shape invariant bounds for reaction diffusion systems with unequal diffusion coefficients abstract estimates by invariant regions or shape invariant bounds are a very useful tool in the study of reaction diffusion systems however the class of the regions available for such estimates is severely restricted if the diffusion coefficients are not equal thus many systems cannot be treated by these methods in the present article we generalize the estimates by shape invariant bounds with arbitrary smooth bounded regions to the case of unequal diffusion coefficients the efficiency of our method increases as the quotient of the largest and the smallest diffusion coefficient approaches 1 the bounds are used to prove existence and stability results

a linear domain decomposition method for partially saturated flow in porous media abstract the richards equation is a nonlinear parabolic equation that is commonly used for modelling saturated unsaturated flow in porous media we assume that the medium occupies a bounded lipschitz domain partitioned into two disjoint subdomains separated by a fixed interface this leads to two problems defined on the subdomains which are coupled through conditions expressing flux and pressure continuity at after an euler implicit discretisation of the resulting nonlinear subproblems a linear iterative l type domain decomposition scheme is proposed the convergence of the scheme is proved rigorously in the last part we present numerical results that are in line with the theoretical finding in particular the convergence of the scheme under mild restrictions on the time step size we further compare the scheme to other approaches not making use of a domain decomposition namely we compare to a newton and a picard scheme we show that the proposed scheme is more stable than the newton scheme while remaining comparable in computational time even if no parallelisation is being adopted after presenting a parametric study that can be used to optimise the proposed scheme we briefly discuss the effect of parallelisation and give an example of a four domain implementation

distributional scaling an algorithm for structure preserving embedding of metric and nonmetric spaces we present a novel approach for embedding general metric and nonmetric spaces into low dimensional euclidean spaces as opposed to traditional multidimensional scaling techniques which minimize the distortion of pairwise distances our embedding algorithm seeks a low dimensional representation of the data that preserves the structure geometry of the original data the algorithm uses a hybrid criterion function that combines the pairwise distortion with what we call the geometric distortion to assess the geometric distortion we explore functions that reflect geometric properties our approach is different from the isomap and lle algorithms in that the discrepancy in distributional information is used to guide the embedding we use clustering algorithms in conjunction with our embedding algorithm to direct the embedding process and improve its convergence properties we test our method on metric and nonmetric data sets and in the presence of noise we demonstrate that our method preserves the structural properties of embedded data better than traditional mds and that its performance is robust with respect to clustering errors in the original data other results of the paper include accelerated algorithms for optimizing the standard mds objective functions and two methods for finding the most appropriate dimension in which to embed a given set of data

l1 contraction and uniqueness for quasilinear elliptic parabolic equations abstract we prove the l 1 contraction principle and uniqueness of solutions for quasilinear elliptic parabolic equations of the form formula where b is monotone nondecreasing and continuous we assume only that u is a weak solution of finite energy in particular we do not suppose that the distributional derivative t b u is a bounded borel measure or a locally integrable function

pointwise bounds for linear reaction diffusion systems and an extension to nonlinear problems abstract we derive pointwise bounds for solutions of linear reaction diffusion systems in cases where estimates by invariant rectangles or two sided bounds are not satisfactory and where other invariance statements or similar estimation techniques cannot be used since the diffusion coefficients are not equal the bounds are explicit if certain green s functions can be estimated suitably via linearization the results are used to obtain error estimations for approximate solutions of nonlinear reaction diffusion systems

degenerate two phase incompressible flow ii regularity stability and stabilization abstract in this paper we analyze a coupled system of highly degenerate elliptic parabolic partial differential equations for two phase incompressible flow in porous media this system involves a saturation and a global pressure or a total flow velocity first we show that the saturation is holder continuous both in space and time and the total velocity is holder continuous in space uniformly in time applying this regularity result we then establish the stability of the saturation and pressure with respect to initial and boundary data from which uniqueness of the solution to the system follows finally we establish a stabilization result on the asymptotic behavior of the saturation and pressure we prove that the solution to the present system converges in appropriate norms to the solution of a stationary system as time goes to infinity an example is given to show typical regularity of the saturation

a nonlinear time compactness result and applications to discretization of degenerate parabolic elliptic pdes abstract we propose a discrete functional analysis result suitable for proving compactness in the framework of fully discrete approximations of strongly degenerate parabolic problems it is based on the original exploitation of a result related to compensated compactness rather than on a classical estimate on the space and time translates in the spirit of simon 49 our approach allows to handle various numerical discretizations both in the space variables and in the time variable in particular we can cope quite easily with variable time steps and with multistep time differentiation methods like e g the backward differentiation formula of order 2 bdf2 scheme we illustrate our approach by proving the convergence of a two point flux finite volume in space and bdf2 in time approximation of the porous medium equation

