perturbations of elliptic operators in 1 sided chord arc domains part i small and large perturbation for symmetric operators the first and third authors acknowledge financial support from the spanish ministry of economy and competitiveness through the severo ochoa programme for centres of excellence in r d sev 2015 0554 they also acknowledge that the research leading to these results has received funding from the european research council under the european union s seventh framework programme fp7 2007 2013 erc agreement no 615112 hapdegmt the second author was supported by nsf grant dms 1664047

multiscale reaction diffusion algorithms pde assisted brownian dynamics two algorithms that combine brownian dynamics bd simulations with mean field partial differential equations pdes are presented this pde assisted brownian dynamics pbd methodology provides exact particle tracking data in parts of the domain whilst making use of a mean field reaction diffusion pde description elsewhere the first pbd algorithm couples bd simulations with pdes by randomly creating new particles close to the interface which partitions the domain and by reincorporating particles into the continuum pde description when they cross the interface the second pbd algorithm introduces an overlap region where both descriptions exist in parallel it is shown that the overlap region is required to accurately compute variances using pbd simulations advantages of both pbd approaches are discussed and illustrative numerical examples are presented

analysis of the two regime method on square meshes the two regime method trm has been recently developed for optimizing stochastic reaction diffusion simulations m flegg j chapman and r erban j roy soc interface 9 2012 pp 859 868 it is a multiscale hybrid algorithm which uses stochastic reaction diffusion models with different levels of detail in different parts of the computational domain the coupling condition on the interface between different modeling regimes of the trm was previously derived for one dimensional models in this paper the trm is generalized to higher dimensional reaction diffusion systems coupling brownian dynamics models with compartment based models on regular square two dimensional lattices is studied in detail in this case the interface between different modeling regimes contains either flat parts or right angle corners both cases are studied in the paper for flat interfaces it is shown that the one dimensional theory can be used along the line perpendicular to the trm interface in the direction tan

harmonic measure and quantitative connectivity geometric characterization of the l p solvability of the dirichlet problem it is well known that quantitative scale invariant absolute continuity more precisely the weak a infty property of harmonic measure with respect to surface measure on the boundary of an open set omega subset mathbb r n 1 with ahlfors david regular boundary is equivalent to the solvability of the dirichlet problem in omega with data in l p partial omega for some p infty in this paper we give a geometric characterization of the weak a infty property of harmonic measure and hence of solvability of the l p dirichlet problem for some finite p this characterization is obtained under background hypotheses an interior corkscrew condition along with ahlfors david regularity of the boundary that are natural and in a certain sense optimal we provide counter examples in the absence of either of them or even one of the two upper or lower ahlfors david bounds moreover the examples show that the upper and lower ahlfors david bounds are each quantitatively sharp

composite media and asymptotic dirichlet forms we introduce a metric topology on the space of dirichlet forms and study compactness and closure properties of families of local and non local forms

a quasi monte carlo approach to particle simulation of the heat equation the convergence of the monte carlo method for numerical integration can often be improved by replacing random numbers with more uniformly distributed numbers known as quasi random in this paper the convergence of monte carlo particle simulation is studied when these quasi random sequences are used for the one dimensional heat equation discretized in both space and time convergence is proved for a quasi random simulation using reordering of the particles according to their position experimental results are presented for the spatially continuous heat equation in one and two dimensions the results indicate that a significant improvement in both magnitude of error and convergence rate can be achieved over standard monte carlo simulations for certain low dimensional problems

simulating mesoscopic reaction diffusion systems using the gillespie algorithm we examine an application of the gillespie algorithm to simulating spatially inhomogeneous reaction diffusion systems in mesoscopic volumes such as cells and microchambers the method involves discretizing the chamber into elements and modeling the diffusion of chemical species by the movement of molecules between neighboring elements these transitions are expressed in the form of a set of reactions which are added to the chemical system the derivation of the rates of these diffusion reactions is by comparison with a finite volume discretization of the heat equation on an unevenly spaced grid the diffusion coefficient of each species is allowed to be inhomogeneous in space including discontinuities the resulting system is solved by the gillespie algorithm using the fast direct method we show that in an appropriate limit the method reproduces exact solutions of the heat equation for a purely diffusive system and the nonlinear reaction rate equation describing the cubic autocatalytic reaction

