existence of weak solutions for a hyperbolic model of chemosensitive movement a hyperbolic model for chemotaxis and chemosensitive movement in one space dimension is considered in contrast to parabolic models for chemotaxis the hyperbolic model allows us to take the dependence of the particle speed on external stimuli explicitly into account this qualitatively covers recent experiments on chemotaxis in which it has been shown that particles adapt their speed to the surrounding environment the model presented here consists of two hyperbolic differential equations of first order coupled with an elliptic equation we assume that the speed depends on the external stimulus only and not on its gradients in that case solutions with steep gradients are expected which have the interpretation of moving swarms a notion of weak solutions for this hyperbolic chemotaxis model is presented and the global existence of weak solutions is shown the proof relies on the vanishing viscosity method i e we obtain the weak solution as the limit of classical solutions of an associated parabolically regularized problem for vanishing viscosity parameter numerical simulations demonstrate phenomena like swarming behaviour and formation of steep gradients

chemotactic traveling waves by metric of food the meaningful distance to biological organisms is not necessarily one measured by the euclidean metric but possibly one by a metric that counts the amount of resources such as food it is assumed in this paper that the distance for biological organisms is measured by the amount of food between two places a new chemotaxis model is introduced as an application of this metric of food it is shown that if the walk length of a random walk system is given by such a metric the well known chemotactic traveling wave phenomena can be obtained without the typical assumption that microscopic scale bacteria may sense the macroscopic scale gradient of a chemical concentration the uniqueness and the existence of a traveling wave solution are obtained

condition for intracellular adaptive dynamics for chemotaxis chemotaxis is ubiquitously performed by bacteria they sense and move toward a region with a higher concentration of an attractive chemical by changing the rate of tumbling for random walk we numerically studied several models with internal adaptive dynamics to examine the validity of the condition for chemotaxis proposed by oosawa and nakaoka j theor biol 66 747 1977 which states that the time scale of tumbling frequency must be smaller than that of adaptation and greater than that of sensing suitable renormalization of the time scales showed that the condition holds for a variety of environments and for both short and long term behavior

global convergence of a kinetic model of chemotaxis to a perturbed keller segel model abstract we consider a class of kinetic models of chemotaxis with two positive non dimensional parameters coupled to a parabolic equation of the chemo attractant if both parameters are set equal zero we have the classical keller segel model for chemotaxis we prove global existence of solutions of this two parameters kinetic model and prove convergence of this model to models of chemotaxis with global existence when one of these two parameters is set equal zero in one case we find as a limit model a kinetic model of chemotaxis while in the other case we find a perturbed keller segel model with global existence of solutions

transport diffusion models with nonlinear boundary conditions and solution by generalized collocation methods this paper deals with the derivation of a class of nonlinear transport and diffusion models implemented with nonlinear boundary conditions mathematical tools to treat the initial boundary value problems are developed based on generalized collocation methods focused on the treatment of nonlinear boundary conditions in one space dimension applications refer to a problem of interest in applied sciences

some examples of kinetic schemes whose diffusion limit is il in s exponential fitting this paper is concerned with diffusive approximations of some numerical schemes for several linear or weakly nonlinear kinetic models which are motivated by wide range applications including radiative transfer or neutron transport run and tumble models of chemotaxis dynamics and vlasov fokker planck plasma modeling the well balanced method applied to such kinetic equations leads to time marching schemes involving a scattering s matrix itself derived from a normal modes decomposition of the stationary solution one common feature these models share is the type of diffusive approximation their macroscopic densities solve drift diffusion systems for which a distinguished numerical scheme is il in scharfetter gummel s exponential fitting discretization we prove that these well balanced schemes relax within a parabolic rescaling towards such type of discretization by means of an appropriate decomposition of the s matrix hence are asymptotic preserving

aggregation blowup and collapse the abc s of taxis in reinforced random walks in many biological systems movement of an organism occurs in response to a diffusible or otherwise transported signal and in its simplest form this can be modeled by diffusion equations with advection terms of the form first derived by patlak bull of math biophys 15 1953 pp 311 338 however other systems are more accurately modeled by random walkers that deposit a nondiffusible signal that modifies the local environment for succeeding passages in these systems one example of which is the myxobacteria the question arises as to whether aggregation is possible under suitable hypotheses on the transition rules and the production of a control species that modulates the transition rates davis probab theory related fields 84 1990 pp 203 229 has studied this question for a certain class of random walks and here we extend this analysis to the continuum limit of such walks we first derive several general classes of partial differential equations that depend on how the movement rules are

krylov methods for the incompressible navier stokes equations methods are presented for time evolution steady state solving and linear stability analysis for the incompressible navier stokes equations at low to moderate reynolds numbers the methods use krylov subspaces constructed by the arnoldi process from actions of the explicit navier stokes right hand side and of its jacobian without inversion of the viscous operator time evolution is performed by a nonlinear extension of the method of exponential propagation steady states are calculated by inexact krylov newton iteration using orthores and gmres linear stability analysis is carried out using an implicitly restarted arnoldi process with implicit polynomial filters a detailed implementation is described for a pseudospectral calculation of the stability of taylor vortices with respect to wavy vortices in the couette taylor problem 9 1994 academic press inc