the richards equation with hysteresis and degenerate capillary pressure abstract we study the richards equation with a dynamic capillary pressure including hysteresis we provide existence and approximation results for degenerate capillary pressure curves p c treating two cases in the first case the permeability function k can be degenerate but the initial saturation does not take the critical values in the second case the permeability function k is strictly positive but the capillary pressure function can be multi valued in both cases the degenerate behavior of p c leads to the physically desired uniform bounds for the saturation variable our approach exploits maximum principles and relies on the corresponding uniform bounds for pressure and saturation a new compactness result for the saturation variable allows to take limits in nonlinear terms the solution concept uses tools of convex analysis

degenerate two phase incompressible flow i existence uniqueness and regularity of a weak solution abstract this is the first paper of a series in which we analyze mathematical properties and develop numerical methods for a degenerate elliptic parabolic partial differential system which describes the flow of two incompressible immiscible fluids in porous media in this paper we first show that this system possesses a weak solution under physically reasonable hypotheses on the data then we prove that this weak solution is unique finally we establish regularity on the weak solution which is needed in the uniqueness proof in particular the holder continuity of the saturation in space and time and the lipschitz continuity of the pressure in space are obtained

a monotone iterative scheme for nonlinear reaction diffusion systems having nonmonotone reaction terms abstract a monotone scheme is proposed for the solution of weakly coupled systems of reaction diffusion equations without any monotonicity property of the nonlinear reaction terms the considerations are taken within the framework of weak solutions

asymptotic behavior of a reaction diffusion system in bacteriology abstract this paper is concerned with the asymptotic behavior of the solution for a coupled system of reaction diffusion equations which describes the bacteria growth and the diffusion of histidine and buffer concentrations under the basic boundary condition of neumann type or mixed type the coupled system can have infinitely many steady state solutions the present paper gives some explicit information on the asymptotic limit of the time dependent solution in relation to these steady states this information exhibits some rather distinct properties of the solutions between the neumann boundary problem and the dirichlet or mixed boundary problem

testing for uniformity in multidimensional data testing for uniformity in multidimensional data is important in exploratory pattern analysis statistical pattern recognition and image processing the goal of this paper is to determine whether the data follow the uniform distribution over some compact convex set in k dimensional space called the sampling window we first provide a simple computationally efficient method for generating a uniformly distributed sample over a set which approximates the convex hul of the data we then test for uniformity by comparing this generated sample to the data by using friedman rafsky s minimal spanning tree mst based test experiments with both simulated and real data indicate that this mst based test is useful in deciding if data are uniform

finite element formulation of unilateral boundary conditions for unsaturated flow in porous continua this paper presents the numerical resolution of unilateral boundary conditions able to effectively model several problems of unsaturated flow as those involving rainfall infiltration and seepage faces besides the penalty technique we also consider the novel regularization of these conditions by means of the more effective augmented lagrangian method the performance of the so obtained finite element method is carefully investigated in terms of accuracy and ill conditioning effects including comparisons with analytical solutions and a complete identification of the analogies with the problem of frictionless contact in this way we provide a priori estimates of optimal and admissible ranges for the penalty coefficient as functions of permeability and spatial discretization the proposed method and the estimated coefficient ranges are validated in further numerical examples involving the propagation of a wetting front due to rainfall and the partial saturation of an aged concrete dam these applications show that the proposed regularizations do not induce any detrimental effect on solution accuracy and on convergence rate of the employed newton raphson method hence the present approach should be preferred to the commonly used iterative switching between the imposed flow and the imposed pressure conditions which often leads to spurious oscillations and convergence failures

a genetic approach to the automatic clustering problem abstract in solving the clustering problem traditional methods for example the k means algorithm and its variants usually ask the user to provide the number of clusters unfortunately the number of clusters in general is unknown to the user therefore clustering becomes a tedious trial and error work and the clustering result is often not very promising especially when the number of clusters is large and not easy to guess in this paper we propose a genetic algorithm for the clustering problem this algorithm is suitable for clustering the data with compact spherical clusters it can be used in two ways one is the user controlled clustering where the user may control the result of clustering by varying the values of the parameter w a small value of w results in a larger number of compact clusters while a large value of w results in a smaller number of looser clusters the other is an automatic clustering where a heuristic strategy is applied to find a good clustering experimental results are given to illustrate the effectiveness of this genetic clustering algorithm

order of convergence estimates for an euler implicit mixed finite element discretization of richards equation we analyze a discretization method for a class of degenerate parabolic problems that includes the richards equation this analysis applies to the pressure based formulation and considers both variably and fully saturated regimes to overcome the difficulties posed by the lack in regularity we first apply the kirchhoff transformation and then integrate the resulting equation in time we state a conformal and a mixed variational formulation and prove their equivalence this will be the underlying idea of our technique to get error estimates a regularization approach is combined with the euler implicit scheme to achieve the time discretization again equivalence between the two formulations is demonstrated for the semidiscrete case the lowest order raviart thomas mixed finite elements are employed for the discretization in space error estimates are obtained showing that the scheme is convergent

