Search

In this paper, we study a parameter identification problem for elliptic partial differential equations. We reconstruct the coefficient with additional boundary measurements, including both Dirichle...

Considered in this paper is an inverse Robin problem governed by a steady-state diffusion equation. By the Robin inverse problem, one wants to recover the unknown Robin coefficient on an inaccessib...

In this paper, we consider an inverse source problem for elliptic partial differential equations with both Dirichlet and Neumann boundary conditions. The unknown source term is to be determined by additional boundary data. This problem is ill-posed since the dimensionality of the boundary is lower than the dimensionality of the inner domain. To overcome the ill-posed nature, using the a priori information (sourcewise representation), and based on the coupled complex boundary method, we propose a coupled complex boundary expanding compacts method (CCBECM). A finite element method is used for the discretization of CCBECM. The regularization properties of CCBECM for both the continuous and discrete versions are proved. Moreover, an a posteriori error estimate of the obtained finite element approximate solution is given and calculated by a projected gradient algorithm. Finally, numerical results show that the proposed method is stable and effective.

Adsorption isotherms are the most important parameters in rigorous models of chromatographic processes. In this paper, in order to recover adsorption isotherms, we consider a coupled complex boundary method (CCBM), which was previously proposed for solving an inverse source problem [2]. With CCBM, the original boundary fitting problem is transferred to a domain fitting problem. Thus, this method has advantages regarding robustness and computation in reconstruction. In contrast to the traditional CCBM, for the sake of the reduction of computational complexity and computational cost, the recovered adsorption isotherm only corresponds to the real part of the solution of a forward complex initial boundary value problem. Furthermore, we take into account the position of the profiles and apply the momentum criterion to improve the optimization progress. Using Tikhonov regularization, the well-posedness, convergence properties and regularization parameter selection methods are studied. Based on an adjoint technique, we derive the exact Jacobian of the objective function and give an algorithm to reconstruct the adsorption isotherm. Finally, numerical simulations are given to show the feasibility and efficiency of the proposed regularization method.

Bioluminescence tomography (BLT) aims at the determination of the distribution of a bioluminescent source quantitatively. The mathematical problem involved is an inverse source problem and is ill-p...

In this paper we propose a new class of iterative regularization methods for solving ill-posed linear operator equations. The prototype of these iterative regularization methods is in the form of second order evolution equation with a linear vanishing damping term, which can be viewed not only as an extension of the asymptotical regularization, but also as a continuous analog of the Nesterov's acceleration scheme. New iterative regularization methods are derived from this continuous model in combination with damped symplectic numerical schemes. The regularization property as well as convergence rates and acceleration effects under the Holder-type source conditions of both continuous and discretized methods are proven. The second part of this paper is concerned with the application of the newly developed accelerated iterative regularization methods to the diffusion-based bioluminescence tomography, which is modeled as an inverse source problem in elliptic partial differential equations with both Dirichlet and Neumann boundary data. A relaxed mathematical formulation is proposed so that the discrepancy principle can be applied to the iterative scheme without the usage of Sobolev embedding constants. Several numerical examples, as well as a comparison with the state-of-the-art methods, are given to show the accuracy and the acceleration effect of the new methods.

We recently proposed in [Cheng, XL et al. A novel coupled complex boundary method for inverse source problems Inverse Problem 2014 30 055002] a coupled complex boundary method (CCBM) for inverse source problems. In this paper, we apply the CCBM to inverse conductivity problems (ICPs) with one measurement. In the ICP, the diffusion coefficient q is to be determined from both Dirichlet and Neumann boundary data. With the CCBM, q is sought such that the imaginary part of the solution of a forward Robin boundary value problem vanishes in the problem domain. This brings in advantages on robustness and computation in reconstruction. Based on the complex forward problem, the Tikhonov regularization is used for a stable reconstruction. Some theoretical analysis is given on the optimization models. Several numerical examples are provided to show the feasibility and usefulness of the CCBM for the ICP. It is illustrated that as long as all the subdomains share some portion of the boundary, our CCBM-based Tikhonov re...