decay rates for inverses of band matrices spectral theory and classical approximation theory are used to give a new proof of the exponential decay of the entries of the inverse of band matrices the rate of decay of a 1 can be bounded in terms of the essential spectrum of aa for general a and in terms of the essential spectrum of a for positive definite a in the positive definite case the bound can be attained these results are used to establish the exponential decay for a class of generalized eigenvalue problems and to establish exponential decay for certain sparse but nonbanded matrices we also establish decay rates for certain generalized inverses

a classification of spikes and plateaus in this paper we classify local maxima into spikes and plateaus we give analytic definitions for spikes and plateaus in terms of a nonlocal gradient and a fourth order derivative in higher dimensions the hesse matrix of delta f x is of relevance this classification is applied to pattern formation models in mathematical physics and mathematical biology including cahn hilliard equations chemotaxis equations reaction diffusion equations gierer meinhardt models and gray scott models we show for some of these examples that the stability of spatial patterns depends on the spike versus plateau type of the solution we prove for example that scalar reaction diffusion equations in any spatial dimension cannot have stable spike steady states

large scale dynamics of the persistent turning walker model of fish behavior this paper considers a new model of individual displacement based on fish motion the so called persistent turning walker ptw model which involves an ornstein uhlenbeck process on the curvature of the particle trajectory the goal is to show that its large time and space scale dynamics is of diffusive type and to provide an analytic expression of the diffusion coefficient two methods are investigated in the first one we compute the large time asymptotics of the variance of the individual stochastic trajectories the second method is based on a diffusion approximation of the kinetic formulation of these stochastic trajectories the kinetic model is a fokker planck type equation posed in an extended phase space involving the curvature among the kinetic variables we show that both methods lead to the same value of the diffusion constant we present some numerical simulations to illustrate the theoretical results

boundary conditions for young van vliet recursive filtering young and van vliet have designed computationally efficient methods for approximating gaussian based convolutions by running a recursive infinite impulse response iir filter forward over the input signal then running a second iir filter backward over the first filter s output to transition between the two filters they use a suboptimal heuristic that produces significant amplitude and phase distortion for all points within about three standard deviations of the right hand boundary in this correspondence a simple linear transition rule that eliminates this distortion is derived

error bounds for the lanczos methods for approximating matrix exponentials in this paper we present new error bounds for the lanczos method and the shift and invert lanczos method for computing e tau a v for a large sparse symmetric positive semidefinite matrix a compared with the existing error analysis for these methods our bounds relate the convergence to the condition numbers of the matrix that generates the krylov subspace in particular we show that the lanczos method will converge rapidly if the matrix a is well conditioned regardless of what the norm of tau a is numerical examples are given to demonstrate the theoretical bounds

well posedness of a system of transport and diffusion equations in space of measures abstract in this paper we establish the well posedness of the following system of two transport equations coupled with a diffusion equation t t 1 v 1 t t 1 n 1 t t t t 2 t 2 n 2 t t t t 3 v 2 t t 3 n 3 t t in r d where t 1 t 2 t 3 are finite signed measures here the vector field v 1 v 2 and the source term n 1 n 2 n 3 depend on the measure valued solution vector t t 1 t 2 t 3

modeling selective local interactions with memory recently we developed a stochastic particle system describing local interactions between cyanobacteria we focused on the common freshwater cyanobacteria synechocystis sp which are coccoidal bacteria that utilize group dynamics to move toward a light source a motion referred to as phototaxis we were particularly interested in the local interactions between cells that were located in low to medium density areas away from the front the simulations of our stochastic particle system in 2d replicated many experimentally observed phenomena such as the formation of aggregations and the quasi random motion of cells in this paper we seek to develop a better understanding of group dynamics produced by this model to facilitate this study we replace the stochastic model with a system of ordinary differential equations describing the evolution of particles in 1d unlike many other models our emphasis is on particles that selectively choose one of their neighbors as the preferred direction of motion furthermore we incorporate memory by allowing persistence in the motion we conduct numerical simulations which allow us to efficiently explore the space of parameters in order to study the stability size and merging of aggregations

reaction advection random motions in inhomogeneous media abstract a probabilistic approach to reaction advection systems with a finite propagation speed is developed the convergence to traveling wave solutions in inhomogeneous media is studied the results are heavily based on the author s previous paper branching random motions nonlinear hyperbolic systems and travelling waves preprint of simon bolivar university caracas venezuela 2003 32 pp submitted for publication

decay properties for functions of matrices over c algebras abstract we extend previous results on the exponential off diagonal decay of the entries of analytic functions of banded and sparse matrices to the case where the matrix entries are elements of a c algebra

residence time and collision statistics for exponential flights the rod problem revisited many random transport phenomena such as radiation propagation chemical biological species migration or electron motion can be described in terms of particles performing exponential flights for such processes we sketch a general approach based on the feynman kac formalism that is amenable to explicit expressions for the moments of the number of collisions and the residence time that the walker spends in a given volume as a function of the particle equilibrium distribution we then illustrate the proposed method in the case of the so called rod problem a one dimensional system and discuss the relevance of the obtained results in the context of monte carlo estimators

