adaptivity in space and time for reaction diffusion systems in electrocardiology the paper introduces and studies numerical methods that are fully adaptive in both three dimensional 3d space and time to challenging multiscale cardiac reaction diffusion models in these methods temporal adaptivity comes via stepsize control in function space oriented linearly implicit time integration while spatial adaptivity is realized within multilevel finite element methods controlled by a posteriori local error estimators in contrast to other recent adaptivity approaches to cardiac modeling that discretize first in space and then in time so called method of lines our method discretizes first in time and then in space so called rothe method an approach that has already proven to be highly efficient in a number of challenging multiscale problems in science and technology kardos code library with this method the evolution of a complete heartbeat from the excitation to the recovery phase is simulated both in the frame of the anisotropic monodomain models and in the more realistic anisotropic bidomain models coupled with either a variant of the simple fitzhugh nagumo model or the more complex phase i luo rudy ionic model the numerical results exhibit a rather satisfactory performance of our adaptive method for complex cardiac reaction diffusion models on 3d domains up to moderate sizes in particular the method accurately resolves the evolution of the intra and extracellular potentials gating variables and ion concentrations during the excitation plateau and recovery phases

fractional calculus and continuous time finance ii the waiting time distribution we complement the theory of tick by tick dynamics of financial markets based on a continuous time random walk ctrw model recently proposed by scalas et al 4 and we point out its consistency with the behaviour observed in the waiting time distribution for bund future prices traded at liffe london

a second order accurate numerical approximation for the fractional diffusion equation fractional order diffusion equations are generalizations of classical diffusion equations treating super diffusive flow processes in this paper we examine a practical numerical method which is second order accurate in time and in space to solve a class of initial boundary value fractional diffusive equations with variable coefficients on a finite domain an approach based on the classical crank nicholson method combined with spatial extrapolation is used to obtain temporally and spatially second order accurate numerical estimates stability consistency and therefore convergence of the method are examined it is shown that the fractional crank nicholson method based on the shifted grunwald formula is unconditionally stable a numerical example is presented and compared with the exact analytical solution for its order of convergence

numerical solution of the space fractional fokker planck equation the traditional second order fokker planck equation may not adequately describe the movement of solute in an aquifer because of large deviation from the dynamics of brownian motion densities of stable type have been used to describe the probability distribution of these motions the resulting governing equation of these motions is similar to the traditional fokker planck equation except that the order of the highest derivative is fractional in this paper a space fractional fokker planck equation sffpe with instantaneous source is considered a numerical scheme for solving sffpe is presented using the riemann liouville and grunwald letnikov definitions of fractional derivatives the sffpe is transformed into a system of ordinary differential equations ode then the ode system is solved by a method of lines numerical results for sffpe with a constant diffusion coefficient are evaluated for comparison with the known analytical solution the numerical approximation of sffpe with a time dependent diffusion coefficient is also used to simulate levy motion with stable densities we will show that the numerical method of sffpe is able to more accurately model these heavy tailed motions

compact difference scheme for the fractional sub diffusion equation with neumann boundary conditions an effective finite difference scheme is considered for solving the time fractional sub diffusion equation with neumann boundary conditions a difference scheme combining the compact difference approach the spatial discretization and l 1 approximation for the caputo fractional derivative is proposed and analyzed although the spatial approximation order at the neumann boundary is one order lower than that for interior mesh points the unconditional stability and the global convergence order o t 2 a h 4 in discrete l 2 norm of the compact difference scheme are proved rigorously where t is the temporal grid size and h is the spatial grid size numerical experiments are included to support the theoretical results and comparison with the related works are presented to show the effectiveness of our method

polynomial acceleration of iterative schemes associated with subproper splittings a subproper splitting of a matrix a is a decomposition a b c such that the kernel of a includes that of b while the range of b includes that of a our purpose in the present work is to extend the convergence analysis of polynomial acceleration to the case of iterative schemes associated with subproper splittings in the case of hermitian matrices and consistent systems briefly stated our conclusions show that the regular theory extends to the subproper case provided that convergence to the solution of ax b is understood as convergence to a solution of ax b while b 1 a is understood as b a 0 where b is the moore penrose inverse of b

a class of second order difference approximations for solving space fractional diffusion equations a class of second order approximations called the weighted and shifted grunwald difference wsgd operators are proposed for riemann liouville fractional derivatives with their effective applications to numerically solving space fractional diffusion equations in one and two dimensions the stability and convergence of our difference schemes for space fractional diffusion equations with constant coefficients in one and two dimensions are theoretically established several numerical examples are implemented to test the efficiency of the numerical schemes and confirm the convergence order and the numerical results for variable coefficients problem are also presented

