Inverse source problems for multi-parameter space-time fractional differential equations with bi-fractional Laplacian operators

Over the last two decades, anomalous diffusion processes in which the mean squares variance grows slower or faster than that in a Gaussian process have found many applications. At a macroscopic level, these processes are adequately described by fractional differential equations, which involves fractional derivatives in time or/and space. The fractional derivatives describe either history mechanism or long range interactions of particle motions at a microscopic level. The new physics can change dramatically the behavior of the forward problems. For example, the solution operator of the time fractional diffusion diffusion equation has only limited smoothing property, whereas the solution for the space fractional diffusion equation may contain weak singularity. Naturally one expects that the new physics will impact related inverse problems in terms of uniqueness, stability, and degree of ill-posedness. The last aspect is especially important from a practical point of view, i.e., stably reconstructing the quantities of interest. In this paper, we employ a formal analytic and numerical way, especially the two-parameter Mittag-Leffler function and singular value decomposition, to examine the degree of ill-posedness of several ‘classical’ inverse problems for fractional differential equations involving a Djrbashian–Caputo fractional derivative in either time or space, which represent the fractional analogues of that for classical integral order differential equations. We discuss four inverse problems, i.e., backward fractional diffusion, sideways problem, inverse source problem and inverse potential problem for time fractional diffusion, and inverse Sturm–Liouville problem, Cauchy problem, backward fractional diffusion and sideways problem for space fractional diffusion. It is found that contrary to the wide belief, the influence of anomalous diffusion on the degree of ill-posedness is not definitive: it can either significantly improve or worsen the conditioning of related inverse problems, depending crucially on the specific type of given data and quantity of interest. Further, the study exhibits distinct new features of ‘fractional’ inverse problems, and a partial list of surprising observations is given below. (a) Classical backward diffusion is exponentially ill-posed, whereas time fractional backward diffusion is only mildly ill-posed in the sense of norms on the domain and range spaces. However, this does not imply that the latter always allows a more effective reconstruction. (b) Theoretically, the time fractional sideways problem is severely ill-posed like its classical counterpart, but numerically can be nearly well-posed. (c) The classical Sturm–Liouville problem requires two pieces of spectral data to uniquely determine a general potential, but in the fractional case, one single Dirichlet spectrum may suffice. (d) The space fractional sideways problem can be far more or far less ill-posed than the classical counterpart, depending on the location of the lateral Cauchy data. In many cases, the precise mechanism of these surprising observations is unclear, and awaits further analytical and numerical exploration, which requires new mathematical tools and ingenuities. Further, our findings indicate fractional diffusion inverse problems also provide an excellent case study in the differences between theoretical ill-conditioning involving domain and range norms and the numerical analysis of a finite-dimensional reconstruction procedure. Throughout we will also describe known analytical and numerical results in the literature.

Most problems in applications are inverse source problems or identification problems. This review deals with a systematic treatment of the nonnegative sources (measures) relative to the Laplace equation and the heat conduction equation producing the same potential outside a domain Ω ⊂ R³. This set is convex and weakly compact. Therefore the basic theorems on convex sets can be applied to such problems. Sets of extremal measures are characterized by geometric conditions. Almost nothing is known about the structure of the sources which produce the earth’s gravitational field. Therefore these investigations may be of interest for the inverse problem in gravimetry. The following question remains unanswered in this review: to find additional conditions to select uniquely a measure producing the earth’s gravitational field. This question is unanswered in most inverse source problems.

The numerical solution of inverse source problem for time fractional diffusion equation was studied: the time fractional derivative was is discretized by L1 algorithm, and the space derivative was approximated by central difference. The L1-finite difference scheme of the inverse source problem was constructed with the accuracy of O(τ 2–α + h 2). By adding a random disturbance to the final value and combining with the discrete scheme of L1-finite difference method, the algorithm flow was given. The effectiveness of L1-finite difference method for solve the inverse problem of time fractional diffusion equation is verified by numerical examples. The method provides an effective reference scheme for the numerical solution of the inverse problem.