hybrid simulations of reaction diffusion systems in porous media hybrid or multiphysics algorithms provide an efficient computational tool for combining micro and macroscale descriptions of physical phenomena their use becomes imperative when microscale descriptions are too computationally expensive to be conducted in the whole domain while macroscale descriptions fail in a small portion of the computation domain we present a hybrid algorithm to model a general class of reaction diffusion processes in granular porous media which includes mixing induced mineral precipitation on or dissolution of the porous matrix these processes cannot be accurately described using continuum darcy scale models the pore scale darcy scale hybrid is constructed by coupling solutions of the reaction diffusion equations rde at the pore scale with continuum darcy level solutions of the averaged rdes the resulting hybrid formulation is solved numerically by employing a multiresolution meshless discretization based on the smoothed particle hydrodynamics method this ensures seamless noniterative coupling of the two components of the hybrid model computational examples illustrate the accuracy and efficiency of the hybrid algorithm

simulation of stochastic reaction diffusion processes on unstructured meshes we model stochastic chemical systems with diffusion by a reaction diffusion master equation on a macroscopic level the governing equation is a reaction diffusion equation for the averages of the chemical species on a mesoscopic level the master equation for a well stirred chemical system is combined with a discretized brownian motion in space to obtain the reaction diffusion master equation the space is covered in our method by an unstructured mesh and the diffusion coefficients on the mesoscale are obtained from a finite element discretization of the laplace operator on the macroscale the resulting method is a flexible hybrid algorithm in that the diffusion can be handled either on the meso or on the macroscale level the accuracy and the efficiency of the method are illustrated in three numerical examples inspired by molecular biology

computing the moments of high dimensional solutions of the master equation derived from the markov character only the master equation of chemical reactions is an accurate stochastic description of quite general systems in chemistry exact solutions of this equation are rare and the most frequently used approximative solution method is to write down the corresponding set of reaction rate equations in many cases this approximation is not valid or only partially so as stochastic effects caused by the natural noise present in the full description of the problem are poorly captured in this paper it is shown how a certain set of higher order equations can be derived it is shown by theory and example that stochastic effects are better captured using this technique while still maintaining the computational advantages of the reaction rate approach

on nonobtuse simplicial partitions this paper surveys some results on acute and nonobtuse simplices and associated spatial partitions these partitions are relevant in numerical mathematics including piecewise polynomial approximation theory and the finite element method special attention is paid to a basic type of nonobtuse simplices called path simplices the generalization of right triangles to higher dimensions in addition to applications in numerical mathematics we give examples of the appearance of acute and nonobtuse simplices in other areas of mathematics

convergence of methods for coupling of microscopic and mesoscopic reaction diffusion simulations in this paper three multiscale methods for coupling of mesoscopic compartment based and microscopic molecular based stochastic reaction diffusion simulations are investigated two of the three methods that will be discussed in detail have been previously reported in the literature the two regime method trm and the compartment placement method cpm the third method that is introduced and analysed in this paper is called the ghost cell method gcm since it works by constructing a ghost cell in which molecules can disappear and jump into the compartment based simulation presented is a comparison of sources of error the convergent properties of this error are studied as the time step t for updating the molecular based part of the model approaches zero it is found that the error behaviour depends on another fundamental computational parameter h the compartment size in the mesoscopic part of the model two important limiting cases which appear in applications are considered i t 0 and h is fixed ii t 0 and h 0 such that t h is fixed the error for previously developed approaches the trm and cpm converges to zero only in the limiting case ii but not in case i it is shown that the error of the gcm converges in the limiting case i thus the gcm is superior to previous coupling techniques if the mesoscopic description is much coarser than the microscopic part of the model

absolute continuity of symmetric markov processes we study girsanov s theorem in the context of symmetric markov processes extending earlier work of fukushima takeda and fitzsimmons on girsanov transformations of gradient type we investigate the most general girsanov transformation leading to another symmetric markov process this investigation requires an extension of the forward backward martingale method of lyons zheng to cover the case of processes with jumps