Considered in this paper is a Cauchy problem governed by an elliptic partial differential equation. In the Cauchy problem, one wants to recover the unknown Neumann and Dirichlet data on a part of the boundary from the measured Neumann and Dirichlet data, usually contaminated with noise, on the remaining part of the boundary. The Cauchy problem is an inverse problem with severe ill-posedness. In this paper, a coupled complex boundary method (CCBM), originally proposed in [Cheng XL, Gong RF, Han W, et al. A novel coupled complex boundary method for solving inverse source problems. Inverse Prob. 2014;30:055002], is applied to solve the Cauchy problem stably. With the CCBM, all the data, including the known and unknown ones on the boundary are used in a complex Robin boundary on the whole boundary. As a result, the Cauchy problem is transferred into a complex Robin boundary problem of finding the unknown data such that the imaginary part of the solution equals zero in the domain. Then the Tikhonov regularizat...

In this paper, we introduce and study a new method for solving inverse source problems, through a working model that arises in bioluminescence tomography (BLT). In the BLT problem, one constructs quantitatively the bioluminescence source distribution inside a small animal from optical signals detected on the animal's body surface. The BLT problem possesses strong ill-posedness and often the Tikhonov regularization is used to obtain stable approximate solutions. In conventional Tikhonov regularization, it is crucial to choose a proper regularization parameter for trade off between the accuracy and stability of approximate solutions. The new method is based on a combination of the boundary condition and the boundary measurement in a parameter-dependent single complex Robin boundary condition, followed by the Tikhonov regularization. By properly adjusting the parameter in the Robin boundary condition, we achieve two important properties for our new method: first, the regularized solutions are uniformly stable with respect to the regularization parameter so that the regularization parameter can be chosen based solely on the consideration of the solution accuracy; second, the convergence order of the regularized solutions reaches one with respect to the noise level. Then, the finite element method is used to compute numerical solutions and a new finite element error estimate is derived for discrete solutions. These results improve related results found in the existing literature. Several numerical examples are provided to illustrate the theoretical results.

We develop Second Order Asymptotical Regularization (SOAR) methods for solving inverse source problems in elliptic partial differential equations with both Dirichlet and Neumann boundary data. We show the convergence results of SOAR with the fixed damping parameter, as well as with a dynamic damping parameter, which is a continuous analog of Nesterov’s acceleration method. Moreover, by using Morozov’s discrepancy principle together with a newly developed total energy discrepancy principle, we prove that the approximate solution of SOAR weakly converges to an exact source function as the measurement noise goes to zero. A damped symplectic scheme, combined with the finite element method, is developed for the numerical implementation of SOAR, which yields a novel iterative regularization scheme for solving inverse source problems. Several numerical examples are given to show the accuracy and the acceleration effect of SOAR. A comparison with the state-of-the-art methods is also provided.

In this paper we propose a Kohn-Vogelius type formulation for an inverse source problem of partial differential equations. The unknown source term is to be determined from both Dirichlet and Neumann boundary conditions. We introduce two different boundary value problems, which depend on two different positive real numbers $\alpha$ and $\beta$, and both of them incorporate the Dirichlet and Neumann data into a single Robin boundary condition. This allows noise in both boundary data. By using the Kohn-Vogelius type Tikhonov regularization, data to be fitted is transferred from boundary into the whole domain, making the problem resolution more robust. More importantly, with the formulation proposed here, satisfactory reconstruction could be achieved for rather small regularization parameter through choosing properly the values of $\alpha$ and $\beta$. This is a desirable property to have since a smaller regularization parameter implies a more accurate approximation of the regularized problem to the original one. The proposed method is studied theoretically. Two numerical examples are provided to show the usefulness of the proposed method.

Bioluminescence tomography (BLT) is a new method in biomedical imaging, with a promising potential in monitoring non-invasively physiological and pathological processes in vivo at the cellular and molecular levels. The goal of BLT is to quantitatively reconstruct a three dimensional bioluminescent source distribution within a small animal from two dimensional optical signals on the surface of the animal body. Mathematically, BLT is an under-determined inverse source problem and is severely ill-posed, making its numerical treatments very challenging. In this paper, we provide a new Tikhonov regularization framework for the BLT problem. Compared with the existing reconstruction methods about BLT, our new method uses an energy functional defined over the whole problem domain for measuring the data fitting, associated with two related but different boundary value problems. Based on the new formulation, a fast solver is introduced by transforming the proposed optimization model into a system of partial differential equations. Moreover, a finite element method is used to obtain a regularized discrete solution. Finally, numerical results show that the fast solver for BLT is feasible and effective.