Two main aims of this paper are to develop a numerical method to solve an inverse source problem for parabolic equations and apply it to solve a nonlinear coefficient inverse problem. The inverse source problem in this paper is the problem to reconstruct a source term from external observations. Our method to solve this inverse source problem consists of two stages. We first establish an equation of the derivative of the solution to the parabolic equation with respect to the time variable. Then, in the second stage, we solve this equation by the quasi-reversibility method. The inverse source problem considered in this paper is the linearization of a nonlinear coefficient inverse problem. Hence, iteratively solving the inverse source problem provides the numerical solution to that coefficient inverse problem. Numerical results for the inverse source problem under consideration and the corresponding nonlinear coefficient inverse problem are presented.

Numerical solution of an inverse source problem for a time-fractional PDE via direct meshless local Petrov–Galerkin method

ABSTRACTThis paper is devoted to solve an inverse time-dependent source problem for a time-fractional diffusion-wave equation. The source function in the time-fractional diffusion-wave equation is assumed as a multiplication of a temporal function and a spatial function. We try to identify the time-dependent source term according to an additional solute concentration distribution measured at a point on the boundary or an inner point in the solution domain. Firstly, we discuss the continuity of the weak solution for the direct problem. Then, we transform the inverse problem into a first kind of Volterra integral equation and show its ill-posedness. We use a generalized Tikhonov regularization to solve the Volterra integral equation. The generalized cross-validation choice rule is applied to find a suitable regularization parameter. Lastly, we test three examples for the inverse time-dependent source problem and show the effectiveness of the proposed regularization method.

x-v PLANE Abstract-The inverse source problem consists of determining the source term of the inhomogeneous wave equation from field measure- ments performed outside the region of space containing the source. The inverse scattering problem consists of deducing a scattering po- tential from the scattered field generated in one or more scattering experiments employing different incident plane waves. In this paper it is shown that both the inverse source and scattering problems have relevance in a number of diverse applications in ultrasonics. Specific applications addressed in the paper are ultrasonic imaging, acoustic holography, and ultrasonic transmission tomography. Ultrasonic im- aging and acoustic holography are shown to be directly related to the inverse source problem while ultrasonic transmission tomography is shown to be related to the inverse scattering problem. A number of examples are given to illustrate the theoretical material presented in the paper.

x-v PLANE Abstract-The inverse source problem consists of determining the source term of the inhomogeneous wave equation from field measure- ments performed outside the region of space containing the source. The inverse scattering problem consists of deducing a scattering po- tential from the scattered field generated in one or more scattering experiments employing different incident plane waves. In this paper it is shown that both the inverse source and scattering problems have relevance in a number of diverse applications in ultrasonics. Specific applications addressed in the paper are ultrasonic imaging, acoustic holography, and ultrasonic transmission tomography. Ultrasonic im- aging and acoustic holography are shown to be directly related to the inverse source problem while ultrasonic transmission tomography is shown to be related to the inverse scattering problem. A number of examples are given to illustrate the theoretical material presented in the paper.

We introduce a level set based approach to Bayesian geometric inverse problems. In these problems the interface between different domains is the key unknown, and is realized as the level set of a function. This function itself becomes the object of the inference. Whilst the level set methodology has been widely used for the solution of geometric inverse problems, the Bayesian formulation that we develop here contains two significant advances: firstly it leads to a well-posed inverse problem in which the posterior distribution is Lipschitz with respect to the observed data, and may be used to not only estimate interface locations, but quantify uncertainty in them; and secondly it leads to computationally expedient algorithms in which the level set itself is updated implicitly via the MCMC methodology applied to the level set function – no explicit velocity field is required for the level set interface. Applications are numerous and include medical imaging, modelling of subsurface formations and the inverse source problem; our theory is illustrated with computational results involving the last two applications.

