A method for solving an inverse biharmonic problem

This paper deals with the problem of determining of an unknown coefficient in an inverse boundary value problem. Using a nonconstant overspecified data, it has been shown that the solution to this inverse problem exists and is unique.     

On an optimal method for solving an inverse Stefan problem

An algorithm optimal in order is proposed for solving an inverse Stefan problem. We also give some exact estimates of accuracy of this method.     

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 gradient descent method for solving an inverse coefficient heat conduction problem

An iterative gradient descent method is applied to solve an inverse coefficient heat conduction problem with overdetermined boundary conditions. Theoretical estimates are derived showing how the target functional varies with varying the coefficient. These estimates are used to construct an approximation for a target functional gradient. In numerical experiments, iteration convergence rates are compared for different descent parameters.     

Numerical method for solving an inverse problem for a population model

The inverse problem of determining the growth rate coefficient of biological objects from additional information on their time-dependent density is considered. Two nonlinear integral equations are derived for the unknown coefficient, which is determined on part of its domain from one equation and on the remaining part from the other equation. The nonlinear integral equations are solved by iterative methods. The convergence conditions for the iterative methods are formulated, and results of numerical experiments are presented.     

Direct numerical method for an inverse problem of hyperbolic equations

A coefficient inverse problem of the one-dimensional hyperbolic equation with overspecified boundary conditions is solved by the finite difference method. The computation is carried out in the x direction instead of the usual t direction. The original boundary condition and the overspecified boundary data are used as the new initial conditions, and the original data at t = 0 are used to compute the coefficient directly. The computation time used by this scheme is almost equal to that for solving the hyperbolic equation in the same region once, even though the inverse problem is essentially nonlinear and hence more difficult to solve. An error estimate is obtained that guarantees the stability of the scheme marching in the x direction. Several numerical experiments are carried out to show the convergence and other properties of the scheme. © 1992 John Wiley & Sons, Inc.     

A penalty function method for solving inverse optimal value problem

In order to consider the inverse optimal value problem under more general conditions, we transform the inverse optimal value problem into a corresponding nonlinear bilevel programming problem equivalently. Using the Kuhn–Tucker optimality condition of the lower level problem, we transform the nonlinear bilevel programming into a normal nonlinear programming. The complementary and slackness condition of the lower level problem is appended to the upper level objective with a penalty. Then we give via an exact penalty method an existence theorem of solutions and propose an algorithm for the inverse optimal value problem, also analysis the convergence of the proposed algorithm. The numerical result shows that the algorithm can solve a wider class of inverse optimal value problem.     

An efficient method for solving an inverse problem for the Richards equation

A parameter identification problem for the hydraulic properties of porous media is considered. Numerically, this inverse problem is solved by minimizing an output least-squares functional. The unknown hydraulic properties which are nonlinear coefficients of a partial differential equation are approximated by spline functions. The identification is embedded into a multi-level algorithm and coupled with a linear sensitivity analysis to describe the ill-posedness of the inverse problem.     

Stably numerical solving inverse boundary value problem for data assimilation

In remote sensing data assimilation, inverse methods are often used to retrieve the boundary conditions from some of the observations in the interior scanning region. Needless to say, the inverse boundary value problem (IBVP) is ill-conditioned in general. Ultimately, the data assimilation must include mechanisms that enable them to overcome the numerical instability for solving IBVP. In this paper, we begin by studying the behavior of ill-conditioned IBVP, and find that the condition number varies with the number of the observations and distribution locations of the observations. Next, we define the number of equivalent independent equations as a novel measurement of ill-conditioning of a problem, which can measure the degree of bad conditions. Furthermore, the novel measure can answer how many additional observations are needed to stabilize the retrieving problem, and where additional observations are fixed up. Finally, we illustrate the proposed methodology by applying it to the study of precipitation data assimilation, with a particular emphasis on the analysis of the effect of the number of observations and their distribution locations. The new methodology appears to be particularly efficient in tackling the instability of the retrieving problem in data assimilation.     

Iterative method for solving an inverse coefficient problem for a hyperbolic equation

For a hyperbolic equation, we consider an inverse coefficient problem in which the unknown coefficient occurs in both the equation and the initial condition. The solution values on a given curve serve as additional information for determining the unknown coefficient. We suggest an iterative method for solving the inverse problem based on reduction to a nonlinear operator equation for the unknown coefficient and prove the uniform convergence of the iterations to a function that is a solution of the inverse problem.     

An order-optimal method for solving an inverse problem for a parabolic equation

An order-optimal method is developed to solve an inverse problem for a parabolic heat conduction equation with a variable coefficient.     

A globally convergent numerical method for an inverse elliptic problem of optical tomography

A new globally convergent numerical method is developed for an inverse problem for the elliptic equation with the unknown potential. The boundary data simulating measurements in optical tomography are generated by the running source. Global convergence analysis is presented along with numerical experiments.     

An order-optimal method of solving an inverse problem for a parabolic equation

An order-optimal method is proposed of approximately solving an inverse problem for a parabolic equation with variable coefficients. We give an order-exact estimate for the error of the method.     

A new method of solving the coefficient inverse problem

This paper is concerned with the new method for solving the coefficient inverse problem in the reproducing kernel space. It is different from the previous studies. This method gives accurate results and shows that it is valid by the numerical example.     

A new regularization method for solving a time-fractional inverse diffusion problem

In this paper, we consider an inverse problem for a time-fractional diffusion equation in a one-dimensional semi-infinite domain. The temperature and heat flux are sought from a measured temperature history at a fixed location inside the body. We show that such problem is severely ill-posed and further apply a new regularization method to solve it based on the solution given by the Fourier method. Convergence estimates are presented under the a priori bound assumptions for the exact solution. Finally, numerical examples are given to show that the proposed numerical method is effective.     

A Tikhonov-type method for solving a multidimensional inverse heat source problem in an unbounded domain

In this study we prove a stability estimate for an inverse heat source problem in the n-dimensional case. We present a revised generalized Tikhonov regularization and obtain an error estimate. Numerical experiments for the one-dimensional and two-dimensional cases show that the revised generalized Tikhonov regularization works well.