Diffuse optical tomography (DOT) and bioluminescence tomography (BLT) are different but complementary modalities in optical molecular imaging. The goal of DOT is to reconstruct optical parameters of a medium from surface measurements induced by external sources, whereas that of BLT is to reconstruct running a bioluminescent source distribution in the medium from surface measurements induced by internal bioluminescent sources. A pre-requisite for BLT reconstruction is knowledge on the distribution of optical parameters within the medium, which is the output of DOT. In this paper, we propose a mathematical model integrating BLT and DOT at the fundamental level; that is, performing the two types of reconstructions simultaneously instead of doing them sequentially. The model is introduced through minimizing the difference between predicted quantities and boundary measurements, as well as incorporating regularization terms. In practice, the optical parameters are assumed to be piecewise constants or piecewise ...

This study considers the inverse source problem for elliptic partial differential equations with both Dirichlet and Neumann boundary data. The unknown source term is to be determined by additional ...

Bioluminescence tomography (BLT) is a promising new area in biomedical imaging. The goal of BLT is to provide quantitative reconstruction of bioluminescent source distribution within a small animal from optical signals on the animal’s body surface. The multispectral version of BLT takes advantage of the measurement information in different spectrum bands. In this paper, we propose a novel approach for the multispectral BLT. The new feature of the mathematical framework is to use numerical prediction results based on two related but distinct boundary value problems. This mathematical framework includes the conventional framework in the study of multispectral BLT. For the new framework introduced here, we establish the solution existence, uniqueness and continuous dependence on data, and characterize the limiting behaviors when the regularization parameter approaches zero or when the penalty parameter approaches infinity. We study two kinds of numerical schemes for multispectral BLT and derive error estimates for the numerical solutions. We also present numerical examples to show the performance of the numerical methods.

In this paper, domain decomposition method (DDM) for numerical solutions of mathematical physics equations is improved into dynamic domain decomposition method (DDDM). The main feature of the DDDM is that the number, shape and volume of the sub-domains are all flexibly changeable during the iterations, so it suits well to be implemented on a reconfigurable parallel computing system. Convergence analysis of the DDDM is given, while an application approach to a weak nonlinear elliptic boundary value problem and a numerical experiment are discussed.

Biomedical imaging has developed into the level of molecular imaging. Bioluminescence tomography (BLT), as an optical imaging modality, is a rapidly developing new and promising field. So far, much of the theoretical analysis of BLT is based on a diffusion approximation equation for media with constant refractive index. In this article, we study the BLT problem for media with spatially varying refractive index. We introduce a general framework with Tikhonov regularization for this purpose, present its well-posedness and establish the error bounds for its numerical solution by the finite element method. Numerical results are reported on simulations of the BLT problem for media with spatially varying refractive index.

Bioluminescence tomography (BLT) is a promising new method in biomedical imaging. The BLT problem is an ill-posed inverse source problem, usually studied through a regularization technique. A new approach is proposed for solving the BLT problem based on an adjoint equation. Numerical examples show that the new formulation allows us to obtain accurate solutions.

Bioluminescence tomography (BLT) is a recently developed area in biomedical imaging. The goal of BLT is to quantitatively reconstruct a bioluminescent source distribution within a small animal from optical signals on the surface of the animal body. While there have been theoretical investigations of the BLT problem in the literature, in this paper, we propose a more general mathematical framework for a study of the BLT problem. For the proposed formulation, we establish a well-posedness result and explore its relation to the formulation studied previously in other papers. We introduce numerical methods for solving the BLT problem, show convergence, and derive error estimates for the discrete solutions. Numerical simulation results are presented showing improvement of solution accuracy with the new general mathematical framework over that with the standard formulation of BLT.

In this paper, we consider an inverse source problem for elliptic partial differential equations with Dirichlet and Neumann boundary data. The unknown source term is to be determined from additional boundary conditions. Unlike the existing methods found in the literature, which usually use some of the boundary conditions to form a boundary value problem for the elliptic partial differential equation and the remaining boundary conditions in the objective functional for optimization to determine the source term, the novel method that we propose here has coupled complex boundary conditions. We use a complex elliptic partial differential equation with a Robin boundary condition coupling the Dirichlet and Neumann boundary data, and optimize with respect to the imaginary part of the solution in the domain to determine the source term. Then, on the basis of the complex boundary value problem, Tikhonov regularization is used to obtain a stable approximate source function and the finite element method is used for discretization. Theoretical analysis is given for both the continuous model and the discrete model. Several numerical examples are provided to show the usefulness of the proposed coupled complex boundary method.