_ estimates via extrapolation of carleson measures and applications to divergence form elliptic operators we revisit the extrapolation method for carleson measures introduced in lm to prove a1 estimates for certain caloric measures and we present a purely real variable version of the method suitable for establishing a1 estimates to illustrate the use of this technique we then reprove a well known result of fkp the results of lm and of hl are examples of carleson a1 extrapolation in which a given non negative measure is shown to belong to a1 or weak a1 using properties of some controlling carleson measure the results of ahlt and of ahmtt involve carleson carleson extrapolation in which a non negative measure in the half space r n 1 is shown to be a carleson measure using properties of another controlling

incorporating diffusion in complex geometries into stochastic chemical kinetics simulations a method is developed for incorporating diffusion of chemicals in complex geometries into stochastic chemical kinetics simulations systems are modeled using the reaction diffusion master equation with jump rates for diffusive motion between mesh cells calculated from the discretization weights of an embedded boundary method since diffusive jumps between cells are treated as first order reactions individual realizations of the stochastic process can be created by the gillespie method numerical convergence results for the underlying embedded boundary method and for the stochastic reaction diffusion method are presented in two dimensions a two dimensional model of transcription translation and nuclear membrane transport in eukaryotic cells is presented to demonstrate the feasibility of the method in studying cell wide biological processes

algorithm refinement for stochastic partial differential equations i linear diffusion a hybrid particle continuum algorithm is formulated for fickian diffusion in the fluctuating hydrodynamic limit the particles are taken as independent random walkers the fluctuating diffusion equation is solved by finite differences with deterministic and white noise fluxes at the interface between the particle and continuum computations the coupling is by flux matching giving exact mass conservation this methodology is an extension of adaptive mesh and algorithm refinement to stochastic partial differential equations results from a variety of numerical experiments are presented for both steady and time dependent scenarios in all cases the mean and variance of density are captured correctly by the stochastic hybrid algorithm for a nonstochastic version i e using only deterministic continuum fluxes the mean density is correct but the variance is reduced except in particle regions away from the interface extensions of the methodology to fluid mechanics applications are discussed

three isomorphic vector spaces ii abstract three isomorphic vector spaces b n k c n k and d n k are defined the interplay of these vector spaces leads to easy proofs for multinomial identities using automorphism each multinomial identity is recast into k 1 1 more identities distributions arising out of some stopping rules in drawing balls of k colors from an urn with and without replacement are connected so that one can easily go from one to the other identities involving joint cdf s of order statistics are generalized to those coming from arbitrary multivariate distributions discrete distribution similar to multinomial is defined where all the calculations can be done on usual multinomial distribution and transferred to the general

the solution of the kato square root problem for second order elliptic operators on rn we prove the kato conjecture for elliptic operators on jfin more precisely we establish that the domain of the square root of a uniformly complex elliptic operator l div av with bounded measurable coefficients in ietn is the

a_p weights and quantitative estimates in the schrodinger setting suppose l delta v is a schrodinger operator on mathbb r n with a potential v belonging to certain reverse holder class rh sigma with sigma ge n 2 the aim of this paper is to study the a p weights associated to l denoted by a p l which is a larger class than the classical muckenhoupt a p weights we first prove the quantitative a p l bound for the maximal function and the maximal heat semigroup associated to l then we further provide the quantitative a p q l bound for the fractional integral operator associated to l we point out that all these quantitative bounds are known before in terms of the classical a p q constant however since a p q subset a p q l the a p q l constants are smaller than a p q constant hence our results here provide a better quantitative constant for maximal functions and fractional integral operators associated to l next we prove two weight inequalities for the fractional integral operator these have been unknown up to this point finally we also have a study on the exp log link between a p l and bmo l the bmo space associated with l and show that for w in a p l log w is in bmo l and that the reverse is not true in general