semi implicit time discretization schemes for the bidomain model the bidomain model is a system of partial differential equations used to model the propagation of electrical potential waves in the myocardium it is composed of coupled parabolic and elliptic partial differential equations as well as at least one ordinary differential equation to model the ion activity through the cardiac cell membranes the purpose of this paper is to propose and analyze several implicit semi implicit and explicit time stepping methods to solve that model in particular to avoid the expensive resolution of a nonlinear system through the newton raphson method we identify necessary stability conditions on the time step delta t for the proposed methods through a theoretical analysis based on energy estimates we next compare the methods for one and two dimensional test cases in terms of both stability and accuracy of the numerical solutions the theoretical stability conditions are seen to be consistent with those observed in practice our analysis allows us to recommend using either the crank nicolson adams bashforth method or the second order semi implicit backward differention method these semi implicit methods produce a good numerical solution unlike the explicit methods their stability does not depend on the spatial grid size and unlike the implicit methods they do not require the resolution of a system of nonlinear equations

a multilevel hybrid newton krylov schwarz method for the bidomain model of electrocardiology abstract a multilevel hybrid newton krylov schwarz nks method is constructed and studied numerically for implicit time discretizations of the bidomain reaction diffusion system in three dimensions this model describes the bioelectrical activity of the heart by coupling two degenerate parabolic equations with a stiff system of ordinary differential equations the nks bidomain solver employs an outer inexact newton iteration to solve the nonlinear finite element system originating at each time step of the implicit discretization the jacobian update during the newton iteration is solved by a krylov method employing a multilevel hybrid overlapping schwarz preconditioner additive within the levels and multiplicative among the levels several parallel tests on linux clusters are performed showing that the convergence of the method is independent of the number of subdomains scalability the discretization parameters and the number of levels optimality

finite difference approximations for fractional advection dispersion flow equations fractional advection dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium in this paper we develop practical numerical methods to solve one dimensional fractional advection dispersion equations with variable coefficients on a finite domain the practical application of these results is illustrated by modeling a radial flow problem use of the fractional derivative allows the model equations to capture the early arrival of tracer observed at a field site

two realizations of schrodinger operators on riemannian manifolds abstract we consider a schrodinger differential expression p m v on a complete riemannian manifold m g with metric g where m is the scalar laplacian on m and v is a real valued locally integrable function on m we study two self adjoint realizations of p in l 2 m and show their equality this is an extension of a result of s agmon

essential self adjointness for semi bounded magnetic schrodinger operators on non compact manifolds abstract we prove essential self adjointness for semi bounded below magnetic schrodinger operators on complete riemannian manifolds with a given positive smooth measure which is fixed independently of the metric some singularities of the scalar potential are allowed this is an extension of the povzner wienholtz simader theorem the proof uses the scheme of wienholtz but requires a refined invariant integration by parts technique as well as the use of a family of cut off functions which are constructed by a non trivial smoothing procedure due to karcher

crank nicolson method for the fractional diffusion equation with the riesz fractional derivative we examine a numerical method to approximate to a fractional diffusion equation with the riesz fractional derivative in a finite domain which has second order accuracy in time and space level in order to approximate the riesz fractional derivative we use the fractional centered derivative approach we determine the error of the riesz fractional derivative to the fractional centered difference we apply the crank nicolson method to a fractional diffusion equation which has the riesz fractional derivative and obtain that the method is unconditionally stable and convergent numerical results are given to demonstrate the accuracy of the crank nicolson method for the fractional diffusion equation with using fractional centered difference approach

the behavior of symmetric krylov subspace methods for solving mx m γi v abstract we analyze the behavior of krylov subspace methods for the solution of the symmetric system mx m i v when is close to some of the extreme eigenvalues of m we show that a stagnation phase may occur if the structure of the right hand side is not taken into account and we analyze the occurrence and persistence of such stagnation a natural alternative strategy is proposed and we show that the new approach provides a better approximation with the same number of matrix vector multiplications numerical experiments are also included

a model based block triangular preconditioner for the bidomain system in electrocardiology we introduce a preconditioner for the solution of the bidomain system governing the propagation of action potentials in the myocardial tissue the bidomain model is a degenerate parabolic set of nonlinear reaction diffusion equations the nonlinear term describes the ion flux at the cellular level the degenerate nature of the problem results in a severe ill conditioning of its discretization our preconditioning strategy is based on a suitable adaptation of the monodomain model a simplified version of the bidomain one which is by far simpler to solve nevertheless is unable to capture significant features of the action potential propagation the monodomain preconditioner application to a non symmetric formulation of the bidomain system results at the algebraic level in a lower block triangular preconditioner we prove optimality of the preconditioner with respect to the mesh size and corroborate our theoretical results with 3d numerical simulations both on idealized and real ventricle geometries