recursive anisotropic 2 d gaussian filtering based on a triple axis decomposition we describe a recursive algorithm for anisotropic 2 d gaussian filtering based on separating the filter into the cascade of three rather two 1 d filters the filters operate along axes obtained by integer horizontal and or vertical pixel shifts this eliminates interpolation which removes spatial inhomogeneity in the filter and produces more elliptically shaped kernels it also results in a more regular filter structure which facilitates implementation in dsp chips finally it improves matching between filters with the same eccentricity and width but different orientations our analysis and experiments indicate that the computational complexity is similar to an algorithm that operates along two axes 11ms for a 512times512 image using a 3 2 ghz pentium 4 pc on the other hand given a limited set of basis filter axes there is an orientation dependent lower bound on the achievable aspect ratios

a restarted krylov subspace method for the evaluation of matrix functions we show how the arnoldi algorithm for approximating a function of a matrix times a vector can be restarted in a manner analogous to restarted krylov subspace methods for solving linear systems of equations the resulting restarted algorithm reduces to other known algorithms for the reciprocal and the exponential functions we further show that the restarted algorithm inherits the superlinear convergence property of its unrestarted counterpart for entire functions and present the results of numerical experiments

implementation of a restarted krylov subspace method for the evaluation of matrix functions a new implementation of restarted krylov subspace methods for evaluating f a b for a function f a matrix a and a vector b is proposed in contrast to an implementation proposed previously it requires constant work and constant storage space per restart cycle the convergence behavior of this scheme is discussed and a new stopping criterion based on an error indicator is given the performance of the implementation is illustrated for three parabolic initial value problems requiring the evaluation of exp a b

collision statistics for random flights with anisotropic scattering and absorption for a broad class of random walks with anisotropic scattering kernels and absorption we derive explicit formulas that allow expressing the moments of the collision number n v performed in a volume v as a function of the particle equilibrium distribution our results apply to arbitrary domains v and boundary conditions and allow assessing the hitting statistics for systems where the typical displacements are comparable to the domain size so that the diffusion limit is possibly not attained an example is discussed for one dimensional random flights with exponential displacements where analytical calculations can be carried out

b spline signal processing ii efficiency design and applications for pt i see ibid vol 41 no 2 p 821 33 1993 a class of recursive filtering algorithms for the efficient implementation of b spline interpolation and approximation techniques is described in terms of simplicity of realization and reduction of computational complexity these algorithms compare favorably with conventional matrix approaches a filtering interpretation low pass filter followed by an exact polynomial spline interpolator of smoothing spline and least squares approximation methods is proposed these techniques are applied to the design of digital filters for cubic spline signal processing an efficient implementation of a smoothing spline edge detector is proposed it is also shown how to construct a cubic spline image pyramid that minimizes the loss of information in passage from one resolution level to the next in terms of common measures of fidelity this data structure appears to be superior to the gaussian laplacian pyramid

on explosions of solutions to a system of partial differential equations modelling chemotaxis a system of partial differential equations modelling chemotactic aggregation is analysed keller segel model conditions on the system of paramaters are given implying global existence of smooth solutions in two space dimensions and radially symmetric situations explosion of the bacteria concentration in finite time is shown for a class of initial values

abstract time dependent transport equations abstract we consider the abstract time dependent linear transport equation as an initial boundary value evolution problem in the banach spaces lp 1 p

multiscale models of taxis driven patterning in bacterial populations spatially distributed populations of various types of bacteria often display intricate spatial patterns that are thought to result from the cellular response to gradients of nutrients or other attractants in the past decade a great deal has been learned about signal transduction metabolism and movement in e coli and other bacteria but translating the individual level behavior into population level dynamics is still a challenging problem however this is a necessary step because it is computationally impractical to use a strictly cell based model to understand patterning in growing populations since the total number of cells may reach 10 12 10 14 in some experiments in the past phenomenological equations such as the patlak keller segel equations have been used in modeling the cell movement that is involved in the formation of such patterns but the question remains as to how the microscopic behavior can be correctly described by a macroscopic equation significant progress has been made for

asymptotic stability of solutions to volterra renewal integral equations with space maps abstract in this paper we consider linear volterra renewal integral equations vies whose solutions depend on a space variable via a map transformation we investigate the asymptotic properties of the solutions and study the asymptotic stability of a numerical method based on direct quadrature in time and interpolation in space we show its properties through test examples

on krylov subspace approximations to the matrix exponential operator krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper we derive error bounds via a functional calculus of arnoldi and lanczos methods that reduces the study of krylov subspace approximations of functions of matrices to that of linear systems of equations as a side result we obtain error bounds for galerkin type krylov methods for linear equations namely the biconjugate gradient method and the full orthogonalization method for krylov approximations to matrix exponentials we show superlinear error decay from relatively small iteration numbers onwards depending on the geometry of the numerical range the spectrum or the pseudospectrum the convergence to exp tau a v is faster than that of corresponding krylov methods for the solution of linear equations i tau a x v which usually arise in the numerical solution of stiff ordinary differential equations odes we therefore propose a new class of time integration methods for large systems of nonlinear differential equations which use krylov approximations to the exponential function of the jacobian instead of solving linear or nonlinear systems of equations in every time step