yin yang zhong grid for numerical simulations inside a sphere an overset grid system yin yang zhong grid is proposed the yin yang zhong grid is an extension of the yin yang grid which is widely used in various simulations in spherical shell geometry the yin yang grid is itself an overset grid system with two component grids and a new component grid called zhong is placed at the center of the yin yang grid the zhong grid component is constructed on cartesian coordinates parallelization is intrinsically embedded in the yin yang zhong grid system because the zhong grid points are defined with cuboid blocks that are decomposed domains for parallelization the computational efficiency approaches the optimum as the process number increases quantitative test simulations are performed for a diffusion problem in a sphere with the yin yang zhong grid correct decay rates are obtained by the simulations two other tests in magnetohydrodynamics mhd in a sphere are also performed one is an mhd dynamo simulation and the other is an mhd relaxation simulation in a sphere

hybrid framework for the simulation of stochastic chemical kinetics abstract stochasticity plays a fundamental role in various biochemical processes such as cell regulatory networks and enzyme cascades isothermal well mixed systems can be modelled as markov processes typically simulated using the gillespie stochastic simulation algorithm ssa 25 while easy to implement and exact the computational cost of using the gillespie ssa to simulate such systems can become prohibitive as the frequency of reaction events increases this has motivated numerous coarse grained schemes where the fast reactions are approximated either using langevin dynamics or deterministically while such approaches provide a good approximation when all reactants are abundant the approximation breaks down when one or more species exist only in small concentrations and the fluctuations arising from the discrete nature of the reactions become significant this is particularly problematic when using such methods to compute statistics of extinction times for chemical species as well as simulating non equilibrium systems such as cell cycle models in which a single species can cycle between abundance and scarcity in this paper a hybrid jump diffusion model for simulating well mixed stochastic kinetics is derived it acts as a bridge between the gillespie ssa and the chemical langevin equation for low reactant reactions the underlying behaviour is purely discrete while purely diffusive when the concentrations of all species are large with the two different behaviours coexisting in the intermediate region a bound on the weak error in the classical large volume scaling limit is obtained and three different numerical discretisations of the jump diffusion model are described the benefits of such a formalism are illustrated using computational examples

hybrid method for the chemical master equation the chemical master equation is solved by a hybrid method coupling a macroscopic deterministic description with a mesoscopic stochastic model the molecular species are divided into one subset where the expected values of the number of molecules are computed and one subset with species with a stochastic variation in the number of molecules the macroscopic equations resemble the reaction rate equations and the probability distribution for the stochastic variables satisfy a master equation the probability distribution is obtained by the stochastic simulation algorithm due to gillespie the equations are coupled via a summation over the mesoscale variables this summation is approximated by quasi monte carlo methods the error in the approximations is analyzed the hybrid method is applied to three chemical systems from molecular cell biology

the theory of weights and the dirichlet problem for elliptic equations in this work we consider questions from the area of partial differential equations and the related questions from harmonic analysis specifically we will study the dirichlet problem for second order divergence form elliptic operators with bounded measurable coefficients roughly speaking we consider the following question suppose that we are given two operators lo and li and that we have good estimates for the dirichlet problem for lo good in this context could mean several different things for instance it might mean that there exists p 1 p oo so that the dirichlet problem is solvable in a suitable sense with lp data what are the optimal conditions on the difference between the coefficients of lo and l1 which guarantee that the dirichlet problem for li also has good estimates this question has been extensively studied recently cf f j k r f and d4 our work uses in a crucial way the results in r f the main results in d4 and r f are proved using an ingenious differential inequality for a family of harmonic measures introduced in d4 another of our goals in this article is to give new direct proofs of the results in d4 and r f without the use of this differential inequality theorems 2 4 2 5 and 2 18 these direct approaches have already had important applications k p2 an interesting class of examples for the theorems quoted above arises when one pulls back the laplacian via a quasiconformal mapping of the plane into itself specializing our theorem to this case suggests a new characterization of the class of am weights introduced in connection with several problems in harmonic analysis by muckenhoupt mul and coifman c fefferman co f we shall present in theorem 2 3 a criterion in terms of carleson measures which is very close to log w e bmo and is necessary and sufficient for w e ad

