On the clustering of stationary points of Tikhonov’s functional for conditionally well-posed inverse problems

In a Hilbert space, we consider a class of conditionally well-posed inverse problems for which the Hölder type estimate of conditional stability on a bounded closed and convex subset holds. We investigate a finite-dimensional version of Tikhonov’s scheme in which the discretized Tikhonov’s functional is minimized over the finite-dimensional section of the set of conditional stability. For this optimization problem, we prove that each its stationary point that is located not too far from the desired solution of the original inverse problem in reality belongs to a small neighborhood of the solution. Estimates for the diameter of this neighborhood in terms of discretization errors and error level in input data are also given.

Seiches in Petrozavodsk bay of the lake Onego

The analysis of observation data on fluctuations in the level and velocity of currents in the Petrozavodsk Bay of Lake Onega, performed in 2016–2017, was carried out. Level oscillations were measured with a discreteness of 10 s using two TDR-2050 devices (RBR Ltd., Canada). Spectral analysis was performed using the simple Fourier transform method after one-minute data averaging to reduce instrument noise. Energy-carrier periods corresponding to the seiches of Petrozavodsk Bay and the Lake Onega are identified. Current velocities were measured with Aquadopp HR-Profiler (Nortek, Norway). Within the framework of the linear long wave theory, seiche oscillations are considered in a model basin approximating the Petrozavodsk Bay with regard to the Ivanovskiye Islands. Using an analytical solution, estimates for the periods of the higher seiche modes and the corresponding maximum wave flow velocities are obtained for the Petrozavodsk Bay. The comparison of theoretical estimates with the data of field observations, was showed satisfactory agreement.

Solution Estimates for the Discrete Lyapunov Equation in a Hilbert Space and Applications to Difference Equations

The paper is devoted to the discrete Lyapunov equation X - A * X A = C , where A and C are given operators in a Hilbert space H and X should be found. We derive norm estimates for solutions of that equation in the case of unstable operator A, as well as refine the previously-published estimates for the equation with a stable operator. By the point estimates, we establish explicit conditions, under which a linear nonautonomous difference equation in H is dichotomic. In addition, we suggest a stability test for a class of nonlinear nonautonomous difference equations in H . Our results are based on the norm estimates for powers and resolvents of non-self-adjoint operators.

Regularisation Technique for a Distributed Parameter Identification Problem

This paper examined the problem of ill-posedness of solution in identifying parameters from a given groundwater flow model. The solution approach to the problem was attempted by the method of Parameter Transformation coupled with Tikhonov Regularisation with and without Truncation which has not been explored. Convergence of the method was assessed by the L-Curve criterion. Numerical examples were presented to illustrate the efficiency of the proposed Regularisation Technique. Tikhonov Regularisation with Truncation turns to give a more realistic solution estimates when examined numerically, compared to that of Regularisation without Truncation.