Numerical solution of an inverse problem for a two-dimensional mathematical model of sorption dynamics

We consider a two-dimensional mathematical model of sorption that allows for inner-diffusion kinetics as well as longitudinal and transverse diffusion. The inverse problem of determining the sorption isotherm from an experimental dynamic output curve is investigated for this model and stable solution methods are proposed for the inverse and the direct problem. The efficiency of the solution methods is explored in computer experiments.     

Unique Solvability of the Inverse Problem with a Time-Dependent Boundary Condition for a Sorption Model

A mixed initial boundary-value problem is considered for nonequilibrium sorption dynamics with inner-diffusion kinetics. The problem allows for convection and longitudinal diffusion and has a time-dependent boundary condition. This condition contains the time derivative of a solution component and constitutes the balance equation for the absorbed mixture near the output boundary of the sorption region—inside the diffusion barrier. Bounds on the solution of the direct problem are obtained: nonnegativity of the solution and its first time derivatives, and boundedness of the solution by known functions. The inverse problem of estimating the nonlinear system parameter—the sorption isotherm—is considered and a uniqueness theorem is proved.     

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.     

The Sinc-Galerkin method for parameter-dependent self-adjoint problems

The Sinc-Galerkin method is being applied to a growing number of diverse problems in ordinary and partial differential equations including both forward and inverse (parameter recovery) problems. As a result of these continuing extensions, the treatment of parameter-dependent problems needs to be thoroughly investigated. Two specific questions considered here are the incorporation of various nonhomogeneous boundary conditions and the treatment of a variable parameter. The latter topic is particularly important for inverse problems that arise when numerically estimating physical parameters. The point of view taken emphasizes the maintenance of the classical exponential convergence rate. The techniques described are suitable both for the direct problem and for the parameter estimation problem. Numerical results are presented to substantiate the accuracy of the method.     

A stochastic model for the direct and inverse problem of adhesive materials

This work deals with the generation of artificial data based on experimental data for adhesive materials and the application of this data to the inverse and the direct problem. In reality there are only a very limited number of experimental data available. Therefore, the prediction of material behaviour is difficult and a statistical analysis with a stochastic proved thesis is nearly impossible. In order to increase the number of tests a method of stochastic simulation based on time series analysis is applied. With artificial data an arbitrary number of data is available and the process of the parameter identification can be statistically analysed. Additionally, one example is shown, which adapts the analysed material parameter to the direct problem. The stochastic finite element method is used to take into account the distribution and deviation of the fracture strain. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

A uniqueness theorem in the inverse problem for the integrodifferential electrodynamics equations

Under study is the problem of finding the kernel and the index of dielectric permeability for the system of integrodifferential electrodynamics equations with wave dispersion. We consider a direct problem in which the external pulse current is a dipole located at a point y on the boundary ?B of the unit ball B. The point y runs over the whole boundary and is a parameter of the problem. The information available about the solution to the direct problem is the trace on ?B of the solution to the Cauchy problem given for the times close to the time when a wave from the dipole source arrives at a point x. The main result of the article consists in obtaining some theorems related to the uniqueness problems for a solution to the inverse problem.     

The Lyapunov-Poincaré method in the theory of periodic motions

The problem of the periodic motions of a system with a small parameter is solved. The non-rough cases, when the problem cannot be solved by a generating system obtained for a zero value of the small parameter, are investigated. Lyapunov's idea of using a new generating system which already contains the small parameter is systematically developed. Systems of general form, inverse systems and systems close to inverse are investigated.     

Practical stability of dynamical systems dependent on a parameter

We consider the problem of ensuring stability of a system under changes in its parameters, which requires algorithms for estimating the deviation of the actual trajectory due to parameter perturbations. The inverse problem is also considered: for given constraints on the admissible deviation of the trajectory or on the performance criterion, determine the tolerances of the system parameters. These problems are solved by practical stability methods. Theorems and algorithms developed in this paper are applied to solve both direct and inverse problems of sensitivity theory. The stability criteria for the sensitivity equations are applied to determine the relationship between initial conditions and parameters in terms of limiting dynamic constraints on the sensitivity functions.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 55, pp. 119–125, 1985.     