switched diffusion processes and systems of elliptic equations a dirichlet space approach the switched diffusion process associated with a weakly coupled system of elliptic equations is studied via a dirichlet space approach and is applied to prove the existence theorem of the cauchy initial problem for the system a representation theorem for the solution of the dirichlet boundary value problem and a generalised skorohod decomposition for the reflecting switched diffusion process are obtained

a first passage kinetic monte carlo method for reaction drift diffusion processes stochastic reaction diffusion models are now a popular tool for studying physical systems in which both the explicit diffusion of molecules and noise in the chemical reaction process play important roles the smoluchowski diffusion limited reaction model sdlr is one of several that have been used to study biological systems exact realizations of the underlying stochastic processes described by the sdlr model can be generated by the recently proposed first passage kinetic monte carlo fpkmc method this exactness relies on sampling analytical solutions to one and two body diffusion equations in simplified protective domains in this work we extend the fpkmc to allow for drift arising from fixed background potentials as the corresponding fokker planck equations that describe the motion of each molecule can no longer be solved analytically we develop a hybrid method that discretizes the protective domains the discretization is chosen so that the drift diffusion of each molecule within its protective domain is approximated by a continuous time random walk on a lattice new lattices are defined dynamically as the protective domains are updated hence we will refer to our method as dynamic lattice fpkmc or dl fpkmc we focus primarily on the one dimensional case in this manuscript and demonstrate the numerical convergence and accuracy of our method in this case for both smooth and discontinuous potentials we also present applications of our method which illustrate the impact of drift on reaction kinetics

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 note on exponential stability of the nonlinear filter for denumerable markov chains abstract asymptotic stability of the optimal filter with respect to its initial conditions is investigated in this paper under the assumption that the observation function is one to one and the observation noise is sufficiently small it is shown that exponential stability of the nonlinear filter holds for a large class of denumerable markov chains including all finite markov chains throughout this paper ergodicity of the signal process is not assumed

reactive boundary conditions as limits of interaction potentials for brownian and langevin dynamics a popular approach to modeling bimolecular reactions between diffusing molecules is through the use of reactive boundary conditions one common model is the smoluchowski partial adsorption condition which uses a robin boundary condition in the separation coordinate between two possible reactants this boundary condition can be interpreted as an idealization of a reactive interaction potential model in which a potential barrier must be surmounted before reactions can occur in this work we show how the reactive boundary condition arises as the limit of an interaction potential encoding a steep barrier within a shrinking region in the particle separation where molecules react instantly upon reaching the peak of the barrier the limiting boundary condition is derived by the method of matched asymptotic expansions and is shown to depend critically on the relative rate of increase of the barrier height as the width of the potential is decreased limiting boundary conditions for the same interaction potential in both the overdamped fokker planck equation brownian dynamics and the kramers equation langevin dynamics are investigated it is shown that different scalings are required in the two models to recover reactive boundary conditions that are consistent in the high friction limit where the kramers equation solution converges to the solution of the fokker planck equation

uniform rectifiability carleson measure estimates and approximation of harmonic functions the first author was supported by nsf grant dms 1361701 the second author was supported in part by mineco grant mtm2010 16518 icmat severo ochoa project sev 2011 0087 he also acknowledges that the research leading to these results has received funding from the european research council under the european union s seventh framework programme fp7 2007 2013 erc agreement no 615112 hapdegmt the third author was supported by the alfred p sloan fellowship the nsf career award dms 1056004 the nsf inspire award dms 1344235 and the nsf materials research science and engineering center seed grant dmr 0212302

extensions of levy khintchine formula and beurling deny formula in semi dirichlet forms setting abstract the levy khintchine formula or more generally courrege s theorem characterizes the infinitesimal generator of a levy process or a feller process on r d for more general markov processes the formula that comes closest to such a characterization is the beurling deny formula for symmetric dirichlet forms in this paper we extend these celebrated structure results to include a general right process on a metrizable lusin space which is supposed to be associated with a semi dirichlet form we start with decomposing a regular semi dirichlet form into the diffusion jumping and killing parts then we develop a local compactification and an integral representation for quasi regular semi dirichlet forms finally we extend the formulae of levy khintchine and beurling deny in semi dirichlet forms setting through introducing a quasi compatible metric

