Home – Home – Jornals – Numerical Analysis and Applications 2024 number 3

Aiming at the problem that the coefficient matrix of multivariate errors-in-variables (MEIV) model contains constant columns, the MEIV model is extended to partial multivariate errors-in-variables (P-MEIV), and the new algorithm of P-MEIV model is proposed based on the principle of partial errors-in-variables (PEIV) model and indirect adjustment. The algorithm is simple and easy to implement. An example of coordinate transformation is used for verifying, and the results are compared with the existing MEIV model algorithm, which shows the effectiveness of the proposed algorithm. Finally, the P-MEIV algorithm is applied to the multi-point grey model (MGM(1,N)) of settlement monitoring. The results show that the P-MEIV model proposed in this paper can better consider the influence of monitoring point errors, and the estimated results are in good agreement with the actual situation.

A unitoid is a square matrix that can be brought to diagonal form by a congruence transformation. Among different diagonal forms of a unitoid A, there is only one, up to the order adopted for the principal diagonal, whose nonzero diagonal entries all have the modulus 1. It is called the congruence canonical form of A, while the arguments of the nonzero diagonal entries are called the canonical angles of A. If A is nonsingular then its canonical angles are closely related to the arguments of the eigenvalues of the matrix A^-∗A, called the cosquare of A. Although the definition of a unitoid reminds the notion of a diagonalizable matrix in the similarity theory, the analogy between these two matrix classes is misleading. We show that the Jordan block J_n(1), which is regarded as an antipode of diagonalizability in the similarity theory, is a unitoid. Moreover, its cosquare C_n(1) has n distinct unimodular eigenvalues. Then we immerse J_n(1) in the family of the Jordan blocks J_n(λ), where λ is varying in the range (0,2]. At some point to the left of 1, J_n(λ) is not a unitoid any longer. Wediscuss this moment in detail in order to comprehend how it can happen. Similar moments with even smaller λ are discussed, and certain remarkable facts about the eigenvalues of cosquares and their condition numbers are pointed out.

The study of the spread of greenhouse gases in space and time, as well as the assessment of fluxes from the Earth's surface of these gases using a data assimilation system is an urgent task of monitoring the state of the environment. One of the approaches to estimating greenhouse gas fluxes is an approach based on the assumption that fluxes are constant in a given subdomain and over a given time interval (on the order of a week). This is due to both the need for an effective implementation of the algorithm and the properties of the observational data used in such problems. Modern problems of estimating greenhouse gas fluxes from the Earth's surface have a large dimension, therefore, a variant is usually considered in which the estimated variable is fluxes, and the transport and diffusion model is included in the observation operator. At the same time, there is a problem of using large assimilation windows, within which the flow values are estimated at several time intervals. The paper considers an algorithm for estimating fluxes based on observations from a given time interval. The algorithm is a variant of the ensemble smoothing algorithm, widely used in such problems. It is shown that when using the assimilation window, in which the flow values are estimated for several time intervals, the algorithm can become unstable, while the observability condition is violated.

This paper considers a new class of nonlinear second degree integro-differential Volterra equations with a convolution kernel. We derive some sufficient conditions to establish the existence and uniqueness of solutions by using the Schauder fixed point theorem. Moreover, the Nystrőm method is applied to obtain an approximate solution of the proposed Volterra equation. Numerical examples are given to validate the adduced results.

Variable order fractional operators can be used in various physical and biological applications where rates of change of the quantity of interest may depend on space and/or time. In this paper, we propose an explicit finite difference approximation for a space-time Riesz-Caputo variable order fractional wave equation with initial and boundary conditions in a finite domain. The proposed scheme is conditionally stable and has global truncation error O(τ²+h²). We also present a numerical experiment to verify the efficiency of the proposed scheme.

In this paper we study the technology of calculating difference problems with internal boundary conditions of flow balance constructed by means of one-sided multipoint difference analogs of first derivatives of arbitrary order of accuracy. The proposed technology is equally suitable for any type of differential equations to be solved and admits a uniform realization at any order of accuracy. It, unlike approximations based on the continued system of equations, does not lead to complications in splitting multidimensional problems into one-dimensional ones. Sufficient conditions of solvability and stability of the realization of algorithms by the run method for boundary conditions of arbitrary order of accuracy are formulated. The proof is based on the reduction of multipoint boundary conditions to a form that does not violate the tridiagonal structure of matrices, and the establishment of the conditions of diagonal dominance in the transformed matrix rows corresponding to the external and internal boundary conditions.

The paper considers an implementation of an adaptive computational grid constructing algorithm inside the numerical solution of the one-dimensional forward magnetotelluric sounding problem (the Tikhonov-Cagniard problem). The numerical solution of the problem is realized by the method of local integral equations, which was proposed by authors earlier. An adaptive computational grid construction is based on geometrical principles, which conduct approximation of the electrical conductivity function via optimization of its’ piecewise-constant interpolant. Numerical experiments are carried out to study and illustrate the effectiveness of the combined method. Approbation was realized on the Kato-Kikuchi model with known exact solution.