two phase flow equations with outflow boundary conditions in the hydrophobic hydrophilic case we introduce an approximation procedure and provide existence results for two phase flow equations in porous media the medium can have hydrophobic and hydrophilic components such that the capillary pressure function is degenerate for extreme saturations our main interest is the outflow boundary condition which models an interface with open space the approximate system introduces standard boundary conditions and can be used in numerical schemes it allows the derivation of maximum principles this is the basis for the derivation of the limiting system in the form of a variational inequality

on nonlinear reaction diffusion systems abstract this paper presents a qualitative analysis for a coupled system of two reaction diffusion equations under various boundary conditions which arises from a number of physical problems the nonlinear reaction functions are classified into three basic types according to their relative quasi monotone property for each type of reaction functions an existence comparison theorem in terms of upper and lower solutions is established for the time dependent system as well as some boundary value problems three concrete physical systems arising from epidemics biochemistry and engineering are taken as representatives of the basic types of reacting problems through suitable construction of upper and lower solutions various qualitative properties of the solution for each system are obtained these include the existence and bounds of time dependent solutions asymptotic behavior of the solution stability and instability of nontrivial steady state solutions estimates of stability regions and finally the blowing up property of the solution special attention is given to the homogeneous neumann boundary condition

a genetic clustering algorithm for data with non spherical shape clusters abstract in solving clustering problem traditional methods for example the k means algorithm and its variants usually ask the user to provide the number of clusters unfortunately the number of clusters in general is unknown to the user the traditional neighborhood clustering algorithm usually needs the user to provide a distance d for the clustering this d is difficult to decide because some clusters may be compact but others may be loose in this paper we propose a genetic clustering algorithm for clustering the data whose clusters are not of spherical shape it can automatically cluster the data according to the similarities and automatically find the proper number of clusters the experimental results are given to illustrate the effectiveness of the genetic algorithm

monotone enclosure for elliptic and parabolic sytems with nonmonotone nonlinearities abstract monotone iteration techniques for weakly coupled systems having nonmonotone nonlinearities have been developed recently by m khavanin and v lakshmikantham and by s carl in this paper an existence and enclosing theorem is given in terms of upper and lower solutions and a new monotone iteration technique which in a natural way extends the monotone iteration technique used in the monotone case to the nonmonotone case considered here is proposed the maximum and minimum functions that occur in the iteration scheme allow one to work under low regularity conditions particularly no local lipschitz conditions on the nonlinearities are required for the iteration process a crucial role in the analysis is played by a measurable selection theorem which is usually known from optimization theory furthermore conditions which ensure that the iteration process converges to the uniquely defined solution are given

error estimates for a mixed finite element discretization of some degenerate parabolic equations we consider a numerical scheme for a class of degenerate parabolic equations including both slow and fast diffusion cases a particular example in this sense is the richards equation modeling the flow in porous media the numerical scheme is based on the mixed finite element method mfem in space and is of one step implicit in time the lowest order raviart thomas elements are used here we extend the results in radu et al siam j numer anal 42 1452 1478 2004 schneid et al numer math 98 353 370 2004 to a more general framework by allowing for both types of degeneracies we derive error estimates in terms of the discretization parameters and show the convergence of the scheme the features of the mfem especially of the lowest order raviart thomas elements are now fully exploited in the proof of the convergence the paper is concluded by numerical examples

threshold validity for mutual neighborhood clustering clustering algorithms have the annoying characteristic of finding clusters in random data a theoretical analysis of the threshold of the mutual neighborhood clustering algorithm mnca under the hypothesis of random data is presented this yields a theoretical minimum value of this threshold below which even unclustered data are broken into separate clusters to derive the threshold a theorem about mutual near neighbors in a poisson process is stated and proved simple experiments demonstrate the usefulness of the theoretical thresholds

flow of oil and water in a porous medium abstract the flow of two immiscible and incompressible fluids in a porous medium is described by a system of quasilinear degenerate partial differential equations in this paper the existence of a weak solution by regularization is shown

regularization of outflow problems in unsaturated porous media with dry regions we study a porous medium with saturated unsaturated and dry regions described by richards equation for the saturation s and the pressure p due to a degenerate permeability coefficient k x s and a degenerate capillary pressure function pc x s the equations may be of elliptic parabolic or of ode type we construct a parabolic regularization of the equations and find conditions that guarantee the convergence of the parabolic solutions to a solution of the degenerate system an example shows that the convergence fails in general our approach provides an existence result for the outflow problem in the case of x dependent coefficients and a method for a numerical approximation

strong solutions to the richards equation in the unsaturated zone abstract we study the unsaturated case of the richards equation in three space dimensions with dirichlet boundary data we first establish an a priori l estimate with its help by means of a fixed point argument we prove global in time existence of a unique weak solution in sobolev spaces finally we are able to improve the regularity of this weak solution in order to gain a strong one