Search

An ill-posed Cauchy problem for a 3D elliptic partial differential equation with variable coefficients is considered. A well-posed quasi-boundary-value (QBV) problem is given to approximate it. Some stability estimates are given. For the numerical implementation, a large sparse system is obtained from discretizing the QBV problem using the finite difference method. A left-preconditioned generalized minimum residual method is used to solve the large system effectively. For the preconditioned system, a fast solver using the fast Fourier transform is given. Numerical results show that the method works well. [ABSTRACT FROM AUTHOR]

We consider the optimal recovery of the th degree of the Laplacian value on a function from the information on its k-plane transform, measured with an error. We present the error of the optimal recovery and the set of optimal methods on classes with the bounded th degree of the Laplacian, where . As a consequence, we give one inequality for the norms of the degree of the Laplace operator and the k-plane transform. Particular cases include new inversion methods and inequalities for the classical Radon and x-ray transforms. [ABSTRACT FROM AUTHOR]