Jozef Kačur

Solution of nonlinear diffusion appearing in image smoothing and edge detection

Abstract A numerical approximation of the nonlinear diffusion problem appearing in image processing is discussed. The mathematical model is proposed by Catte, Lions, Morel and Coll and represents an improvement of the original model of Perona and Malik. The scheme is linear, based on Rothes approximation in time and on the finite element approach in space. The approximating solutions converge strongly in C(I, L2)∩L2(I, V) space to the variational solution.

Solution of Some Free Boundary Problems by Relaxation Schemes

A numerical method is proposed to solve a large number of free boundary problems modeled by degenerate doubly nonlinear parabolic, parabolic-elliptic equations and systems. The method is related to one proposed in [W. Jager and J. Kacur, Math. Modeling Numer. Anal., 29 (1995), pp. 605--627] and is based on a nonstandard time discretization including two relaxation functions by means of which the diffusion degeneracies (slow and fast) are controlled.

Solution of Degenerate Convection-Diffusion Problems by the Method of Characteristics

A new approximation scheme is proposed to solve nonlinear degenerate convection-diffusion problems. The method is based on a relaxation scheme developed for degenerate parabolic problems in [W. Jager and J. Kacur, Numer. Math., 60 (1991), pp. 407--427; M2AN Math. Model. Numer. Anal., 29 (1995), pp. 605--627], and on the method of characteristics [J. Douglas and T. I. Russell, SIAM J. Numer. Anal., 19 (1982), pp. 871--885], [O. Pironneau, Numer. Math., 38 (1982), pp. 309--332].

Parameter identification by a single injection?extraction well

In this paper, we present numerical modelling techniques supporting the determination of parameters for the contaminant transport by underground water flow. The parameter identification is based on measurements obtained by a single injection–extraction well. The underground water flow is modelled using a Dupuit–Forchheimer approximation for the unsaturated–saturated aquifer.

Determination of Soil Parameters via the Solution of Inverse Problems in Infiltration

In this paper, we propose an efficient method for the identification of soil parameters in unsaturated porous media, using measurements from infiltration experiments. The infiltration is governed by Richards nonlinear equation expressed in terms of effective saturation. The soil retention and hydraulic permeability functions are expressed using the Van Genuchten-Mualem ansatz in terms of the soil parameters. The mathematical algorithm is based on a transformation of Richards equation to a system of ordinary differential equations completed by the governing equation for the movement of the wetness front. This system can be efficiently solved by specialized packages for the solution of stiff systems of ODE. The unknown parameters are determined using the optimization approach of minimizing a cost functional for the discrepancy between the model output and the measurements. The gradient and Hessian of the solution with respect to soil parameter vector are determined using automatic differentiation. Several numerical experiments are included.

Approximation of degenerate parabolic systems by nondegenerate elliptic and parabolic systems

A degenerate, doubly nonlinear parabolic system is approximated by a nondegenerate one. The proposed type of approximation is effective from numerical point of view. The convergence of approximate solutions is proved for a rather general mathematical model.

Slowed Anisotropic Diffusion

A generalization of the regularized (in the sense of Catte, Lions, Morel and Coll) Perona-Malik anisotropic diffusion equation is proposed for image analysis. We present a numerical method for solving the above nonlinear degenerate diffusion problem, together with existence and convergence results. Numerical experiments are disscussed.

A Benchmark Solution for Infiltration and Adsorption of Polluted Water Into Unsaturated–Saturated Porous Media

We discuss the numerical modeling of the infiltration of contaminated water into unsaturated porous media. A system with contaminant transport, dispersion, and adsorption is considered. The mathematical model for unsaturated flow is based on Richards nonlinear and degenerate equation. Nonlinear adsorption is represented by adsorption isotherms and kinetic rates. An accurate numerical method is constructed in 1D which can be a good candidate for the solution of inverse problems to determine model parameters in the adsorption part of the model. Our numerical solution is based on the method of lines (MOL method) where space discretization leads to the corresponding system of ODEs. We substantially use the numerical modeling of interfaces, separating fully saturated, partially saturated, and dry zones in the underground. Finally, in a series of numerical experiments and in comparisons with HYDRUS (Šimunek et al., The HYDRUS-1D software package for simulating the one-dimensional movement of water, heat, and multiple solutes in variably/saturated media, version 2.0, Rep. IGWMC-TPS-70, 202 pp., Int. Groundwater Model. Cent., Colo. Sch of Mines, Golden, Colo), we demonstrate the effectiveness of our method.

On a numerical model for diffusion-controlled growth and dissolution of spherical precipitates

This paper deals with a numerical model for the kinetics of some diffusion-limited phase transformations. For the growth and dissolution processes in 3D we consider a single spherical precipitate at a constant and uniform concentration, centered in a finite spherical cell of a matrix, at the boundary of which there is no mass transfer, see also Asthana and Pabi [1] and Caers [2].

Computation and sensitivity analysis of the pricing of American call options

Abstract We present a new numerical scheme for the approximate solution of the Black–Scholes partial differential equation describing the pricing of American call options. The corresponding mathematical problem is a free boundary problem for a convection–diffusion equation. The concept of our numerical solution is suitable for sensitivity analysis with respect to all parameters, for which we have used automatic differentiation implemented in the ODE solver LSODA-C.