# Solodkii S. G.

### Approximate and information aspects of the numerical solution of unstable integral and pseudodifferential equations

Semenova E. V., Solodkii S. G.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 3. - pp. 429-444

We present a review of the latest results obtained in the field of numerical solution of unstable integral and pseudodifferential equations. New versions of fully discrete projection and collocation methods are constructed and justified. It is shown that these versions are characterized by the optimal accuracy and cost efficiency, as far as the use of computational resources is concerned.

### Hyperbolic cross and complexity of various classes of linear ill-posed problems

Ukr. Mat. Zh. - 2017. - 69, № 7. - pp. 951-963

The present paper is a survey of the latest results obtained in the fields of information and algorithmiс complexity of severely ill-posed problems.

### Volodymyr Leonidovych Makarov (on his 75th birthday)

Korolyuk V. S., Lukovsky I. O., Nesterenko B. B., Nikitin A. G., Perestyuk N. A., Samoilenko A. M., Solodkii S. G., Trohimchuk Yu. Yu

Ukr. Mat. Zh. - 2016. - 68, № 12. - pp. 1715-1717

### Optimal discretization of Ill-posed problems

Pereverzev S. V., Solodkii S. G.

Ukr. Mat. Zh. - 2000. - 52, № 1. - pp. 106-121

We present a survey of results on the optimal discretization of ill-posed problems obtained in the Institute of Mathematics of the Ukrainian National Academy of Sciences.

### Optimization of projection schemes of digitization of ill-posed problems

Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1398–1410

We construct new projection schemes of digitization of ill-posed problems, which are optimal in the sense of the amount of discrete information used. We establish that the application of self-adjoint projection schemes to digitization of equations with self-adjoint operators is not optimal.

### Information complexity of projection algorithms for the solution of Fredholm equations of the first kind. II

Ukr. Mat. Zh. - 1998. - 50, № 6. - pp. 838–844

The optimal order of information complexity is found for certain classes of ill-posed problems.

### Information complexity of projection algorithms for the solution of Fredholm equations of the first kind. I

Ukr. Mat. Zh. - 1998. - 50, № 5. - pp. 699–711

We construct a new system of discretization of the Fredholm integral equations of the first kind with linear compact operators *A* and free terms from the set Range (*A(A*A*)^{V}), v > 1/2. The approach proposed enables one to obtain the optimal order of error on such classes of equations by using a considerably smaller amount of discrete information as compared with standard schemes.

### On information complexity of certain classes of operator equations

Ukr. Mat. Zh. - 1997. - 49, № 9. - pp. 1271–1277

For some classes of operator equations of the second kind, we obtain an estimate of information complexity exact in order. We construct a new projection-type method that realizes the optimal estimate.

### On the optimization of projection-iterative methods for the approximate solution of ill-posed problems

Pereverzev S. V., Solodkii S. G.

Ukr. Mat. Zh. - 1996. - 48, № 11. - pp. 1530-1537

We consider a new version of the projection-iterative method for the solution of operator equations of the first kind. We show that it is more economical in the sense of amount of used discrete information.

### Complexity of projective methods for the solution of ill-posed problems

Ukr. Mat. Zh. - 1996. - 48, № 8. - pp. 1114-1124

We consider the problem of finite-dimensional approximation for solutions of equations of the first kind and propose a modification of the projective scheme for solving ill-posed problems. We show that this modification allows one to obtain, for many classes of equations of the first kind, the best possible order of accuracy for the Tikhonov regularization by using an amount of information which is far less than for the standard projective technique.

### Complexity of fredholm equations of the second kind with kernels from anisotropic classes of differentiable functions

Ukr. Mat. Zh. - 1996. - 48, № 4. - pp. 525-534

We establish the exact order of complexity of the approximate solution of Fredholm equations with periodic kernels with dominant mixed partial derivative.

### On one approach to the discretization of the lavrent’ev method

Pereverzev S. V., Solodkii S. G.

Ukr. Mat. Zh. - 1996. - 48, № 2. - pp. 212-219

We propose a new scheme of discretization of the Lavrent’ev method for operator equations of the first kind with self-adjoint nonnegative operators of certain “smoothness.” This scheme is more economical in the sense of the amount of used discrete information as compared with traditional approaches.

### On direct methods for solution of regularized equations

Pereverzev S. V., Solodkii S. G.

Ukr. Mat. Zh. - 1995. - 47, № 9. - pp. 1231–1242

We prove that the application of so-called adaptive direct methods to approximation of Fredholm equations of the first kind leads to a more economical way of finite-dimensional approximation as compared with traditional approaches.

### Optimization of algorithms for the approximate solution of the Volterra equations with infinitely differentiable kernels

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1534–1545

For the Volterra equations with analytic kernels, we establish the exact power order of complexity of their approximate solutions and show that the optimal power order is realized by the method of simple iterations based on the use of information in the form of the values of kernels and free terms at certain points. In addition, for the Volterra equations with infinitely differentiable kernels, we determine the minimal order of the error of direct methods and construct a method which realizes this order.

### Optimization of adaptive direct methods for the solution of operator equations in Hilbert space

Ukr. Mat. Zh. - 1990. - 42, № 1. - pp. 95–102