a fourier method for the fractional diffusion equation describing sub diffusion in this paper a fractional partial differential equation fpde describing sub diffusion is considered an implicit difference approximation scheme idas for solving a fpde is presented we propose a fourier method for analyzing the stability and convergence of the idas derive the global accuracy of the idas and discuss the solvability finally numerical examples are given to compare with the exact solution for the order of convergence and simulate the fractional dynamical systems

localized self adjointness of schrodinger type operators on riemannian manifolds abstract we prove self adjointness of the schrodinger type operator h v v where is a hermitian connection on a hermitian vector bundle e over a complete riemannian manifold m with positive smooth measure d which is fixed independently of the metric and v lloc1 ende is a hermitian bundle endomorphism self adjointness of hv is deduced from the self adjointness of the corresponding localized operator this is an extension of a result by cycon the proof uses the scheme of cycon but requires a refined integration by parts technique as well as the use of a family of cut off functions which are constructed by a non trivial smoothing procedure due to karcher

on a class of non linear delay distributed order fractional diffusion equations in this paper we consider a numerical scheme for a class of non linear time delay fractional diffusion equations with distributed order in time this study covers the unique solvability convergence and stability of the resulted numerical solution by means of the discrete energy method the derivation of a linearized difference scheme with convergence order o 4 h 4 in l norm is the main purpose of this study numerical experiments are carried out to support the obtained theoretical results

new methods for linear inequalities abstract the iterative method of cimmino for solving linear equations is generalized to linear inequalities we also present a richardson type iterative method for solving the inequality problem which includes the generalized cimmino scheme convergence proofs are provided

the convergence of linear stationary iterative processes for solving singular unstructured systems of linear equations this paper reviews the convergence properties of linear stationary iterative processes for solving large unstructured systems of linear equations special attention is given to the case where the system to be solved is singular and possibly inconsistent the review starts by investigating the iteration bf x k 1 h bf x k bf f k 0 1 2 cdots when the system i h bf x bf f is inconsistent in this case the sequence left bf x k right always diverges nevertheless when h is convergent there exists a vector bf q in text null i h such that the sequence hat bf x k bf x k k bf q converges the limit of this sequence hat bf x solves a certain least squares problem and the sequence of residual vectors bf r k i h bf x k bf f converges to i h hat bf x bf f furthermore if h is similar to a real diagonal matrix then the corresponding norms of bf r k and hat bf x k hat bf x decr

implicit explicit methods for time dependent partial differential equations implicit explicit imex schemes have been widely used especially in conjunction with spectral methods for the time integration of spatially discretized partial differential equations pdes of diffusion convection type typically an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term reaction diffusion problems can also be approximated in this manner in this work we systematically analyze the performance of such schemes propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods for the prototype linear advection diffusion equation a stability analysis for first second third and fourth order multistep imex schemes is performed stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified numerical experiments demonstrate that weak decay

efficient algebraic solution of reaction diffusion systems for the cardiac excitation process in this paper we deal with the problem of solving the large algebraic linear system arising in the numerical solution of a reaction diffusion r d system associated with myocardial excitation process modeling we show that an ad hoc preconditioning technique can be devised so as to efficiently and simultaneously handle the differential equations of the r d system with no additional memory requirements two different formulations are commonly considered for the theoretical and numerical analyses respectively we observe that the formulation employed for the theoretical analysis of the problem actually yields the best numerical performance when compared with the usual numerical scheme

application of a fractional advection dispersion equation abstract a transport equation that uses fractional order dispersion derivatives hasfundamental solutions that are le vy s a stable densities these densities represent plumesthat spread proportional to time 1 a have heavy tails and incorporate any degree ofskewness the equation is parsimonious since the dispersion parameter is not a functionof time or distance the scaling behavior of plumes that undergo le vy motion isaccounted for by the fractional derivative a laboratory tracer test is described by adispersion term of order 1 55 while the cape cod bromide plume is modeled by anequation of order 1 65 to 1 8 1 introduction anomalous or non fickian dispersion has been an activearea of research in the physics community since the introduc tion of continuous time random walks ctrw by montrolland weiss 1965 these random walks extended the predictivecapability of models built on the stochastic process of brown ian motion which is the basis for the classical advection dispersion equation ade the ctrw assign a joint space time distribution called the transition density to individualparticle motions when the tails are heavy enough i e powerlaw non fickian dispersion results for all time scales andspace scales

numerical schemes with high spatial accuracy for a variable order anomalous subdiffusion equation in this paper we consider a variable order anomalous subdiffusion equation a numerical scheme with first order temporal accuracy and fourth order spatial accuracy for the equation is proposed the convergence stability and solvability of the numerical scheme are discussed via the technique of fourier analysis another improved numerical scheme with second order temporal accuracy and fourth order spatial accuracy is also proposed some numerical examples are given and the results demonstrate the effectiveness of theoretical analysis

