An inverse problem for pseudoparabolic equation of filtration: the stabilization

In this article, the properties of the solution to the inverse problem on the identification of the leading coefficient of the multi-dimensional pseudoparabolic equation are studied. The stabilization of its strong solution to the solution of the inverse problem for an elliptic equation is established.     

Numerical method for the simultaneous determination of the hydraulic properties of unsaturated porous media

This paper presents a numerical algorithm for solving the inverse coefficient problem for nonlinear parabolic equations. This problem arises in simultaneous determination of the hydraulic properties of unsaturated porous media from a simple outflow experiment. The novel feature of the method is that it is not based on output least squares. In this method, the unknown functions are represented as polygons (continuous and piecewise linear functions) every new linear pieces that are determined in each time step by using information based only on previous time intervals. The results of some numerical experiments are displayed.     

Criteria for the functional controllability and invertibility of non-linear systems

In the development of investigations on inverse problems [1, 2], criteria for the functional controllability and invertibility of non-linear systems of equations with an output are obtained. The solution is based on the construction of an inverse system for which the input action of the initial system is the output. An identification problem is considered which corresponds to the problem of invertibility with an unknown initial state. The properties of λ-invertibility and λ-identifiability, which arise in cases when the output signal is known in a set of trajectories, are investigated.     

State-constrained optimization for identification of small inclusions

The inverse problem of identification of small geometric objects (defects, inclusions) of unknown topological properties is under the investigation. This problem is treated within the state-constrained optimization framework. Using topological sensitivity analysis and methods of singular perturbations, a proper approximation by the asymptotic model is justified rigorously. The underlying parametric optimization problem is solved semi-analytically by a variational calculus. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Inverse problems for quasi-variational inequalities

In this short note, our aim is to investigate the inverse problem of parameter identification in quasi-variational inequalities. We develop an abstract nonsmooth regularization approach that subsumes the total variation regularization and permits the identification of discontinuous parameters. We study the inverse problem in an optimization setting using the output-least squares formulation. We prove the existence of a global minimizer and give convergence results for the considered optimization problem. We also discretize the identification problem for quasi-variational inequalities and provide the convergence analysis for the discrete problem. We give an application to the gradient obstacle problem.     

Uniqueness for an inverse problem for a nonlinear parabolic system with an integral term by one-point Dirichlet data

We consider an inverse problem arising in laser-induced thermotherapy, a minimally invasive method for cancer treatment, in which cancer tissues is destroyed by coagulation. For the dosage planning quantitatively reliable numerical simulation are indispensable. To this end the identification of the thermal growth kinetics of the coagulated zone is of crucial importance. Mathematically, this problem is a nonlinear and nonlocal parabolic inverse heat source problem. We show in this paper that the temperature dependent thermal growth parameter can be identified uniquely from a one-point measurement.     

Application of Sinc-collocation method for solving an inverse problem

In this article, the identification of an unknown time-dependent source term in an inverse problem of parabolic type with nonlocal boundary conditions is considered. The main approach is to change the inverse problem to a system of Volterra integral equations. The resulting integral equations are convolution-type, which by using Sinc-collocation method, are replaced by a system of linear algebraic equations. The convergence analysis is included, and it is shown that the error in the approximate solution is bounded in the infinity norm by the norm of the inverse of the coefficient matrix multiplied by a factor that decays exponentially with the size of the system. To show the efficiency of the present method, an example is presented. The method is easy to implement and yields very accurate results.     

A New Energy Inversion for Parameter Identification in Saddle Point Problems with an Application to the Elasticity Imaging Inverse Problem of Predicting Tumor Location

The primary objective of this work is a detailed theoretical and computational study of the elasticity imaging inverse problem for tumor identification within the human body. Apart from this inverse problem's important and interesting application, it also poses noteworthy mathematical challenges since the underlying mathematical model is a system of elasticity involving incompressibility. This gives rise to the “locking” effect and special treatment is necessary for both the direct and inverse problems. To study the inverse problem in an optimization framework, we introduce a general computational scheme for handling parameter identification in saddle point problems along with the introduction and analysis of a new energy output least-squares objective functionals. We also present a treatment of the identification of discontinuous elasticity coefficients using the total variation regularization method. General formulas for the computation of the coefficient-to-solution map and a complete convergence analysis are given for the continuous problem as well as for its discrete analogue. Discrete formulas and implementation issues are discussed in detail and numerical examples for smooth and discontinuous coefficients are given.     

On a multiscale analysis of an inverse problem of nonlinear transfer law identification in periodic microstructure

This paper is devoted to the multiscale analysis of a homogenization inverse problem of the heat exchange law identification, which is governed by parabolic equations with nonlinear transmission conditions in a periodic heterogeneous medium. The aim of this work is to transform this inverse problem with nonlinear transmission conditions into a new one governed by a less complex nonlinear parabolic equation, while preserving the same form and physical properties of the heat exchange law that it will be identified, based on periodic homogenization theory. For this, we reformulate first the encountered homogenization inverse problem to an optimal control one. Then, we study the well-posedness of the state problem using the Leray–Schauder topological degrees and we also check the existence of the solution for the obtained optimal control problem. Finally, using the periodic homogenization theory and priori estimates, with justified choise of test functions, we reduce our inverse problem to a less complex one in a homogeneous medium.     

