Volume 24, Issue 5

The inverse problem under investigation consists of the boundary data completion in a deoxygenation-reaeration model in stream-waters. The unidimensional model we deal with is based on the one introduced by Streeter and Phelps, augmented by Taylor dispersion terms. The missing boundary condition is the load or/and the flux of the biochemical oxygen demand indicator at the upstream point. The counterpart is the availability of two boundary conditions on the dissolved oxygen tracer at the same point. The major consequence of these non-standard boundary conditions is that dispersion-reaction equations on both oxygen indicators are strongly coupled and the resulting system becomes ill-posed. The main purpose here is a finite element space-discretization of the variational problem. Mixed finite elements turn out to be well fitted and yield a non-symmetric saddle point system. The obtained semi-discrete problem is a differential algebraic equation that needs specific tools for its analysis. Combining analytical calculations and theoretical justifications, we try to elucidate the main properties of this ill-posed dynamical problem and understand its mathematical structure.

In this paper, we consider the modified inhomogeneous Helmholtz equation Δ u ( x , y ) -$-$k2$k^{2}$ u ( x , y ) =$=$ f ( x , y ), x ∈$\in$ℝ$\mathbb{R}$, 0 <$<$ y <$<$ 1, with inhomogeneous Cauchy data being given at y =$=$ 0. The problem is known to be ill-posed, as the solution (if exists) does not depend continuously on the given data. We propose a regularization method to obtain a stable approximate solution of the problem and get some error estimates. Finally, a numerical example shows the effectiveness of the proposed method.

We consider the inverse scattering problem of determining the shape of a crack with impedance boundary condition on one side from the complex conjugate of point sources placed on a closed curve which contains the crack. The near field factorization method is established to reconstruct the crack from the measurements on the same curve. Then, we deduce an inversion algorithm and present some numerical examples to show the viability of our method. So far as we know, when the incident waves are the point sources, the near field operator cannot be directly decomposed into the form that the factorization method required. However, we overcome this difficulty with the complex conjugate of point sources which are recently used in [Inverse Problems 30 (2014), Article ID 095005], [Inverse Problems 30 (2014), Article ID 045008] for the justification of the factorization method from near field data.

This work is dedicated to the numerical solution of a class of parametric identification problems for dynamic objects. The process is described by a system of loaded ordinary differential equations. Observations over the object have integral (interval) and point characters, at which the results of the observations are given in a summarized non-separated form. We propose a technique that reduces the solution of the initial problem to the solution of specially-built supplementary Cauchy problems. The results of some numerical experiments are also given.

In this paper, an inverse problem of determining the unknown coefficient of a multidimensional nonlinear time-dependent Schrödinger equation that has a complex number at nonlinear part is considered. The inverse problem is reformulated as a variational one which aims to minimize the observation functional. This paper presents existence and uniqueness theorems of solutions of the constituted variational problem, the gradient of the observation functional and a necessary condition for the solution of the variational problem.

The goal of the present study is to derive an integral identity for a class of ill-posed problems generated by a parabolic equation. The obtained result enables us to reduce the original ill-posed problem directly to the first-kind Fredholm equation with translation kernel. Various real world applications to a different fields of knowledge, including ecology and finances, are presented.

In this paper, we consider an inverse heat conduction problem (IHCP). First, the existence, uniqueness and unstability solution of the inverse problem will be studied. Due to the ill-posedness of the inverse problem, we propose a meshless numerical approach based on basis function to solve this problem in the presence of noisy data. The Tikhonov regularization method with generalized discrepancy principle is applied to obtain a stable numerical approximation to the solution. The effectiveness of the algorithm is illustrated by some numerical examples.

In this paper an inverse problem is considered for a non-coercive partial differential equation, issued from a mass conservation cavitation model for a slightly compressible fluid. The cavitation phenomenon and compressibility take place and are described by the Elrod model. The existence of an optimal solution is proven. Optimality conditions are derived and some numerical results are given.

The relations between stability and accuracy of the operator marching method (OMM) are usually conflicting in waveguides with strong range dependence. To explain this phenomenon, this study intends to present an error estimate for the OMM in range dependent waveguides. We utilize “approximation level” to measure truncation error for a marching method in various range step sizes. Then, the error estimate is developed to analyze the performances of the OMM. Through an error analysis, we verify the following features of the OMM: (i) it is valid to apply the OMM in slowly varying waveguides with very large range step sizes; (ii) the OMM may blow up suddenly when the range dependence is strong and the step size is extremely small in the same time. We also develop a three-number set to describe the stability and accuracy level of a general marching method for computing wave propagation in a waveguide. In the end, extensive numerical experiments are implemented to verify the correctness of the error analysis.