Narrow Results

The Cauchy problem for the heat equation in which Cauchy data are prescribed on the outer boundary of a domain with cavity and no data are given on the inner boundary is known to be ill-posed. By a slight modification of the boundary conditions a new problem is introduced whose solution depends continuously on the data in L₂.

In this paper, we investigate a terminal value problem for stochastic fractional diffusion equations with Caputo–Fabrizio derivative. The stochastic noise we consider here is the white noise taken value in the Hilbert space $W$. The main contribution is to investigate the well-posedness and ill-posedness of such problem in two distinct cases of the smoothness of the Hilbert scale space $W_{\nu }$ (see Assumption 3.1), which is a subspace of $W$. When $W_{\nu }$ is smooth enough, i.e. the parameter $\nu$ is sufficiently large, our problem is well-posed and it has a unique solution in the space of Hölder continuous functions. In contract, in the different case when $\nu$ is smaller, our problem is ill-posed; therefore, we construct a regularization result.

We consider linear differential equations with bounded time delay driven by additive white noise. Our aim is the estimation of the maximal delay time from observations of one realisation of the solution process X under nonparametric drift assumptions. In the stationarity case the covariance function has a jump in the third derivative according to the location of the delay time. Based on this result, the delay time estimator is obtained from a singularity detection in the covariance function using a multiresolution framework. It is proved that the estimator attains the rate T^-1/3 for observation times T→∞, which corresponds to change point detection in an ill-posed setting.

In this paper, we study the performance of some parallel iterative regularization methods for solving large overdetermined systems of linear equations.

In this paper, we consider a conductivity interface problem to recover the salient features of the inclusion within a body from noisy observation data on the boundary. Based on integral equations, we propose iterative algorithms to detect the location, the size and the shape of the conductivity inclusion. This problem is severely ill-posed and nonlinear, thus we should consider regularization techniques in order to improve the corresponding approximation. We give several examples to show the viability of our proposed reconstruction algorithms.

This paper develops a new method to deal with the problem of identifying the unknown source in the Poisson equation. We obtain the regularization solution by the Tikhonov regularization method with a super-order penalty term. The order optimal error bounds can be obtained for various smooth conditions when we choose the regularization parameter by a discrepancy principle and the solution process of the new method is uniform. Numerical examples show that the proposed method is effective and stable.

In this paper, we consider an inverse time problem for a nonlinear parabolic equation in the form u_t + Au(t) = f(t, u(t)), u(T) = φ, where A is a positive self-adjoint unbounded operator and f is a Lipschitz function. As known, it is ill-posed. Using a quasi-reversibility method, we shall construct regularization solutions depended on a small parameter ϵ. We show that the regularized problem is well-posed and that their solution u^ϵ(t) converges on [0, T] to the exact solution u(t). This paper extends the work by Dinh Nho Hao et al. [8] to nonlinear ill-posed problems. Some numerical tests illustrate that the proposed method is feasible and effective.

A preconditioning version of the Kozlov–Maz’ya iteration method for the stable identification of missing boundary data is presented for an ill-posed problem governed by generalized elliptic equations. The ill-posed data identification problem is reformulated as a sequence of well-posed fractional elliptic equations in infinite domain. Moreover, some convergence results are established. Finally, numerical results are included showing the accuracy and efficiency of the proposed method.

The regularization of ill-posed problems has become a useful tool in studying initial value problems that do not adhere to certain desired properties such as continuous dependence of solutions on initial data. Because direct computation of the solution becomes difficult in this situation, many authors have alternatively approximated the solution by the solution of a closely defined well posed problem. In this paper, we demonstrate this process of regularization for nonautonomous ill-posed problems including the backward heat equation with a time-dependent diffusion coefficient. In the process, we provide two different approximate well posed models and numerically compare convergence rates of their solutions to a known solution of the original ill-posed problem.

This paper is concerned with an inverse scattering problem in frequency domain, when the scattered field is governed by the Helmholtz equation. The algorithm is iterative in nature. It introduces a new approach which we refer to as proper solution space. It assumes an initial guess for the unknown function and obtains corrections to the guessed value. The updating stage is accomplished by generating a set of functions that satisfy some of the required boundary conditions. We refer to this space as proper solution space. The correction to the assumed value can then be obtained by imposing the remaining boundary conditions. A number of numerical examples are used to study the applicability and effectiveness of the new approach.

In this paper, an inverse heat conduction problem will be considered. By reducing this inverse problem to an integral equation and using an overspecified condition, it is shown that the solution to this problem exists, and this solution is unique.

Performing a proper regularization requires first a good understanding of the nature of the ill-posedness and then an efficient strategy on how to tread stability with efficiency and accuracy in solving the ill-posed problems. In this paper, a simple one-dimensional mechanics problem is presented to clearly and explicitly reveal the nature of ill-posedness in the inverse problem of external force estimation using a discretized (FEM) forward model. Also a novel regularization method with discretization-based filter is proposed to obtain a much more stable and accurate solution for the inverse problem. Estimation with noise-contaminated inputs is carried out, and the efficiency and accuracy of this regularization method are demonstrated. The concept of this new regularization method is applicable for all inverse problems based on methods of domain discretization. In addition, the present regularization method requires no information on the noise source.

Poggio and Edelman have shown that for each object there exists a smooth mapping from an arbitrary view to its standard view and that the mapping can be learnt from a sparse data set. They have demonstrated that such a mapping function can be well approximated by Gaussian GRBF(Genaralized Radial Basis Function) network. In this paper, we have applied this network to the recognition of three kinds of hand shape changes, such as grasp, stroke and flap. Also during learning stage, we have introduced a structural learning algorithm to GRBF network in order to explore the small-sized and essential network structure for the recognition. Finally, we have designed a system to capture motion of the hand with this GRBF by using data glove.

In this paper we consider the regularization of the Cauchy problem for a system of second order differential equations with constant coefficients.

The problem of finding the temperature u satisfying