Stability rates for linear ill-posed problems with compact and non-compact operators.
Stable evaluation of differential operators and linear and nonlinear multi-scale filtering.
Structure et estimations de coefficients d'erreurs
Structure et estimations de coefficients d'erreurs
Su una estensione di un teorema di Tikhonov.
Sur les bornes d'erreur a posteriori pour les éléments propres d'opérateurs linéaires.
The combination of approximate solutions for accelerating the convergence
The fourth order accuracy decomposition scheme for an evolution problem
In the present work, the symmetrized sequential-parallel decomposition method with the fourth order accuracy for the solution of Cauchy abstract problem with an operator under a split form is presented. The fourth order accuracy is reached by introducing a complex coefficient with the positive real part. For the considered scheme, the explicit a priori estimate is obtained.
The fourth order accuracy decomposition scheme for an evolution problem
In the present work, the symmetrized sequential-parallel decomposition method with the fourth order accuracy for the solution of Cauchy abstract problem with an operator under a split form is presented. The fourth order accuracy is reached by introducing a complex coefficient with the positive real part. For the considered scheme, the explicit a priori estimate is obtained.
The instability of some gradient methods for ill-posed problems.
The norm convergence of a Magnus expansion method
We consider numerical approximation to the solution of non-autonomous evolution equations. The order of convergence of the simplest possible Magnus method is investigated.
Three remarks on the use of Čebyšev polynomials for solving equations of the second kind
Time discretizations for evolution problems
The aim of this work is to give an introductory survey on time discretizations for liner parabolic problems. The theory of stability for stiff ordinary differential equations is explained on this problem and applied to Runge-Kutta and multi-step discretizations. Moreover, a natural connection between Galerkin time discretizations and Runge-Kutta methods together with order reduction phenomenon is discussed.
Tractability of multivariate problems for weighted spaces of functions
We survey recent results on tractability of multivariate problems. We mainly restrict ourselves to linear multivariate problems studied in the worst case setting. Typical examples include multivariate integration and function approximation for weighted spaces of smooth functions.
Two step extrapolation and optimum choice of relaxation factor of the extrapolated S.O.R. method
Limits of the extrapolation coefficients are rational functions of several poles with the largest moduli of the resolvent operator and therefore good estimates of these poles could be calculated from these coefficients. The calculation is very easy for the case of two coefficients and its practical effect in finite dimensional space is considerable. The results are used for acceleration of S.O.R. method.
Über die Konvergenzgeschwindigkeit projektiver Verfahren. II.
Unconditional stability of difference formulas
The paper concerns the solution of partial differential equations of evolution type by the finite difference method. The author discusses the general assumptions on the original equation as well as its discretization, which guarantee that the difference scheme is unconditionally stable, i.e. stable without any stability condition for the time-step. A new notion of the -acceptability of the integration formula is introduced and examples of such formulas are given. The results can be applied to ordinary...
Uniformly convergent adaptive methods for a class of parametric operator equations∗
We derive and analyze adaptive solvers for boundary value problems in which the differential operator depends affinely on a sequence of parameters. These methods converge uniformly in the parameters and provide an upper bound for the maximal error. Numerical computations indicate that they are more efficient than similar methods that control the error in a mean square sense.
Uniformly convergent adaptive methods for a class of parametric operator equations∗
We derive and analyze adaptive solvers for boundary value problems in which the differential operator depends affinely on a sequence of parameters. These methods converge uniformly in the parameters and provide an upper bound for the maximal error. Numerical computations indicate that they are more efficient than similar methods that control the error in a mean square sense.
Universality of the best determined terms method
The properties are studied of the best determined terms method with respect to an a priori decomposition . The universal approximation to the normal solution of the first kind Fredholm integral equation is found.