row action methods for huge and sparse systems and their applications this paper brings together and discusses theory and applications of methods identified and labelled as row action methods for linear feasibility problems find x in bf r n such that ax leqq b linearly constrained optimization problems minimize f x subject to ax leqq b and some interval convex programming problems minimize f x subject to c leqq ax leqq b the main feature of row action methods is that they are iterative procedures which without making any changes to the original matrix a use the rows of a one row at a time such methods are important and have demonstrated effectiveness for problems with large or huge matrices which do not enjoy any detectable or usable structural pattern apart from a high degree of sparaseness fields of application where row action methods are used in various ways include image reconstruction from projection operations research and game theory learning theory pattern recognition and transportation theory a row action method for the

a fully implicit finite element method for bidomain models of cardiac electromechanics we propose a novel monolithic and unconditionally stable finite element algorithm for the bidomainbased approach to cardiac electromechanics we introduce the transmembrane potential the extracellular potential and the displacement field as independent variables and extend the common two field bidomain formulation of electrophysiology to a three field formulation of electromechanics the intrinsic coupling arises from both excitation induced contraction of cardiac cells and the deformation induced generation of intra cellular currents the coupled reaction diffusion equations of the electrical problem and the momentum balance of the mechanical problem are recast into their weak forms through a conventional isoparametric galerkin approach as a novel aspect we propose a monolithic approach to solve the governing equations of excitation contraction coupling in a fully coupled implicit sense we demonstrate the consistent linearization of the resulting set of non linear residual equations to assess the algorithmic performance we illustrate characteristic features by means of representative three dimensional initial boundary value problems the proposed algorithm may open new avenues to patient specific therapy design by circumventing stability and convergence issues inherent to conventional staggered solution schemes

finite difference approximations for a fractional advection diffusion problem the use of the conventional advection diffusion equation in many physical situations has been questioned by many investigators in recent years and alternative diffusion models have been proposed fractional space derivatives are used to model anomalous diffusion or dispersion where a particle plume spreads at a rate inconsistent with the classical brownian motion model when a fractional derivative replaces the second derivative in a diffusion or dispersion model it leads to enhanced diffusion also called superdiffusion we consider a one dimensional advection diffusion model where the usual second order derivative gives place to a fractional derivative of order a with 1 a 2 we derive explicit finite difference schemes which can be seen as generalizations of already existing schemes in the literature for the advection diffusion equation we present the order of accuracy of the schemes and in order to show its convergence we prove they are stable under certain conditions in the end we present a test problem

bpx preconditioners for the bidomain model of electrocardiology the aim of this work is to develop a bpx preconditioner for the bidomain model of electrocardiology this model describes the bioelectrical activity of the cardiac tissue and consists of a system of a non linear parabolic reaction diffusion partial differential equation pde and an elliptic linear pde modeling at macroscopic level the evolution of the transmembrane and extracellular electric potentials of the anisotropic cardiac tissue the evolution equation is coupled through the non linear reaction term with a stiff system of ordinary differential equations the so called membrane model describing the ionic currents through the cellular membrane the discretization of the coupled system by finite elements in space and semi implicit finite differences in time yields at each time step the solution of an ill conditioned linear system the goal of the present study is to construct analyze and numerically test a bpx preconditioner for the linear system arising from the discretization of the bidomain model optimal convergence rate estimates are established and verified by two and three dimensional numerical tests on both structured and unstructured meshes moreover in a full heartbeat simulation on a three dimensional wedge of ventricular tissue the bpx preconditioner is about 35 faster in terms of cpu times than ilu 0 and an algebraic multigrid preconditioner

a splitting preconditioner for a block two by two linear system with applications to the bidomain equations we construct an alternating splitting iteration scheme for solving and preconditioning a block two by two linear system arising from numerical discretizations of the bidomain equations the convergence theory of this class of splitting iteration methods is established and some useful properties of the preconditioned matrix are analyzed the potential of this approach is illustrated by numerical experiments

maximum norm error analysis of difference schemes for fractional diffusion equations in this study we present several numerical approximation methods for solving the fractional diffusion equations with the riesz space fractional derivative based on the fractional centered difference approximation to the riesz fractional derivative an explicit and an implicit finite difference schemes for solving the space fractional diffusion equations are obtained the stability and global convergence of these two schemes in the maximum norm are analyzed rigorously the convergence order is o h 2 where is the temporal grid size and h is spatial grid size respectively furthermore a numerical scheme for the one dimensional multi term time space fractional diffusion equations is obtained the unconditionally stability and convergence of the scheme in maximum norm are established the convergence orders are 2 in the temporal direction and two in the spatial direction where is the maximum time fractional derivative orders in the equation in addition the extension to the two dimensional case is also discussed finally several numerical examples are provided to show the effectiveness and accuracy of our methods