Modified auxiliary boundary conditions method for an ill-posed problem for the homogeneous biharmonic equation

In this paper, we are interested in the inverse problem for the biharmonic equation posed on a rectangle, which is of great importance in many areas of industry and engineering. We show that the problem under consideration is ill-posed; therefore, to solve it, we opted for a regularization method via modified auxiliary boundary conditions. The numerical implementation is based on the application of the semidiscrete finite difference method for a sequence of well-posed direct problems depending on a small parameter of regularization. Numerical results are performed for a rectangle domain showing the effectiveness of the proposed method.     

Numerical solution of an inverse problem of nonisothermal sorption dynamics

We consider the inverse problem for a mathematical model of sorption dynamics that incorporates diffusion, intradiffusion kinetics, and the heat balance. Two numerical methods are proposed. Their efficiency is investigated in a computer experiment. Translated from Prikladnaya Matematika i Informatika, No. 29, pp. 56–63, 2008.     

The Complexity Analysis of the Inverse Center Location Problem

Given a feasible solution, the inverse optimization problem is to modify some parameters of the original problem as little as possible, and sometimes also with bound restrictions on these adjustments, to make the feasible solution become an optimal solution under the new parameter values. So far it is unknown that for a problem which is solvable in polynomial time, whether its inverse problem is also solvable in polynomial time. In this note we answer this question by considering the inverse center location problem and show that even though the original problem is polynomially solvable, its inverse problem is NP–hard.     

Determination of precession and dissipation parameters in micromagnetism

The precession β and the dissipation parameter α of a ferromagnetic material can be considered microscopically space dependent. Their space distribution is difficult to obtain by direct measurements. In this article we consider an inverse problem, where we aim at recovering α and β from space measurements of the magnetization. The evolution of the magnetization in micromagnetism is governed by the Landau-Lifshitz (LL) equation. We first study the sensitivity of the LL equation. We derive the existence, uniqueness and stability results for the LL equation and the corresponding sensitivity equations. On the basis of the results we analyze the inverse problem. We employ the energy method and we minimize the underlying cost functional by means of the steepest descent method. We derive a convergence result for the proposed algorithm. The presented numerical examples support the theoretical results.     

Meta-control of an interacting-particle algorithm for global optimization

A common issue for stochastic global optimization algorithms is how to set the parameters of the sampling distribution (e.g. temperature, mutation/cross-over rates, selection rate, etc.) so that the samplings converge to the optimum effectively and efficiently. We consider an interacting-particle algorithm and develop a meta-control methodology which analytically guides the inverse temperature parameter of the algorithm to achieve desired performance characteristics (e.g. quality of the final outcome, algorithm running time, etc.). The main aspect of our meta-control methodology is to formulate an optimal control problem where the fractional change in the inverse temperature parameter is the control variable. The objectives of the optimal control problem are set according to the desired behavior of the interacting-particle algorithm. The control problem considers particles’ average behavior, rather than treating the behavior of individual particles. The solution to the control problem provides feedback on the inverse temperature parameter of the algorithm.     

A Stability Estimate for a Solution in the Inverse Problem of Finding the Coefficients of Lower Derivatives

We consider the problem of finding the coefficients of the first derivatives in a second-order hyperbolic equation. The additional information is the trace of a solution and its normal derivative on the lateral surface of the cylindrical domain of some direct problem for the original equation. The impulse point source lies outside the domain in which the sought coefficients are determined and is a parameter of the problem. We suppose that the number of sources for which the trace of a solution is given coincides with the number of the coefficients to be determined. The main result of this article is a stability estimate for a solution to the inverse problem under consideration.     

