Adaptivity in 3D image processing

We present an adaptive numerical scheme for computing the nonlinear partial differential equations arising in 3D image multiscale analysis. The scheme is based on a semi-implicit scale discretization and on an adaptive finite element method in 3D-space. Successive coarsening of the computational grid is used for increasing the efficiency of the numerical procedure. L_∞-stability of the semi–discrete scheme is proved and computational results related to 3D nonlinear image filtering are discussed. Received: 15 December 1999 / Accepted: 8 June 2001     

Analysis of systems of singular ordinary differential equations

Systems of ordinary differential equations with a small parameter at the derivative and specific features of the construction of their periodic solution are considered. Sufficient conditions of existence and uniqueness of the periodic solution are presented. An iterative procedure of construction of the steady-state solution of a system of differential equations with a small parameter at the derivative is proposed. This procedure is reduced to the solution of a system of nonlinear algebraic equations and does not involve the integration of the system of differential equations. Problems of numerical calculation of the solution are considered based on the procedure proposed. Some sources of its divergence are found, and the sufficient conditions of its convergence are obtained. The results of numerical experiments are presented and compared with theoretical ones. Translated from Kibemetika i Sistemnyi Analiz, No. 5, pp. 103–110, September–October, 1999.     

Image segmentation using a multilayer level-set approach

We propose an efficient multilayer segmentation method based on implicit curve evolution and on variational approach. The proposed formulation uses the minimal partition problem as formulated by D. Mumford and J. Shah, and can be seen as a more efficient extension of the segmentation models previously proposed in Chan and Vese (Scale-Space Theories in Computer Vision, Lecture Notes in Computer Science, Vol. 1682, pp. 141–151, 1999, IEEE Trans Image Process 10(2):266–277, 2001), and Vese and Chan (Int J Comput Vis 50(3):271–293, 2002). The set of unknown discontinuities is represented implicitly by several nested level lines of the same function, as inspired from prior work on island dynamics for epitaxial growth (Caflisch et al. in Appl Math Lett 12(4):13, 1999; Chen et al. in J Comput Phys 167:475, 2001). We present the Euler–Lagrange equations of the proposed minimizations together with theoretical results of energy decrease, existence of minimizers and approximations. We also discuss the choice of the curve regularization and conclude with several experimental results and comparisons for piecewise-constant segmentation of gray-level and color images.     

Discretization of the Navier–Stokes equations on non-smooth grids using auxiliary points

The colocated scheme for the incompressible Navier–Stokes equations is improved on structured non-Cartesian grids. The method relies on a finite volume discretization and on the use of auxiliary points to locally approximate gradients following a two-point discretization. Enhanced accuracy is demonstrated for two-dimensional cases on strongly distorted meshes by computing Poiseuille flow and a flow in a differentially heated cavity. Received: 23 February 1999 / Accepted: 17 June 1999     

Linearly implicit schemes for a class of dispersive–dissipative systems

We consider initial value problems for semilinear parabolic equations, which possess a dispersive term, nonlocal in general. This dispersive term is not necessarily dominated by the dissipative term. In our numerical schemes, the time discretization is done by linearly implicit schemes. More specifically, we discretize the initial value problem by the implicit–explicit Euler scheme and by the two-step implicit–explicit BDF scheme. In this work, we extend the results in Akrivis et al. (Math. Comput. 67:457–477, 1998; Numer. Math. 82:521–541, 1999), where the dispersive term (if present) was dominated by the dissipative one and was integrated explicitly. We also derive optimal order error estimates. We provide various physically relevant applications of dispersive–dissipative equations and systems fitting in our abstract framework.     

Methods of solution and criteria of consistency of systems of linear diophantine equations over the set of natural numbers1

Some methods and algorithms of solution of systems of linear Diophantine equations over the naturals are briefly reviewed. Criteria and an incremental algorithm of efficient solution of the problem of consistency for a system of linear Diophantine equations and inequalities over the naturals are given. This research was supported under grant INTAS-RFBR 95-0095. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 12–36, July–August, 1999.     

Solution of systems of linear algebraic equations with preliminary flattening

A method of improving computing properties of matrices of systems of linear algebraic equations is considered. Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 144–149, September–October, 1999.     

State estimation and filtering in stochastic systems using adequate simplification

The method of order reduction in solving stochastic problems of state estimation and filtering is considered. The method presented concerns the case where mathematical models of objects being studied are defined by systems of nonstationary differential equations. Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 98–102, September–October, 1999.     

A finite element/Monte–Carlo method for polymer dilute solutions

A finite element/Monte–Carlo method is proposed for solving the flow of a polymer dilute solution. The mass and momentum equations are supplemented with stochastic differential equations which model the dynamics of the polymer chains. Finite elements and a Monte–Carlo method are used to solve the problem. A theoretical analysis is performed on a simplified, deterministic, Oldroyd-B problem. Numerical results are compared to experimental ones. Received: 22 December 2000 / Accepted: 30 May 2001