Approximated solution algorithms for Urysohn-type equations

Abstract

Problems of remote sensing and gravity, magnetic, seismic, geological prospecting use systems of indirect data measurement, which can be modeled with non-linear Urysohn equations. The paper presents equations of Urysohn type and their operator analogues. A number of algorithms for their solution has been developed based on the availability of a priori information, asymptotic properties of integral operators, and specific features of a model. It has been formulated a theorem on constructing a solution of the original equation based on the neighboring one with an error estimate. Both the original and neighboring equations are taken as the regularized equations. The proposed approach allows for a variety of algorithms, depending on the type of regularization and iteration schemes, in particular, a modified version of the Levenberg-Marquardt algorithm. Additionally, the algorithm for searching characteristic points of a given function based on the asymptotics of an integral Urysohn operator is provided.

Application of Wavelet Transform to Urysohn-Type Equations

“…and valid is Theorem 2 from [21] on error estimation. In Equation (20), for K and K in terms of α(z n − z m ) and α( zn − zm ), instead of the parameters α and α, you can use the new parameters β and β, which are responsible for the proximity of the solution z n to the specified approximation z m ( zn and zm ).…”

“…To obtain k(τ) we can also use the representation (5) (via the FT). Approximate solutions of the integral equation of the first kind (9) obtained using the regularization method with a priori information are applicable to iterative algorithms and close solution theorems proposed earlier in the paper [21].…”

“…Urysohn equations of the first kind emerge in various applied problems [21]. Depending on the availability of a priori information, the model's structure, and classes of functions, a variety of approaches are used to obtain approximate solutions based on asymptotic and regularization methods [20,21,26,32].…”

“…The results previously obtained by authors [20,21,32] on iterative algorithms based on the use of the close equation à z = ũ can be extended to the Urysohn-type equation Az = u (18). Depending on the availability of a priori and other information about the solution and the indirect measurement model, one can use the whole range of known regularization algorithms.…”

“…In [20,21], the authors analyze the problem of a solution reconstruction for Urysohn equations under the assumption that a priori information on the solution and additional information on the solution of close equations is known. Hence, two independent problems of processing and interpreting indirectly obtained data arise.…”