maxwellian decay for well balanced approximations of a super characteristic chemotaxis model we focus on the numerical simulation of a one dimensional model of chemotaxis dynamics proposed by greenberg and alt trans amer math soc 300 1987 pp 235 258 in a bounded domain by means of a previously introduced well balanced wb and asymptotic preserving ap scheme l gosse j math anal appl 2011 we are especially interested in studying the decay onto numerical steady states for two reasons 1 conventional upwind schemes have been shown to stabilize onto spurious non maxwellian states with a very big mass flux see e g f guarguaglini et al discrete contin dyn syst ser b 12 2009 pp 39 76 and 2 the initial data lead to a dynamic which is mostly super characteristic in the sense of s jin and m katsoulakis siam j appl math 61 2000 pp 273 292 thus the stability results of gosse do not apply a reflecting boundary condition which is compatible with the well balanced character is presented a mass preservation property is proved and some results on super characteristic relaxation are recalled numerical experiments with coarse computational grids are presented in detail they illustrate the bifurcation diagrams of guarguaglini et al which relate the total initial mass of cells with the time asymptotic values of the chemoattractant substance on each side of the domain it is shown that the wb scheme stabilizes correctly onto zero mass flow rate hence maxwellian steady states which agree with the aforementioned bifurcation diagrams the evolution in time of residues is commented for every considered test case

on the derivation of macroscopic hyperbolic equations for binary multicellular growing mixtures this paper presents an asymptotic theory based on the hyperbolic scaling for a large class of boltzmann type equations suitable to model the evolution of a binary mixture of multicellular systems in biology the modelling approach is that of the kinetic theory for active particles time scaling related to cell movement and biological activity are shown to play a crucial role in determining the macroscopic equations corresponding to each case applications are focused on the immune competition

simulating diffusions with piecewise constant coefficients using a kinetic approximation using a kinetic approximation of a linear diffusion operator we propose an algorithm that allows one to deal with the simulation of a multi dimensional stochastic process in a media which is locally isotropic except on some surface where the diffusion coefficient presents some discontinuities numerical examples are given in dimensions one to three on pdes or stochastic pdes with or without source terms

from a discrete model of chemotaxis with volume filling to a generalized patlak keller segel model we present a discrete model of chemotaxis whereby cells responding to a chemoattractant are seen as individual agents whose movement is described through a set of rules that result in a biased rand

from individual to collective behavior in bacterial chemotaxis bacterial chemotaxis is widely studied from both the microscopic cell and macroscopic population points of view and here we connect these very different levels of description by deriving the classical macroscopic description for chemotaxis from a microscopic model of the behavior of individual cells the analysis is based on the velocity jump process for describing the motion of individuals such as bacteria wherein each individual carries an internal state that evolves according to a system of ordinary differential equations forced by a time and or space dependent external signal in the problem treated here the turning rate of individuals is a functional of the internal state which in turn depends on the external signal using moment closure techniques in one space dimension we derive and analyze a macroscopic system of hyperbolic differential equations describing this velocity jump process using a hyperbolic scaling of space and time we obtain a single second order hyperbolic equation for the

krylov implicit integration factor methods for spatial discretization on high dimensional unstructured meshes application to discontinuous galerkin methods abstract integration factor methods are a class of exactly linear part time discretization methods in q nie y t zhang r zhao efficient semi implicit schemes for stiff systems journal of computational physics 214 2006 521 537 a class of efficient implicit integration factor iif methods were developed for solving systems with both stiff linear and nonlinear terms arising from spatial discretization of time dependent partial differential equations pdes with linear high order terms and stiff lower order nonlinear terms the tremendous challenge in applying iif temporal discretization for pdes on high spatial dimensions is how to evaluate the matrix exponential operator efficiently for spatial discretization on unstructured meshes to solve pdes on complex geometrical domains how to efficiently apply the iif temporal discretization was open in this paper we solve this problem by applying the krylov subspace approximations to the matrix exponential operator then we apply this novel time discretization technique to discontinuous galerkin dg methods on unstructured meshes for solving reaction diffusion equations numerical examples are shown to demonstrate the accuracy efficiency and robustness of the method in resolving the stiffness of the dg spatial operator for reaction diffusion pdes application of the method to a mathematical model in pattern formation during zebrafish embryo development shall be shown

asymptotic series for singularly perturbed kolmogorov fokker planck equations we derive limit theorems for the transition densities of diffusion processes and develop asymptotic expansions for solutions of a class of singularly perturbed kolmogorov fokker planck equations the model under consideration can be viewed as a markov process having two time scales one of them is a rapidly changing scale and the other is a slowly varying one the study is motivated by a wide range of applications involving singularly perturbed markov processes in manufacturing systems reliability analysis queueing networks statistical physics population biology financial economics and many other related fields in this work the asymptotic expansion is constructed explicitly it is shown that the initial layer terms in the expansion decay at an exponential rate error bounds on the remainder terms also are obtained the validity of the expansion is rigorously justified

on the convergence of krylov subspace methods for matrix mittag leffler functions in this paper we analyze the convergence of some commonly used krylov subspace methods for computing the action of matrix mittag leffler functions as is well known such functions find application in the solution of fractional differential equations we illustrate the theoretical results by some numerical experiments

partial moment minimum entropy models for kinetic chemotaxis equations in one and two dimensions the aim of this work is to investigate the application of partial moment approximations to kinetic chemotaxis equations in one and two spatial dimensions starting with a kinetic equation for the cell densities we apply a half quarter moments method with different closure relations to derive macroscopic equations appropriate numerical schemes are presented as well as numerical results for several test cases the resulting solutions are compared to kinetic reference solutions and solutions computed using a full moment method with a linear superposition strategy

the derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many particle systems the chemotaxis equations are a well known system of partial differential equations describing aggregation phenomena in biology in this paper they are rigorously derived from an interacting stochastic many particle system where the interaction between the particles is rescaled in a moderate way as population size tends to infinity the novelty of this result is that in all previous applications of this kind of limiting procedure the principal part of the system is assumed to fulfill an ellipticity condition which is not satisfied in our case new techniques which deal with this difficulty are presented

invariance principles for hyperbolic random walk systems abstract reaction random walk systems are hyperbolic models for the description of spatial motion in one dimension and reaction of particles in contrast to reaction diffusion equations particles have finite propagation speed for parabolic systems invariance results and maximum principles are well known a convex set is positively invariant if at each boundary point an outer normal is a left eigenvector of the diffusion matrix and if the vector field defined by the pure reaction equation points inward at the boundary here we show a corresponding result for random walk systems the model parameters are the particle speeds the rates of change in direction and the reaction vector field a convex domain is invariant if at each boundary point an outer normal is a left eigenvector of the speed matrix and if a vector field given by the reaction equation combined with the turning rates points inward finally a positivity result is shown

hyperbolic versus parabolic asymptotics in kinetic theory toward fluid dynamic models in this work we are interested in the hyperbolic limits in kinetic theory we propose a nonstandard scaling to be understood as a sort of intermediate hyperbolic limit which connects the macroscopic hyperbolic limiting behavior of the physical system with the microscopic properties usually obtained under parabolic scalings we present our main result by means of a general kinetic frame for the intermediate hyperbolic limit which covers some well known examples in kinetic theory vlasov poisson fokker planck systems and linear relaxation for boltzmann type equations in semiconductor theory among others we will also apply our methods to deal with the kac approach to boltzmann operators

going from microscopic to macroscopic on nonuniform growing domains throughout development chemical cues are employed to guide the functional specification of underlying tissues while the spatiotemporal distributions of such chemicals can be influenced by the growth of the tissue itself these chemicals termed morphogens are often modeled using partial differential equations pdes the connection between discrete stochastic and deterministic continuum models of particle migration on growing domains was elucidated by baker yates and erban bull math biol 72 719 2010 in which the migration of individual particles was modeled as an on lattice position jump process we build on this work by incorporating a more physically reasonable description of domain growth instead of allowing underlying lattice elements to instantaneously double in size and divide we allow incremental element growth and splitting upon reaching a predefined threshold size such a description of domain growth necessitates a nonuniform partition of the domain we first demonstrate that an individual based stochastic model for particle diffusion on such a nonuniform domain partition is equivalent to a pde model of the same phenomenon on a nongrowing domain providing the transition rates which we derive are chosen correctly and we partition the domain in the correct manner we extend this analysis to the case where the domain is allowed to change in size altering the transition rates as necessary through application of the master equation formalism we derive a pde for particle density on this growing domain and corroborate our findings with numerical simulations

a branching random evolution and a nonlinear hyperbolic equation particles moving at constant speed along the real line and reversing direction as a poisson process constitute a random evolution if the particles simultaneously undergo an independent branching p

superdiffusion of a random walk driven by an ergodic markov process with switching we propose a markov model with an ergodic two component switching mechanism that dynamically generates anomalous diffusion the first component plays the role of a hidden parameter the second one is the switching component generating the superdiffusion of a random walker and is itself non markovian the model is studied numerically using the monte carlo technique

the diffusion limit of transport equations ii chemotaxis equations in this paper we use the diffusion limit expansion of transport equations developed earlier t hillen and h g othmer siam j appl math 61 2000 pp 751 775 to study the limiting equation under a variety of external biases imposed on the motion when applied to chemotaxis or chemokinesis these biases produce modification of the turning rate the movement speed or the preferred direction of movement depending on the strength of the bias it leads to anisotropic diffusion to a drift term in the flux or to both in the parabolic limit we show that the classical chemotaxis equation which we call the patlak keller segel alt pksa equation arises only when the bias is sufficiently small using this general framework we derive phenomenological models for chemotaxis of flagellated bacteria of slime molds and of myxobacteria we also show that certain results derived earlier for one dimensional motion can easily be generalized to two or three dimensional motion as well

analysis of some krylov subspace approximations to the matrix exponential operator in this note a theoretical analysis of some krylov subspace approximations to the matrix exponential operation exp a v is presented and a priori and a posteriors error estimates are established several such approximations are considered the main idea of these techniquesis to approximately project the exponential operator onto a small krylov subspace and to carry out the resulting small exponential matrix computation accurately this general approach which has been used with success in several applications provides a systematic way of defining high order explicit type schemes for solving systems of ordinary differential equations or time dependent partial differential equations