THE QUASI-BOUNDARY VALUE METHOD FOR IDENTIFYING THE INITIAL VALUE OF THE SPACE-TIME FRACTIONAL DIFFUSION EQUATION

In this article, we consider to solve the inverse initial value problem for an inhomogeneous space-time fractional diffusion equation. This problem is ill-posed and the quasi-boundary value method is proposed to deal with this inverse problem and obtain the series expression of the regularized solution for the inverse initial value problem. We prove the error estimates between the regularization solution and the exact solution by using an a priori regularization parameter and an a posteriori regularization parameter choice rule. Some numerical results in one-dimensional case and two-dimensional case show that our method is effcient and stable.     

Inverse unitary eigenproblems and related orthogonal functions

Summary. This paper explores the relationship between certain inverse unitary eigenvalue problems and orthogonal functions. In particular, the inverse eigenvalue problems for unitary Hessenberg matrices and for Schur parameter pencils are considered. The Szeg? recursion is known to be identical to the Arnoldi process and can be seen as an algorithm for solving an inverse unitary Hessenberg eigenvalue problem. Reformulation of this inverse unitary Hessenberg eigenvalue problem yields an inverse eigenvalue problem for Schur parameter pencils. It is shown that solving this inverse eigenvalue problem is equivalent to computing Laurent polynomials orthogonal on the unit circle. Efficient and reliable algorithms for solving the inverse unitary eigenvalue problems are given which require only O() arithmetic operations as compared with O() operations needed for algorithms that ignore the structure of the problem. Received April 3, 1995 / Revised version received August 29, 1996     

Quasi-classical approach to the inverse scattering problem for the KdV equation,and solution of the Whitham modulation equations

We consider an initial value problem for the KdV equation in the limit of weak dispersion. This model describes the formation and evolution in time of a nondissipative shock wave in plasma. Using the perturbation theory in power series of a small dispersion parameter, we arrive at the Riemann simple wave equation. Once the simple wave is overturned, we arrive at the system of Whitham modulation equations that describes the evolution of the resulting nondissipative shock wave. The idea of the approach developed in this paper is to study the asymptotic behavior of the exact solution in the limit of weak dispersion, using the solution given by the inverse scattering problem technique. In the study of the problem, we use the WKB approach to the direct scattering problem and use the formulas for the exact multisoliton solution of the inverse scattering problem. By passing to the limit, we obtain a finite set of relations that connects the space-time parameters x, t and the modulation parameters of the nondissipative shock wave.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 1, pp. 44–61, January, 1996.     

On Determining Sources with Compact Supports in a Bounded Plane Domain for the Heat Equation

The inverse problem of determining the source for the heat equation in a bounded domain on the plane is studied. The trace of the solution of the direct problem on two straight line segments inside the domain is given as overdetermination (i.e., additional information on the solution of the direct problem). A Fredholm alternative theorem for this problem is proved, and sufficient conditions for its unique solvability are obtained. The inverse problem is considered in classes of smooth functions whose derivatives satisfy the Hölder condition.     

Solving a nonlinear inverse Robin problem through a linear Cauchy problem

ABSTRACT

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 inaccessible boundary from Cauchy data measured on the accessible boundary. In this paper, instead of reconstructing the Robin coefficient directly, we compute first the Cauchy data on the inaccessible boundary which is a linear inverse problem, and then compute the Robin coefficient through Newton's law. For the Cauchy problem, a parameter-dependent coupled complex boundary method (CCBM) is applied. The CCBM has its own merits, and this is particularly true when it is applied to the Cauchy problem. With the introduction of a positive parameter, we can prove the regularized solution is uniformly bounded with respect to the regularization parameter which is a very good property because the solution can now be reconstructed for a rather small value of the regularization parameter. For the problem of computing the Robin coefficient from the recovered Cauchy data, a least square output Tikhonov regularization method is applied to Newton's law to obtain a stable approximate Robin coefficient. Numerical results are given to show the feasibility and effectiveness of the proposed method.