coarse graining and hybrid methods for efficient simulation of stochastic multi scale models of tumour growth the development of hybrid methodologies is of current interest in both multi scale modelling and stochastic reaction diffusion systems regarding their applications to biology we formulate a hybrid method for stochastic multi scale models of cells populations that extends the remit of existing hybrid methods for reaction diffusion systems such method is developed for a stochastic multi scale model of tumour growth i e population dynamical models which account for the effects of intrinsic noise affecting both the number of cells and the intracellular dynamics in order to formulate this method we develop a coarse grained approximation for both the full stochastic model and its mean field limit such approximation involves averaging out the age structure which accounts for the multi scale nature of the model by assuming that the age distribution of the population settles onto equilibrium very fast we then couple the coarse grained mean field model to the full stochastic multi scale model by doing so within the mean field region we are neglecting noise in both cell numbers population and their birth rates structure this implies that in addition to the issues that arise in stochastic reaction diffusion systems we need to account for the age structure of the population when attempting to couple both descriptions we exploit our coarse graining model so that within the mean field region the age distribution is in equilibrium and we know its explicit form this allows us to couple both domains consistently as upon transference of cells from the mean field to the stochastic region we sample the equilibrium age distribution furthermore our method allows us to investigate the effects of intracellular noise i e fluctuations of the birth rate on collective properties such as travelling wave velocity we show that the combination of population and birth rate noise gives rise to large fluctuations of the birth rate in the region at the leading edge of front which cannot be accounted for by the coarse grained model such fluctuations have non trivial effects on the wave velocity beyond the development of a new hybrid method we thus conclude that birth rate fluctuations are central to a quantitatively accurate description of invasive phenomena such as tumour growth

excluded volume effects in the diffusion of hard spheres excluded volume effects can play an important role in determining transport properties in diffusion of particles here the diffusion of finite sized hard core interacting particles in two or three dimensions is considered systematically using the method of matched asymptotic expansions the result is a nonlinear diffusion equation for the one particle distribution function with excluded volume effects enhancing the overall collective diffusion rate an expression for the effective collective diffusion coefficient is obtained stochastic simulations of the full particle system are shown to compare well with the solution of this equation for two examples

on the existence of maximum principles in parabolic finite element equations in 1973 h fujii investigated discrete versions of the maximum principle for the model heat equation using piecewise linear finite elements in space in particular he showed that the lumped mass method allows a maximum principle when the simplices of the triangulation are acute and this is known to generalize in two space dimensions to triangulations of delauney type in this note we consider more general parabolic equations and first show that a maximum principle cannot hold for the standard spatially semidiscrete problem we then show that for the lumped mass method the above conditions on the triangulation are essentially sharp this is in contrast to the elliptic case in which the requirements are weaker we also study conditions for the solution operator acting on the discrete initial data with homogeneous lateral boundary conditions to be a contraction or a positive operator

flexible single molecule simulation of reaction diffusion processes an algorithm is developed for simulation of the motion and reactions of single molecules at a microscopic level the molecules diffuse in a solvent and react with each other or a polymer and molecules can dissociate such simulations are of interest e g in molecular biology the algorithm is similar to the green s function reaction dynamics gfrd algorithm by van zon and ten wolde where longer time steps can be taken by computing the probability density functions pdfs and then sample from the distribution functions our computation of the pdfs is much less complicated than gfrd and more flexible the solution of the partial differential equation for the pdf is split into two steps to simplify the calculations the sampling is without splitting error in two of the coordinate directions for a pair of molecules and a molecule polymer interaction and is approximate in the third direction the pdf is obtained either from an analytical solution or a numerical discretization the errors due to the operator splitting the partitioning of the system and the numerical approximations are analyzed the method is applied to three different systems involving up to four reactions comparisons with other mesoscopic and macroscopic models show excellent agreement