Volume 24, Issue 1 - Special Issue: The Third International Workshop on Computational Inverse Problems and Applications (2013)

In this paper, an inverse Robin problem for an elliptic equation is investigated. The Robin coefficient is assumed to be a small perturbation from a reference coefficient, hence a sparse structure is a priori known. An L q -regularized output least square formulation is considered with q ∈ (0,1). The existence of the minimizer is proved. The convergence of the finite element discretization with respect to mesh size is shown. A modified Newton algorithm is applied to solve the smoothing first order optimality system. Two-dimensional and three-dimensional numerical examples demonstrate the efficiency of the proposed method.

The inverse scattering problem considered in this paper is to reconstruct multiple sound-soft obstacles from one incident wave and its far field pattern. Based on the ideas of the Kirsch–Kress method and the hybrid Newton method, three variant Newton-type methods are developed for reconstructing the shape of multiple obstacles. The proposed hybrid Newton-type methods I and II can choose auxiliary curves adaptively, and do not require them to be contained in the unknown multiple obstacles. The proposed hybrid Newton-type method III is simpler than the hybrid Newton method developed by Kress in terms of computational complexity since it adopts quasi-Newton iterations in numerical reconstructions. Results of numerical examples show that the proposed methods, especially the one with both adaptively choosing auxiliary curves and quasi-Newton iterations, are more efficient and stable for reconstructing multiple obstacles.

This paper deals with numerical solution and parameters inversion for a one-dimensional non-symmetric two-sided fractional advection-dispersion equation (FADE) with zero Neumann boundary condition in a finite domain. A fully discretized finite difference scheme is set forth based on Grünwald–Letnikov's definition of the fractional derivative, and its stability and convergence are proved by estimating the spectral radius of the coefficient matrix of the scheme. Furthermore, an inverse problem of simultaneously determining the fractional order and the dispersion coefficients is investigated, and numerical inversions are carried out by using the optimal perturbation regularization algorithm. The inversion results show that the fractional order and the dispersion coefficients in the FADE can be determined successfully by the final observations.

In this article, we consider a mathematical modelling problem in engineering of uranium ores heap leaching. Firstly, we deduce a mathematical model of uranium ores heap leaching by combining solute transportation equations with microbial chemical reactions. Secondly, an inverse problem, which is solved by the optimal perturbation method together with the Tikhonov regularization, is considered for identifying the parameters of the proposed mathematical model. Finally, numerical simulations are given for the forward problem and the inverse problem to show the pattern of uranium ores microbial heap leaching and verify the effectiveness of parameters identification, respectively.

We consider an inverse problem for a time-fractional advection-dispersion equation, where the measured data is given at x = 1 and the solution is sought in the interval 0 ≤ x < 1. Such a problem is obtained from the classical advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α ∈ (0,1). We show that the inverse problem for a time-fractional advection-dispersion equation is severely ill-posed and we further apply an optimal filtering regularization method to solve it, based on the solution in the frequency domain. The corresponding convergence estimates are provided. To illustrate the results, an example is constructed to show the feasibility and efficiency of the proposed method.

The thickness, thermal conductivity and porosity of textile material are three key factors which determine the heat-moisture comfort level of the human body to a large extent based on the heat and moisture transfer process in the human body-clothing-environment system. This paper puts forward an Inverse Problem of Textile Thickness-Heat conductivity-Porosity Determination (IPT(THP)D) based on the steady-state model of heat and moisture transfer and the heat-moisture comfort indexes. Adopting the idea of the weighted least-squares method, we formulate IPT(THP)D into a function minimization problem. We employ the Particle Swarm Optimization (PSO) method to stochastically search the optimal solution of the objective function. We put the optimal solution into the corresponding direct problem to verify the effectiveness of the proposed numerical algorithms and the validity of the IPT(THP)D.

In the matrix completion problem, most methods to solve the nuclear norm model are relaxing it to the nuclear norm regularized least squares problem. In this paper, we propose a new unconstraint model for matrix completion problem based on nuclear norm and indicator function and design a proximal point algorithm (PPA-IF) to solve it. Then the convergence of our algorithm is established strictly. Finally, we report numerical results for solving noiseless and noisy matrix completion problems and image reconstruction.

In this paper, we aim to calibration pricing models from market prices. We investigate the problem of calibrating the parameters using the trust region method from given price data. We start with the Hull–White model and formulate the problem by obtaining the first kind integral equation, and then consider the parameter recovery problem of the Black–Scholes model. We apply trust region algorithm for numerical retrieval problems. Numerical simulations are given to illustrate the feasibility of our proposed method.

In this paper, the adaptive multigrid conjugate gradient method is applied to the inversion of the space-dependent permeability function in a nonlinear diffusion equation, which plays an important role in promoting the permeability estimation within multiphase porous media flow. The forward problem is discretized using finite-difference methods and the estimation is formulated as a least-square minimization problem with a regularization term. Firstly, the conjugate gradient method for optimization is used to solve this parameter identification problem. Secondly, to reduce the amount of calculation, the idea of multigrid inversion and the necessary condition of “the optimal solution should be the fixed point of multigrid inversion” is considered. Then, by combining the multigrid multiscale method with the conjugate gradient method, a new inversion method called the adaptive multigrid conjugate gradient method is designed. The results of numerical simulations indicate that the proposed method can effectively reduce the required computation burden and have good ability in anti-noise.

Observed polar motion consists of uniform circular motions at both positive (prograde) and negative (retrograde) frequencies. Generalized Euler–Liouville equations of Bizouard, taking into account Earth's triaxiality and asymmetry of the ocean tide, show that the corresponding retrograde and prograde circular excitations are coupled at any frequency. In this work, we reconstructed the polar motion excitation in the Chandler band (prograde and retrograde). Then we compared it with geophysical excitation, filtered out in the same way from the series of the Oceanic Angular Momentum (OAM) and Atmospheric Angular Momentum (AAM) for the period 1960–2000. The agreement was found to be better in the prograde band than in the retrograde one.