the diffusion limit of transport equations derived from velocity jump processes in this paper we study the diffusion approximation to a transport equation that describes the motion of individuals whose velocity changes are governed by a poisson process we show that under an appropriate scaling of space and time the asymptotic behavior of solutions of such equations can be approximated by the solution of a diffusion equation obtained via a regular perturbation expansion in general the resulting diffusion tensor is anisotropic and we give necessary and sufficient conditions under which it is isotropic we also give a method to construct approximations of arbitrary high order for large times in a second paper part ii we use this approach to systematically derive the limiting equation under a variety of external biases imposed on the motion depending on the strength of the bias it may lead to an anisotropic diffusion equation to a drift term in the flux or to both our analysis generalizes and simplifies previous derivations that lead to the classical patlak keller segel alt

asymptotic preserving and well balanced schemes for the 1d cattaneo model of chemotaxis movement in both hyperbolic and diffusive regimes abstract the original well balanced wb framework of greenberg and leroux 1996 24 and gosse 2002 18 relying on non conservative nc products see lefoch and tzavaras 1999 41 is set up in order to efficiently treat the so called cattaneo model of chemotaxis in 1d see hillen and stevens 2000 29 it proceeds by concentrating the source terms onto dirac masses this allows to handle them by nc jump relations based on steady state equations which can be integrated explicitly a riemann solver is deduced and the corresponding wb godunov scheme completed with the standard hoff smoller theory see hillen and stevens 2000 29 for the diffusion reaction equation ruling the evolution of the chemoattractant concentration is studied in detail later following former results of gosse and toscani 2002 2003 21 22 a simple rewriting of the nc jump relations allows to generate another version of the same godunov scheme which is well adapted to the parabolic scaling involving a small parameter e the standard bv framework is used to study the uniform stability of this asymptotic preserving ap scheme with respect to e allows to pass to the limit and derive a simple centered discretization of the keller segel model finally results by filbet 2006 14 permit to pass to the complementary limit when the space step h is sent to zero numerical results are included to illustrate the feasibility and the efficiency of the method

b spline signal processing i theory the use of continuous b spline representations for signal processing applications such as interpolation differentiation filtering noise reduction and data compressions is considered the b spline coefficients are obtained through a linear transformation which unlike other commonly used transforms is space invariant and can be implemented efficiently by linear filtering the same property also applies for the indirect b spline transform as well as for the evaluation of approximating representations using smoothing or least squares splines the filters associated with these operations are fully characterized by explicitly evaluating their transfer functions for b splines of any order applications to differentiation filtering smoothing and least squares approximation are examined the extension of such operators for higher dimensional signals such as digital images is considered

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

an optimal nonorthogonal separation of the anisotropic gaussian convolution filter we give an analytical and geometrical treatment of what it means to separate a gaussian kernel along arbitrary axes in ropfn and we present a separation scheme that allows us to efficiently implement anisotropic gaussian convolution filters for data of arbitrary dimensionality based on our previous analysis we show that this scheme is optimal with regard to the number of memory accesses and interpolation operations needed the proposed method relies on nonorthogonal convolution axes and works completely in image space thus it avoids the need for a fast fourier transform fft subroutine depending on the accuracy and speed requirements different interpolation schemes and methods to implement the one dimensional gaussian finite impulse response and infinite impulse response can be integrated special emphasis is put on analyzing the performance and accuracy of the new method in particular we show that without any special optimization of the source code it can perform anisotropic gaussian filtering faster than methods relying on the fft

modeling a self propelled autochemotactic walker we develop a minimal model for the stochastic dynamics of microorganisms where individuals communicate via autochemotaxis this means that microorganisms such as bacteria amoebae or cells follow the gradient of a chemical that they produce themselves to attract or repel each other a microorganism is represented as a self propelled particle or walker with constant speed while its velocity direction diffuses on the unit circle we study the autochemotactic response of a single self propelled walker whose dynamics is non markovian we show that its long time dynamics is always diffusive by deriving analytic expressions for its diffusion coefficient in the weak and strong coupling case we confirm our findings by numerical simulations

a convergence analysis for nonsymmetric lanczos algorithms a convergence analysis for the nonsymmetric lanczos algorithm is presented by using a tridiagonal structure of the algorithm some identities concerning ritz values and ritz vectors are established and used to derive approximation bounds in particular the analysis implies the classical results for the symmetric lanczos algorithm

from microscopic to macroscopic description of multicellular systems and biological growing tissues this paper presents an asymptotic theory for a large class of boltzmann type equations suitable to model the evolution of multicellular systems in biology the mathematical approach described herein shows how various types of diffusion phenomena linear and nonlinear can be obtained in suitable asymptotic limits time scaling related to cell movement and biological activity are shown to play a crucial role in determining the macroscopic equations corresponding to each case

lattice free models of directed cell motility directed cell migration often occurs when individual cells move in response to an external chemical stimulus cells can respond by moving in either the direction of increasing chemoattraction or decreasing chemorepulsion concentration many previous models of directed cell migration use a lattice based framework where agents undergo a lattice based random walk and the direction of nearest neighbour motility events is biased in a preferred direction such lattice based models can lead to unrealistic configurations of agents since the agents always move on an artificial lattice structure which is never observed experimentally we present a lattice free model of directed cell migration that incorporates two key features first agents move on a continuous domain with the possibility that there is some preferred direction of motion second to be consistent with experimental observations we enforce a crowding mechanism so that motility events that would lead to agent overlap are not permitted we compare simulation data from the new lattice free model with a more traditional lattice based model to provide additional insight into the lattice free model we construct an approximate conservation statement which corresponds to a nonlinear advection diffusion equation in the continuum limit the solution of this mean field model compares well with averaged data from the individual based model

