5,449 research outputs found
On surface completion and image inpainting by biharmonic functions: Numerical aspects
Numerical experiments with smooth surface extension and image inpainting
using harmonic and biharmonic functions are carried out. The boundary data used
for constructing biharmonic functions are the values of the Laplacian and
normal derivatives of the functions on the boundary. Finite difference schemes
for solving these harmonic functions are discussed in detail.Comment: Revised 21 July, 2017. Revised 12 January, 2018. To appear in
International Journal of Mathematics and Mathematical Science
The Dynamical Systems Method for solving nonlinear equations with monotone operators
A review of the authors's results is given. Several methods are discussed for
solving nonlinear equations , where is a monotone operator in a
Hilbert space, and noisy data are given in place of the exact data. A
discrepancy principle for solving the equation is formulated and justified.
Various versions of the Dynamical Systems Method (DSM) for solving the equation
are formulated. These methods consist of a regularized Newton-type method, a
gradient-type method, and a simple iteration method. A priori and a posteriori
choices of stopping rules for these methods are proposed and justified.
Convergence of the solutions, obtained by these methods, to the minimal norm
solution to the equation is proved. Iterative schemes with a
posteriori choices of stopping rule corresponding to the proposed DSM are
formulated. Convergence of these iterative schemes to a solution to equation
is justified. New nonlinear differential inequalities are derived and
applied to a study of large-time behavior of solutions to evolution equations.
Discrete versions of these inequalities are established.Comment: 50p
Dynamical systems method for solving linear finite-rank operator equations
A version of the Dynamical Systems Method (DSM) for solving ill-conditioned
linear algebraic systems is studied in this paper. An {\it a priori} and {\it a
posteriori} stopping rules are justified. An iterative scheme is constructed
for solving ill-conditioned linear algebraic systems.Comment: 16 pages, 1 table, 1 figur
Some nonlinear inequalities and applications
Sufficient conditions are given for the relation
to hold, where is a continuous nonnegative function on
satisfying some nonlinear inequalities. The results are used for a study of
large time behavior of the solutions to nonlinear evolution equations. Example
of application is given for a solution to some evolution equation with a
nonlinear partial differential operator.Comment: 16 page
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