5,544 research outputs found

    On surface completion and image inpainting by biharmonic functions: Numerical aspects

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    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

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    A review of the authors's results is given. Several methods are discussed for solving nonlinear equations F(u)=fF(u)=f, where FF 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 F(u)=fF(u)=f 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 F(u)=fF(u)=f 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

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    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

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    Sufficient conditions are given for the relation lim⁑tβ†’βˆžy(t)=0\lim_{t\to\infty}y(t) = 0 to hold, where y(t)y(t) is a continuous nonnegative function on [0,1)[0,1) 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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