modeling tumor cell migration from microscopic to macroscopic models it has been shown experimentally that contact interactions may influence the migration of cancer cells previous works have modelized this thanks to stochastic discrete models cellular automata at the cell level however for the study of the growth of real size tumors with several million cells it is best to use a macroscopic model having the form of a partial differential equation pde for the density of cells the difficulty is to predict the effect at the macroscopic scale of contact interactions that take place at the microscopic scale to address this we use a multiscale approach starting from a very simple yet experimentally validated microscopic model of migration with contact interactions we derive a macroscopic model we show that a diffusion equation arises as is often postulated in the field of glioma modeling but it is nonlinear because of the interactions we give the explicit dependence of diffusivity on the cell density and on a parameter governing cell cell interactions we discuss in detail the conditions of validity of the approximations used in the derivation and we compare analytic results from our pde to numerical simulations and to some in vitro experiments we notice that the family of microscopic models we started from includes as special cases some kinetically constrained models that were introduced for the study of the physics of glasses supercooled liquids and jamming systems

on transition densities of singularly perturbed diffusions with fast and slow components we derive asymptotic properties of transition densities for singularly perturbed diffusion processes with fast and slow components our study focuses on the kolmogorov fokker planck equations the model can be viewed as a diffusion process having two time scales and is motivated by a wide variety of applications involving singularly perturbed markov processes in manufacturing systems homogenization reliability analysis queueing networks statistical physics population biology financial economics and many other related fields by virtue of the methods of matched singular perturbation asymptotic expansion is constructed for the transition density the expansion includes both regular part and boundary layer corrections detailed justification of the asymptotic expansion is given and error bounds are also provided

advection diffusion equations for internal state mediated random walks in many biological examples of biased random walks movement statistics are determined by state dynamics that are internalto the organism or cell and that mediate responses to variable environments internal state dynamics are an essential element of such behaviors because they provide a memory mechanism for retaining effects of previous environments over multiple iterations of the random walk and because they impose constraints on the gradient climbing movements that can result when spatial gradients in individual density and in environmental properties are small the timescales characterizing redistribution of populations over large spatial scales are typically much larger than those characterizing equilibration in the internal state and in orientation taking advantage of this difference in timescales an advection diffusion equation is derived that approximates the long term spatial redistribution of individuals performing a velocity jump process in which turning probabilities are mediated by comp

branching exponential flights travelled lengths and collision statistics the evolution of several physical and biological systems ranging from neutron transport in multiplying media to epidemics or population dynamics can be described in terms of branching exponential flights a stochastic process which couples a galton watson birth death mechanism with random spatial displacements within this context one is often called to assess the length lv that the process travels in a given region v of the phase space or the number of visits nv to this same region in this paper we address this issue by resorting to the feynman kac formalism which allows characterizing the full distribution of lv and nv and in particular deriving explicit moment formulas some other significant physical observables associated to lv and nv such as the survival probability are discussed as well and results are illustrated by revisiting the classical example of the rod model in nuclear reactor physics

hermite approximation of a hyperbolic fokker planck optimality system to control a piecewise deterministic process abstractthe hermite spectral approximation of a hyperbolic fokker planck fp optimality system arising in the control of an unbounded piecewise deterministic process pdp is discussed to control the probability density function pdf corresponding to the pdp process an optimal control based on an fp strategy is considered the resulting optimality system consists of a hyperbolic system with opposite time orientation and an integral optimality condition equation a hermite spectral discretisation is investigated to approximate solutions to the optimality system in unbounded domains it is proven that the proposed scheme satisfies the conservativity requirement of the pdfs the spectral convergence rate of the discretisation scheme is proved and validated by numerical experiments

aging in mortal superdiffusive levy walkers a growing body of literature examines the effects of superdiffusive subballistic movement premeasurement aging or time lag on observations arising from single particle tracking a neglected aspect is the finite lifetime of these levy walkers be they proteins cells or larger structures we examine the effects of aging on the motility of mortal walkers and discuss the means by which permanent stopping of walkers may be categorized as arising from natural death or experimental artifacts such as low photostability or radiation damage this is done by comparison of the walkers mean squared displacement msd with the front velocity of propagation of a group of walkers which is found to be invariant under time lags for any running time distribution of a mortal random walker the msd is tempered by the stopping rate this provides a physical interpretation for truncated heavy tailed diffusion processes and serves as a tool by which to better classify the underlying running time distributions of random walkers tempering of aged msds raises the issue of misinterpreting superdiffusive motion which appears brownian or subdiffusive over certain time scales

using nonorthogonal lanczos vectors in the computation of matrix functions the lanczos algorithm uses a three term recurrence to construct an orthonormal basis for the krylov space corresponding to a symmetric matrix a and a nonzero starting vector varphi the vectors and recurrence coefficients produced by this algorithm can be used for a number of purposes including solving linear systems au varphi and computing the matrix exponential e ta varphi although the vectors produced in finite precision arithmetic are not orthogonal we show why they can still be used effectively for these purposes