<abstract><p>In this paper, an inverse problem of determining the space-dependent volatility from the observed market prices of options with different strikes is studied. Being different from other inverse volatility problem with classical parabolic equations, we apply the linearization method and introduce some variable substitutions to convert the original problem into an inverse source problem in a degenerate parabolic equation in a bounded area, from which an unknown volatility can be recovered and deficiencies caused by artificial truncation can be solved. Based on the optimal control framework, the problem is transformed into an optimization problem and the existence of the minimizer is established. After the necessary conditions are deduced, the uniqueness and stability of the minimizer are proved. Then, the Landweber iterative method is used to obtain a stable numerical solution of the inverse problem and some numerical experiments are also performed. The numerical results show that the algorithm which we proposed is robust and the unknown coefficient is recovered quite well.</p></abstract>

Two types of inverse source problems of identifying asynchronously distributed spatial loads governed by the Euler–Bernoulli beam equation , with hinged–clamped ends ( ), are studied. Here are linearly independent functions, describing an asynchronous temporal loading, and are the spatial load distributions. In the first identification problem the values , of the deflection , are assumed to be known, as measured output data, in a neighbourhood of the finite set of points , corresponding to the internal points of a continuous beam, for all . In the second identification problem the values , of the slope , are assumed to be known, as measured output data in a neighbourhood of the same set of points P for all . These inverse source problems will be defined subsequently as the problems ISP1 and ISP2. The general purpose of this study is to develop mathematical concepts and tools that are capable of providing effective numerical algorithms for the numerical solution of the considered class of inverse problems. Note that both measured output data and contain random noise. In the first part of the study we prove that each measured output data and can uniquely determine the unknown functions . In the second part of the study we will introduce the input–output operators , and , , corresponding to the problems ISP1 and ISP2, and then reformulate these problems as the operator equations: and , where and . Since both measured output data contain random noise, we use the most prominent regularisation method, Tikhonov regularisation, introducing the regularised cost functionals and . Using a priori estimates for the weak solution of the direct problem and the Tikhonov regularisation method combined with the adjoint problem approach, we prove that the Fréchet gradients and of both cost functionals can explicitly be derived via the corresponding weak solutions of adjoint problems and the known temporal loads . Moreover, we show that these gradients are Lipschitz continuous, which allows the use of gradient type iteration convergent algorithms. Two applications of the proposed theory are presented. It is shown that solvability results for inverse source problems related to the synchronous loading case, with a single interior measured data, are special cases of the obtained results for asynchronously distributed spatial load cases.

A numerical algorithm in reproducing kernel-based approach for solving the inverse source problem of the time–space fractional diffusion equation

An inverse source problem for non-smooth multiterm time Caputo fractional diffusion with fractional order designed as β0 < β1 < ⋯ < βM < 1 is the case of study in a bounded Lipschitz domain in $\mathbb {R}^{d}$ . The missing solely time-dependent source function is reconstructed from an additional integral measurement. The existence, uniqueness and regularity of a weak solution for the inverse source problem is investigated. We design a numerical algorithm based on Rothe’s method over graded meshes, derive a priori estimates and prove convergence of iterates towards the exact solution. An essential feature of the multiterm time Caputo fractional subdiffusion problem is that the solution possibly lacks the smoothness near the initial time, although it would be smooth away from t = 0. In this contribution, we will establish an extension of Grönwall’s inequalities for multiterm fractional operators. This extension will be crucial for showing the existence of a unique solution to the inverse problem. The theoretical obtained results are supported by some numerical experiments.

A new approach for space-time fractional partial differential equations by residual power series method

This paper is concerned with the mathematical analysis of an inverse random source problem for the time fractional diffusion equation, where the source is driven by a fractional Brownian motion. Given the random source, the direct problem is to study the stochastic time fractional diffusion equation. The inverse problem is to determine the statistical properties of the source from the expectation and variance of the final time data. For the direct problem, we show that it is well-posed and has a unique mild solution under a certain condition. For the inverse problem, the uniqueness is proved and the instability is characterized. The major ingredients of the analysis are based on the properties of the Mittag–Leffler function and the stochastic integrals associated with the fractional Brownian motion.