnetic fields. It is shown that the nontrivial limit of diffusion-controlled reaction can be directly calculated by a careful choice of boundary conditions.

Optimal regularized image restoration algorithm based on backward diffusion equation

Based on the interpretation of the restoration process as a backward diffusion process, an optimal filtering in terms of the spectral characteristics of an image is proposed. The restored image is given as the solution of the optimal backward diffusion equation (BDE). Several aspects of applying the backward diffusion on the blurred image are given. The ill-posedness of the backward diffusion is subdued by regularizing the speed at which the high-frequency components should be recovered by the use of a decreasing function. In manipulating the recovery speed, the spectral characteristics of an image are taken into account. The proposed scheme uses an exponentially decreasing speed function on the recovery of high-frequency components. The use of the decreasing function makes it possible to include more significant low-frequency components without failing the system, and thus yields results of better quality than the constant bounded scheme.

Young measure solutions of a class of forward–backward diffusion equations

This paper is devoted to Young measure solutions of a class of forward–backward diffusion equations. Inspired by the idea from a recent work of Demoulini, we first discuss the regular case by introducing the Young measure solutions and prove the existence for such solutions, and then approximate the extreme case by the approach of regularization and establish the existence of Young measure solutions in the class of functions with bounded variation.