exact representation of the asymptotic drift speed and diffusion matrix for a class of velocity jump processes abstract this paper examines a class of linear hyperbolic systems which generalizes the goldstein kac model to an arbitrary finite number of speeds v i with transition rates i j under the basic assumptions that the transition matrix is symmetric and irreducible and the differences v i v j generate all the space the system exhibits a large time behavior described by a parabolic advection diffusion equation the main contribution is to determine explicit formulas for the asymptotic drift speed and diffusion matrix in term of the kinetic parameters v i and i j establishing a complete connection between microscopic and macroscopic coefficients it is shown that the drift speed is the arithmetic mean of the velocities v i the diffusion matrix has a more complicate representation based on the graph with vertices the velocities v i and arcs weighted by the transition rates i j the approach is based on an exhaustive analysis of the dispersion relation and on the application of a variant of the kirchoff s matrix tree theorem from graph theory

model reduction of multi scale chemical langevin equations this paper addresses the model reduction problem for a class of stiff chemical langevin equations that arise as models of biomolecular networks with fast and slow reactions and can be described as continuous markov processes initially a coordinate transformation is sought that allows the decoupling of fast and slow variables in the model equations necessary and sufficient conditions are derived for such a linear transformation to exist along with an explicit change of variables which achieves the desired decoupling for the systems for which this step is applicable the method of adiabatic elimination is applied to determine a representation of the slow dynamics theoretical concepts and results are illustrated with simple examples

averaging principle and systems of singularly perturbed stochastic differential equations by developing certain auxiliary results a modified version of the stochastic averaging principle is developed to investigate dynamical systems consisting of fast and slow phenomena moreover an attempt is made to establish a relationship between the averaging assumption and certain ergodic type properties of the random process determined by an auxiliary system of stochastic differential equations finally an example is given to illustrate the scope of the results

the systematic adiabatic elimination of fast variables from a many dimensional fokker planck equation the chapman enskog method for the adiabatic elimination of fast variables is applied to a general fokker planck equation linear in the fast variables this equation is the counterpart of the generalized haken zwanzig model a system of coupled langevin equations often encountered in quantum optics and in the theory of non equilibrium phase transitions after a few equilibration times for the fast variables the time dependence of smooth solutions of this fokker planck equation is completely governed by the reduced distribution function of the slow variables which obeys a closed evolution equation we obtain an explicit perturbation series for the generator of this reduced evolution the system considered here is the most general system for which the chapman enskog hierarchy can be solved explicitly by exploiting an analogy between the unperturbed operator and the hamiltonian for coupled harmonic oscillators

transport equations and indices for random and biased cell migration based on single cell properties a mathematical modeling and analysis framework is presented to predict quantitative indices for random and biased cell migration based on mechanistic parameters describing the receptor mediated motility of an individual cell a general set of stochastic differential equations is derived to model cell movement on the time scale of the molecular processes that govern cell locomotion then by adiabatic elimination of the fast variables with projector operator formalism we derive approximate fokker planck equations fpes for the resultant cell movement on longer time scales analysis of these fpes provides expressions for statistical indices that are commonly used to characterize cell movement such as root mean squared cell speed directional persistence time mean squared displacement random motility coefficient and drift velocity in terms of the mechanistic parameters as specific examples we apply this approach to adhesion mediated directed cell migration haptotaxis and chemoattractant mediated d

singularity patterns in a chemotaxis model the authors study a chemotactic model under certain assumptions and obtain the existence of a class of solutions which blow up at the center of an open disc in finite time such a finite time blow up of solutions implies chemotactic collapse namely concentration of species to form sporae the model studied is the limiting case of a basic chemotactic model when diffusion of the chemical approaches infinity which has the form ut u u v 0 v u 1 on r2 where is an open disc with no flux homogeneous neumann boundary conditions the initial conditions are continuous functions u x 0 u0 x 0 v x 0 v0 x 0 for x under these conditions the authors prove there exists a radially symmetric solution u r t which blows up at r 0 t t a specific description of such a solution is presented the authors also discuss the strong similarity between the chemotactic model they study and the classical stefan problem

hydrodynamic singular regimes in 1 1 kinetic models and spectral numerical methods classical results from spectral theory of stationary linear kinetic equations are applied to efficiently approximate two physically relevant weakly nonlinear kinetic models a model of chemotaxis involving a biased velocity redistribution integral term and a vlasov fokker planck vfp system both are coupled to an attractive elliptic equation producing corresponding mean field potentials spectral de compositions of stationary kinetic distributions are recalled based on a variation of case s elementary solutions for the first model and on a sturm liouville eigenvalue problem for the second one well balanced godunov schemes with strong stability properties are deduced moreover in the stiff hydrodynamical scaling an hybridized algorithm is set up for which asymptotic preserving properties can be established under mild restrictions on the computational grid several numerical validations are displayed including the consistency of the vfp model with burgers hopf dynamics on the velocity field after blowup of the macroscopic density into dirac masses

generalized linear boltzmann equation describing non classical particle transport and related asymptotic solutions for small mean free paths abstract in classical kinetic models a particle free path distribution is exponential but this is more likely to be an exception than a rule in this paper we derive a generalized linear boltzmann equation glbe for a general free path distribution in the framework of alt s model in the case that the free path distribution has at least first and second finite moments we construct an asymptotic solution to the initial value problem for the glbe for small mean free paths in the special case of the one speed transport problem the asymptotic solution results in a diffusion approximation to the glbe