A direct identification method for current dipoles from electromagnetic fields

We discuss the identification problem for current dipoles in a spherically symmetric conductor. This mathematical model is used for a biomedical inverse problem such as the source current identification for the human brain activity. We have already proposed a direct identification method for this inverse source problem using observations of the magnetic fields outside of the conductor. One of the difficulties of current dipole identification using the magnetic fields is caused by the fact that magnetic field does not include any information about the radial component of dipole moments. In this paper, we consider an improvement of the direct method to identify both radial and tangential components of current dipole moments by combining electric and magnetic observation data. Furthermore, our approach is effective in the case where the number of dipoles is unknown.     

Unique solvability of the inverse problem for the heat equation with nonlocal boundary conditions

We consider the inverse problem of source identification for the heat conduction problem. The neoclassical formulation of the direct problem with integral boundary condition is used. Conditions for unique solvability of the inverse problem are obtained. __________ Translated from Prikladnaya Matematika i Informatika, No. 23, pp. 36–50, 2006.     

An Inverse Problem for a Model of a Hierarchical Structure

We consider the inverse problem for the identification of the coefficient in a parabolic equation. The model is applied to describe the functioning of a hierarchical structure [1]; it is also relevant for heat-conduction theory [2]. Unique solvability of the inverse problem is proved.     

Identification of constitutive parameters for fractional viscoelasticity

This paper develops a numerical model to identify constitutive parameters in the fractional viscoelastic field. An explicit semi-analytical numerical model and a finite difference (FD) method based numerical model are derived for solving the direct homogenous and regionally inhomogeneous fractional viscoelastic problems, respectively. A continuous ant colony optimization (ACO) algorithm is employed to solve the inverse problem of identification. The feasibility of the proposed approach is illustrated via the numerical verification of a two-dimensional identification problem formulated by the fractional Kelvin–Voigt model, and the noisy data and regional inhomogeneity etc. are taken into account.     

Legendre multiscaling functions for solving the one‐dimensional parabolic inverse problem

An inverse problem concerning diffusion equation with a source control parameter is investigated. The approximation of the problem is based on the Legendre multiscaling basis. The properties of Legendre multiscaling functions are first presented. These properties together with Galerkin method are then utilized to reduce the inverse problem to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the new technique. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Coefficient identification in Euler-Bernoulli equation from over-posed data

A method for solving the inverse problem for coefficient identification in the Euler-Bernoulli equation from over-posed data is presented. The original inverse problem is replaced by a minimization problem. The method is applied to the problem for identifying the coefficient in the case when it is a piece-wise polynomial function. Several examples are elaborated and the numerical results confirm that the solution of the imbedding problem coincides with the direct simulation of the original problem within the second order of approximation.     

A numerical method for solving an inverse thermoacoustic problem

In this paper, we consider an inverse problem of determining the initial condition of an initial boundary value problem for the wave equation with some additional information about solving a direct initial boundary value problem. The information is obtained from measurements at the boundary of the solution domain. The purpose of our paper is to construct a numerical algorithm for solving the inverse problem by an iterative method called a method of simple iteration (MSI) and to study the resolution quality of the inverse problem as a function of the number and location of measurement points. Three two-dimensional inverse problem formulations are considered. The results of our numerical calculations are presented. It is shown that the MSI decreases the objective functional at each iteration step. However, due to the ill-posedness of the inverse problem the difference between the exact and approximate solutions decreases up to some fixed number k _min, and then monotonically increases. This shows the regularizing properties of the MSI, and the iteration number can be considered a regularization parameter.     

二维恒定各向同性介质渗透系数反演的遗传算法

给出了利用遗传算法求解二维恒定各项同性介质渗透系数反演的一种新方法,该方法把参数反演问题转化为优化问题通过遗传算法求解.数值模拟结果表明:该方法具有精度高、收敛速度快、编程简单、易于计算机实现等优点,值得在实际工作采用.     

一类随机对流扩散方程的反源问题

考虑了一类由分数阶Brown运动驱动的随机对流扩散方程的源项反演问题。正问题部分首先利用分离变量法,得出了方程的温和解,进一步在期望的意义下,讨论了正问题的适定性。反问题部分研究了由终止时刻的随机数据来反演随机源项的部分统计量,并证明了相应的唯一性和不稳定性。最后进行了一些数值模拟,验证了相应的理论结果。     

Identification of the Exchange Coefficient from Indirect Data for a Coupled Continuum Pipe-Flow Model

Calibration and identification of the exchange effect between the karst aquifers and the underlying conduit network are important issues in order to gain a better understanding of these hydraulic systems. Based on a coupled continuum pipe-flow （CCPF for short） model describing flows in karst aquifers, this paper is devoted to the identification of an exchange rate function, which models the hydraulic interaction between the fissured volume （matrix） and the conduit, from the Neumann boundary data, i.e., matrix/conduit seepage velocity. The authors formulate this parameter identification problem as a nonlinear operator equation and prove the compactness of the forward mapping. The stable approximate solution is obtained by two classic iterative regularization methods, namely, the Landweber iteration and Levenberg-Marquardt method. Numerical examples on noisefree and noisy data shed light on the appropriateness of the proposed approaches.     

Inverse spectral problems for differential pencils on the half-line with turning points

The inverse spectral problem of recovering pencils of second-order differential operators on the half-line with turning points is studied. We establish properties of the spectral characteristics, give a formulation of the inverse problem, prove a uniqueness theorem and provide a constructive procedure for the solution of the inverse problem.